free_module.py

r"""
Free modules

Sage supports computation with free modules over an arbitrary commutative ring.
Nontrivial functionality is available over `\ZZ`, fields, and some principal
ideal domains (e.g. `\QQ[x]` and rings of integers of number fields). All free
modules over an integral domain are equipped with an embedding in an ambient
vector space and an inner product, which you can specify and change.

Create the free module of rank `n` over an arbitrary commutative ring `R` using
the command ``FreeModule(R,n)``. Equivalently, ``R^n`` also creates that free
module.

The following example illustrates the creation of both a vector space and a
free module over the integers and a submodule of it.  Use the functions
``FreeModule``, ``span`` and member functions of free modules to create free
modules.  *Do not use the FreeModule_xxx constructors directly.*

EXAMPLES::

    sage: V = VectorSpace(QQ,3)
    sage: W = V.subspace([[1,2,7], [1,1,0]])
    sage: W
    Vector space of degree 3 and dimension 2 over Rational Field
    Basis matrix:
    [ 1  0 -7]
    [ 0  1  7]
    sage: C = VectorSpaces(FiniteField(7))
    sage: C
    Category of vector spaces over Finite Field of size 7
    sage: C(W)
    Vector space of degree 3 and dimension 2 over Finite Field of size 7
    Basis matrix:
    [1 0 0]
    [0 1 0]

::

    sage: M = ZZ^3
    sage: C = VectorSpaces(FiniteField(7))
    sage: C(M)
    Vector space of dimension 3 over Finite Field of size 7
    sage: W = M.submodule([[1,2,7], [8,8,0]])
    sage: C(W)
    Vector space of degree 3 and dimension 2 over Finite Field of size 7
    Basis matrix:
    [1 0 0]
    [0 1 0]

We illustrate the exponent notation for creation of free modules.

::

    sage: ZZ^4
    Ambient free module of rank 4 over the principal ideal domain Integer Ring
    sage: QQ^2
    Vector space of dimension 2 over Rational Field
    sage: RR^3
    Vector space of dimension 3 over Real Field with 53 bits of precision

Base ring::

    sage: R.<x,y> = QQ[]
    sage: M = FreeModule(R,2)
    sage: M.base_ring()
    Multivariate Polynomial Ring in x, y over Rational Field

::

    sage: VectorSpace(QQ, 10).base_ring()
    Rational Field

TESTS:

We intersect a zero-dimensional vector space with a
1-dimension submodule.

::

    sage: V = (QQ^1).span([])
    sage: W = ZZ^1
    sage: V.intersection(W)
    Free module of degree 1 and rank 0 over Integer Ring
    Echelon basis matrix:
    []

We construct subspaces of real and complex double vector spaces and
verify that the element types are correct::

    sage: V = FreeModule(RDF, 3); V
    Vector space of dimension 3 over Real Double Field
    sage: V.0
    (1.0, 0.0, 0.0)
    sage: type(V.0)
    <type 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
    sage: W = V.span([V.0]); W
    Vector space of degree 3 and dimension 1 over Real Double Field
    Basis matrix:
    [1.0 0.0 0.0]
    sage: type(W.0)
    <type 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
    sage: V = FreeModule(CDF, 3); V
    Vector space of dimension 3 over Complex Double Field
    sage: type(V.0)
    <type 'sage.modules.vector_complex_double_dense.Vector_complex_double_dense'>
    sage: W = V.span_of_basis([CDF.0 * V.1]); W
    Vector space of degree 3 and dimension 1 over Complex Double Field
    User basis matrix:
    [  0.0 1.0*I   0.0]
    sage: type(W.0)
    <type 'sage.modules.vector_complex_double_dense.Vector_complex_double_dense'>

Basis vectors are immutable::

    sage: A = span([[1,2,3], [4,5,6]], ZZ)
    sage: A.0
    (1, 2, 3)
    sage: A.0[0] = 5
    Traceback (most recent call last):
    ...
    ValueError: vector is immutable; please change a copy instead (use copy())

Among other things, this tests that we can save and load submodules
and elements::

    sage: M = ZZ^3
    sage: TestSuite(M).run()
    sage: W = M.span_of_basis([[1,2,3],[4,5,19]])
    sage: TestSuite(W).run()
    sage: v = W.0 + W.1
    sage: TestSuite(v).run()

AUTHORS:

- William Stein (2005, 2007)

- David Kohel (2007, 2008)

- Niles Johnson (2010-08): (:trac:`3893`) ``random_element()`` should pass on ``*args`` and ``**kwds``.

- Simon King (2010-12):
  :trac:`8800` : Fixing a bug in ``denominator()``.

- Simon King (2010-12), Peter Bruin (June 2014):
  :trac:`10513` : New coercion model and category framework.

"""

###########################################################################
#       Copyright (C) 2005, 2007 William Stein <wstein@gmail.com>
#       Copyright (C) 2007, 2008 David Kohel <kohel@iml.univ-mrs.fr>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
###########################################################################
from __future__ import print_function, absolute_import

from . import free_module_element
import sage.matrix.matrix_space
import sage.misc.latex as latex

from sage.modules.module import Module
import sage.rings.finite_rings.finite_field_constructor as finite_field
import sage.rings.ring as ring
import sage.rings.integer_ring
import sage.rings.rational_field
import sage.rings.finite_rings.integer_mod_ring
import sage.rings.infinity
import sage.rings.integer
from sage.categories.principal_ideal_domains import PrincipalIdealDomains
from sage.misc.randstate import current_randstate
from sage.structure.sequence import Sequence

from sage.misc.cachefunc import cached_method

from warnings import warn

###############################################################################
#
# Constructor functions
#
###############################################################################

from sage.structure.factory import UniqueFactory

class FreeModuleFactory(UniqueFactory):
    r"""
    Create the free module over the given commutative ring of the given
    rank.

    INPUT:


    -  ``base_ring`` - a commutative ring

    -  ``rank`` - a nonnegative integer

    -  ``sparse`` - bool; (default False)

    -  ``inner_product_matrix`` - the inner product
       matrix (default None)


    OUTPUT: a free module

    .. note::

       In Sage it is the case that there is only one dense and one
       sparse free ambient module of rank `n` over `R`.

    EXAMPLES:

    First we illustrate creating free modules over various base fields.
    The base field affects the free module that is created. For
    example, free modules over a field are vector spaces, and free
    modules over a principal ideal domain are special in that more
    functionality is available for them than for completely general
    free modules.

    ::

        sage: FreeModule(Integers(8),10)
        Ambient free module of rank 10 over Ring of integers modulo 8
        sage: FreeModule(QQ,10)
        Vector space of dimension 10 over Rational Field
        sage: FreeModule(ZZ,10)
        Ambient free module of rank 10 over the principal ideal domain Integer Ring
        sage: FreeModule(FiniteField(5),10)
        Vector space of dimension 10 over Finite Field of size 5
        sage: FreeModule(Integers(7),10)
        Vector space of dimension 10 over Ring of integers modulo 7
        sage: FreeModule(PolynomialRing(QQ,'x'),5)
        Ambient free module of rank 5 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
        sage: FreeModule(PolynomialRing(ZZ,'x'),5)
        Ambient free module of rank 5 over the integral domain Univariate Polynomial Ring in x over Integer Ring

    Of course we can make rank 0 free modules::

        sage: FreeModule(RealField(100),0)
        Vector space of dimension 0 over Real Field with 100 bits of precision

    Next we create a free module with sparse representation of
    elements. Functionality with sparse modules is *identical* to dense
    modules, but they may use less memory and arithmetic may be faster
    (or slower!).

    ::

        sage: M = FreeModule(ZZ,200,sparse=True)
        sage: M.is_sparse()
        True
        sage: type(M.0)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>

    The default is dense.

    ::

        sage: M = ZZ^200
        sage: type(M.0)
        <type 'sage.modules.vector_integer_dense.Vector_integer_dense'>

    Note that matrices associated in some way to sparse free modules
    are sparse by default::

        sage: M = FreeModule(Integers(8), 2)
        sage: A = M.basis_matrix()
        sage: A.is_sparse()
        False
        sage: Ms = FreeModule(Integers(8), 2, sparse=True)
        sage: M == Ms  # as mathematical objects they are equal
        True
        sage: Ms.basis_matrix().is_sparse()
        True

    We can also specify an inner product matrix, which is used when
    computing inner products of elements.

    ::

        sage: A = MatrixSpace(ZZ,2)([[1,0],[0,-1]])
        sage: M = FreeModule(ZZ,2,inner_product_matrix=A)
        sage: v, w = M.gens()
        sage: v.inner_product(w)
        0
        sage: v.inner_product(v)
        1
        sage: w.inner_product(w)
        -1
        sage: (v+2*w).inner_product(w)
        -2

    You can also specify the inner product matrix by giving anything
    that coerces to an appropriate matrix. This is only useful if the
    inner product matrix takes values in the base ring.

    ::

        sage: FreeModule(ZZ,2,inner_product_matrix=1).inner_product_matrix()
        [1 0]
        [0 1]
        sage: FreeModule(ZZ,2,inner_product_matrix=[1,2,3,4]).inner_product_matrix()
        [1 2]
        [3 4]
        sage: FreeModule(ZZ,2,inner_product_matrix=[[1,2],[3,4]]).inner_product_matrix()
        [1 2]
        [3 4]

    .. todo::

        Refactor modules such that it only counts what category the base
        ring belongs to, but not what is its Python class.

    """
    def create_key(self, base_ring, rank, sparse=False, inner_product_matrix=None):
        """
        TESTS::

            sage: loads(dumps(ZZ^6)) is ZZ^6
            True
            sage: loads(dumps(RDF^3)) is RDF^3
            True

        TODO: replace the above by ``TestSuite(...).run()``, once
        :meth:`_test_pickling` will test unique representation and not
        only equality.
        """
        rank = int(sage.rings.integer.Integer(rank))

        if not (inner_product_matrix is None):
            inner_product_matrix = sage.matrix.matrix_space.MatrixSpace(base_ring, rank)(inner_product_matrix)
            inner_product_matrix.set_immutable()

        return (base_ring, rank, sparse, inner_product_matrix)

    def create_object(self, version, key):

        base_ring, rank, sparse, inner_product_matrix = key

        if inner_product_matrix is not None:
            from .free_quadratic_module import FreeQuadraticModule
            return FreeQuadraticModule(base_ring, rank, inner_product_matrix=inner_product_matrix, sparse=sparse)

        if not isinstance(sparse,bool):
            raise TypeError("Argument sparse (= %s) must be True or False" % sparse)

        if not (hasattr(base_ring,'is_commutative') and base_ring.is_commutative()):
            warn("""You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.""")

        #            raise TypeError, "The base_ring must be a commutative ring."

        try:
            if not sparse and isinstance(base_ring,sage.rings.real_double.RealDoubleField_class):
                return RealDoubleVectorSpace_class(rank)

            elif not sparse and isinstance(base_ring,sage.rings.complex_double.ComplexDoubleField_class):
                return ComplexDoubleVectorSpace_class(rank)

            elif base_ring.is_field():
                return FreeModule_ambient_field(base_ring, rank, sparse=sparse)

            elif base_ring in PrincipalIdealDomains():
                return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)

            elif isinstance(base_ring, sage.rings.number_field.order.Order) \
                and base_ring.is_maximal() and base_ring.class_number() == 1:
                return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)

            elif isinstance(base_ring, ring.IntegralDomain) or base_ring.is_integral_domain():
                return FreeModule_ambient_domain(base_ring, rank, sparse=sparse)

            else:
                return FreeModule_ambient(base_ring, rank, sparse=sparse)
        except NotImplementedError:
            return FreeModule_ambient(base_ring, rank, sparse=sparse)


FreeModule = FreeModuleFactory("FreeModule")


def VectorSpace(K, dimension, sparse=False, inner_product_matrix=None):
    """
    EXAMPLES:

    The base can be complicated, as long as it is a field.

    ::

        sage: V = VectorSpace(FractionField(PolynomialRing(ZZ,'x')),3)
        sage: V
        Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in x over Integer Ring
        sage: V.basis()
        [
        (1, 0, 0),
        (0, 1, 0),
        (0, 0, 1)
        ]

    The base must be a field or a ``TypeError`` is raised.

    ::

        sage: VectorSpace(ZZ,5)
        Traceback (most recent call last):
        ...
        TypeError: Argument K (= Integer Ring) must be a field.
    """
    if not K.is_field():
        raise TypeError("Argument K (= %s) must be a field." % K)
    if not sparse in (True,False):
        raise TypeError("Argument sparse (= %s) must be a boolean."%sparse)
    return FreeModule(K, rank=dimension, sparse=sparse, inner_product_matrix=inner_product_matrix)

###############################################################################
#
# The span of vectors
#
###############################################################################

def span(gens, base_ring=None, check=True, already_echelonized=False):
    r"""
    Return the span of the vectors in ``gens`` using scalars from ``base_ring``.

    INPUT:

    - ``gens`` - a list of either vectors or lists of ring elements
      used to generate the span

    - ``base_ring`` - default: ``None`` - a principal ideal domain
      for the ring of scalars

    - ``check`` - default: ``True`` - passed to the ``span()`` method
      of the ambient module

    - ``already_echelonized`` - default: ``False`` - set to ``True``
      if the vectors form the rows of a matrix in echelon form, in
      order to skip the computation of an echelonized basis for the
      span.

    OUTPUT:

    A module (or vector space) that is all the linear combinations of the
    free module elements (or vectors) with scalars from the
    ring (or field) given by ``base_ring``.  See the examples below
    describing behavior when the base ring is not specified and/or
    the module elements are given as lists that do not carry
    explicit base ring information.

    EXAMPLES:

    The vectors in the list of generators can be given as
    lists, provided a base ring is specified and the elements of the list
    are in the ring (or the fraction field of the ring).  If the
    base ring is a field, the span is a vector space.  ::

        sage: V = span([[1,2,5], [2,2,2]], QQ); V
        Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [ 1  0 -3]
        [ 0  1  4]

        sage: span([V.gen(0)], QuadraticField(-7,'a'))
        Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 7
        Basis matrix:
        [ 1  0 -3]

        sage: span([[1,2,3], [2,2,2], [1,2,5]], GF(2))
        Vector space of degree 3 and dimension 1 over Finite Field of size 2
        Basis matrix:
        [1 0 1]

    If the base ring is not a field, then a module is created.
    The entries of the vectors can lie outside the ring, if they
    are in the fraction field of the ring.  ::

        sage: span([[1,2,5], [2,2,2]], ZZ)
        Free module of degree 3 and rank 2 over Integer Ring
        Echelon basis matrix:
        [ 1  0 -3]
        [ 0  2  8]

        sage: span([[1,1,1], [1,1/2,1]], ZZ)
        Free module of degree 3 and rank 2 over Integer Ring
        Echelon basis matrix:
        [  1   0   1]
        [  0 1/2   0]

        sage: R.<x> = QQ[]
        sage: M= span( [[x, x^2+1], [1/x, x^3]], R); M
        Free module of degree 2 and rank 2 over
        Univariate Polynomial Ring in x over Rational Field
        Echelon basis matrix:
        [          1/x           x^3]
        [            0 x^5 - x^2 - 1]
        sage: M.basis()[0][0].parent()
        Fraction Field of Univariate Polynomial Ring in x over Rational Field

    A base ring can be inferred if the generators are given as a
    list of vectors. ::

        sage: span([vector(QQ, [1,2,3]), vector(QQ, [4,5,6])])
        Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [ 1  0 -1]
        [ 0  1  2]
        sage: span([vector(QQ, [1,2,3]), vector(ZZ, [4,5,6])])
        Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [ 1  0 -1]
        [ 0  1  2]
        sage: span([vector(ZZ, [1,2,3]), vector(ZZ, [4,5,6])])
        Free module of degree 3 and rank 2 over Integer Ring
        Echelon basis matrix:
        [1 2 3]
        [0 3 6]

    TESTS::

        sage: span([[1,2,3], [2,2,2], [1,2/3,5]], ZZ)
        Free module of degree 3 and rank 3 over Integer Ring
        Echelon basis matrix:
        [  1   0  13]
        [  0 2/3   6]
        [  0   0  14]
        sage: span([[1,2,3], [2,2,2], [1,2,QQ['x'].gen()]], ZZ)
        Traceback (most recent call last):
        ...
        ValueError: The elements of gens (= [[1, 2, 3], [2, 2, 2], [1, 2, x]]) must be defined over base_ring (= Integer Ring) or its field of fractions.

    For backwards compatibility one can also give the base ring as the
    first argument.  ::

        sage: span(QQ,[[1,2],[3,4]])
        Vector space of degree 2 and dimension 2 over Rational Field
        Basis matrix:
        [1 0]
        [0 1]

    The base ring must be a principal ideal domain (PID).  ::

        sage: span([[1,2,3]], Integers(6))
        Traceback (most recent call last):
        ...
        TypeError: The base_ring (= Ring of integers modulo 6)
        must be a principal ideal domain.

    Fix :trac:`5575`::

        sage: V = QQ^3
        sage: span([V.0, V.1])
        Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [1 0 0]
        [0 1 0]

    Improve error message from :trac:`12541`::

        sage: span({0:vector([0,1])}, QQ)
        Traceback (most recent call last):
        ...
        TypeError: generators must be lists of ring elements
        or free module elements!
    """
    if ring.is_Ring(gens):
        # we allow the old input format with first input the base_ring.
        # Do we want to deprecate it?..
        base_ring, gens = gens, base_ring

    try:
        if base_ring is None:
            gens = Sequence(gens)
            R = gens.universe().base_ring()
        else:
            gens = list(gens)
            R = base_ring
    except TypeError:
        raise TypeError("generators must be given as an iterable structure!")

    if R not in PrincipalIdealDomains():
        raise TypeError("The base_ring (= %s) must be a principal ideal "
                        "domain." % R)
    if len(gens) == 0:
        return FreeModule(R, 0)
    else:
        x = gens[0]
        if free_module_element.is_FreeModuleElement(x):
            M = x.parent()
        else:
            try:
                x = list(x)
            except TypeError:
                raise TypeError("generators must be lists of ring elements or "
                                "free module elements!")
            M = FreeModule(R, len(x))
            try:
                gens = [M(_) for _ in gens]
            except TypeError:
                R = R.fraction_field()
                M = FreeModule(R, len(x))
                try:
                    gens = [M(_) for _ in gens]
                except TypeError:
                    raise ValueError("The elements of gens (= %s) must be "
                                     "defined over base_ring (= %s) or its "
                                     "field of fractions." % (gens, base_ring))
        return M.span(gens=gens, base_ring=base_ring, check=check,
                      already_echelonized=already_echelonized)

###############################################################################
#
# Base class for all free modules
#
###############################################################################

def is_FreeModule(M):
    """
    Return True if M inherits from FreeModule_generic.

    EXAMPLES::

        sage: from sage.modules.free_module import is_FreeModule
        sage: V = ZZ^3
        sage: is_FreeModule(V)
        True
        sage: W = V.span([ V.random_element() for i in range(2) ])
        sage: is_FreeModule(W)
        True
    """
    return isinstance(M, FreeModule_generic)

class FreeModule_generic(Module):
    """
    Base class for all free modules.

    TESTS:

    Check that :trac:`17576` is fixed::

        sage: V = VectorSpace(RDF, 3)
        sage: v = vector(RDF, [1, 2, 3, 4])
        sage: v in V
        False
    """
    def __init__(self, base_ring, rank, degree, sparse=False,
                 coordinate_ring=None, category=None):
        """
        Create the free module of given rank ``rank`` over the given base
        ring ``base_ring``.

        INPUT:

        - ``base_ring`` -- a commutative ring

        - ``rank`` -- a non-negative integer

        - ``degree`` -- a non-negative integer

        - ``sparse`` -- bool (default: False)

        - ``coordinate_ring`` -- a ring containing ``base_ring``
          (default: equal to ``base_ring``)

        - ``category`` -- category (default: None)

        If ``base_ring`` is a field, then the default category is the
        category of finite-dimensional vector spaces over that field;
        otherwise it is the category of finite-dimensional free modules
        over that ring.  In addition, the category is intersected with the
        category of finite enumerated sets if the ring is finite or the
        rank is 0.

        EXAMPLES::

            sage: PolynomialRing(QQ,3,'x')^3
            Ambient free module of rank 3 over the integral domain Multivariate Polynomial Ring in x0, x1, x2 over Rational Field

            sage: FreeModule(GF(7),3).category()
            Category of finite enumerated finite dimensional vector spaces with basis over
             (finite enumerated fields and subquotients of monoids and quotients of semigroups)
            sage: V = QQ^4; V.category()
            Category of finite dimensional vector spaces with basis over
             (quotient fields and metric spaces)
            sage: V = GF(5)**20; V.category()
            Category of finite enumerated finite dimensional vector spaces with basis over (finite enumerated fields and subquotients of monoids and quotients of semigroups)
            sage: FreeModule(ZZ,3).category()
            Category of finite dimensional modules with basis over
             (euclidean domains and infinite enumerated sets
              and metric spaces)
            sage: (QQ^0).category()
            Category of finite enumerated finite dimensional vector spaces with basis over (quotient fields and metric spaces)

        TESTS::

            sage: M = FreeModule(ZZ,20,sparse=False)
            sage: x = M.random_element()
            sage: type(x)
            <type 'sage.modules.vector_integer_dense.Vector_integer_dense'>
            sage: M.element_class
            <type 'sage.modules.vector_integer_dense.Vector_integer_dense'>

            sage: N = FreeModule(ZZ,20,sparse=True)
            sage: y = N.random_element()
            sage: type(y)
            <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
            sage: N.element_class
            <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>

        """
        if not base_ring.is_commutative():
            warn("""You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.""")

        if coordinate_ring is None:
            coordinate_ring = base_ring

        if not hasattr(self, 'Element'):
            self.Element = element_class(coordinate_ring, sparse)

        rank = sage.rings.integer.Integer(rank)
        if rank < 0:
            raise ValueError("rank (=%s) must be nonnegative"%rank)
        degree = sage.rings.integer.Integer(degree)
        if degree < 0:
            raise ValueError("degree (=%s) must be nonnegative"%degree)

        if category is None:
            from sage.categories.all import FreeModules
            category = FreeModules(base_ring.category()).FiniteDimensional()
            try:
                if base_ring.is_finite() or rank == 0:
                    category = category.Enumerated().Finite()
            except:
                pass

        super(FreeModule_generic, self).__init__(base_ring, category=category)
        self.__coordinate_ring = coordinate_ring
        self.__uses_ambient_inner_product = True
        self.__rank = rank
        self.__degree = degree
        self.__is_sparse = sparse
        self._gram_matrix = None

    def construction(self):
        """
        The construction functor and base ring for self.

        EXAMPLES::

            sage: R = PolynomialRing(QQ,3,'x')
            sage: V = R^5
            sage: V.construction()
            (VectorFunctor, Multivariate Polynomial Ring in x0, x1, x2 over Rational Field)
        """
        from sage.categories.pushout import VectorFunctor
        if hasattr(self,'_inner_product_matrix'):
            return VectorFunctor(self.rank(), self.is_sparse(),self.inner_product_matrix()), self.base_ring()
        return VectorFunctor(self.rank(), self.is_sparse()), self.base_ring()

    # FIXME: what's the level of generality of FreeModuleHomspace?
    # Should there be a category for free modules accepting it as hom space?
    # See similar method for FreeModule_generic_field class
    def _Hom_(self, Y, category):
        from .free_module_homspace import FreeModuleHomspace
        return FreeModuleHomspace(self, Y, category)

    def dense_module(self):
        """
        Return corresponding dense module.

        EXAMPLES:

        We first illustrate conversion with ambient spaces::

            sage: M = FreeModule(QQ,3)
            sage: S = FreeModule(QQ,3, sparse=True)
            sage: M.sparse_module()
            Sparse vector space of dimension 3 over Rational Field
            sage: S.dense_module()
            Vector space of dimension 3 over Rational Field
            sage: M.sparse_module() == S
            True
            sage: S.dense_module() == M
            True
            sage: M.dense_module() == M
            True
            sage: S.sparse_module() == S
            True

        Next we create a subspace::

            sage: M = FreeModule(QQ,3, sparse=True)
            sage: V = M.span([ [1,2,3] ] ); V
            Sparse vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
            sage: V.sparse_module()
            Sparse vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
        """
        if self.is_sparse():
            return self._dense_module()
        return self

    def _dense_module(self):
        """
        Creates a dense module with the same defining data as self.

        N.B. This function is for internal use only! See dense_module for
        use.

        EXAMPLES::

            sage: M = FreeModule(Integers(8),3)
            sage: S = FreeModule(Integers(8),3, sparse=True)
            sage: M is S._dense_module()
            True
        """
        A = self.ambient_module().dense_module()
        return A.span(self.basis())

    def sparse_module(self):
        """
        Return the corresponding sparse module with the same defining
        data.

        EXAMPLES:

        We first illustrate conversion with ambient spaces::

            sage: M = FreeModule(Integers(8),3)
            sage: S = FreeModule(Integers(8),3, sparse=True)
            sage: M.sparse_module()
            Ambient sparse free module of rank 3 over Ring of integers modulo 8
            sage: S.dense_module()
            Ambient free module of rank 3 over Ring of integers modulo 8
            sage: M.sparse_module() is S
            True
            sage: S.dense_module() is M
            True
            sage: M.dense_module() is M
            True
            sage: S.sparse_module() is S
            True

        Next we convert a subspace::

            sage: M = FreeModule(QQ,3)
            sage: V = M.span([ [1,2,3] ] ); V
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
            sage: V.sparse_module()
            Sparse vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
        """
        if self.is_sparse():
            return self
        return self._sparse_module()

    def _sparse_module(self):
        """
        Creates a sparse module with the same defining data as self.

        N.B. This function is for internal use only! See sparse_module for
        use.

        EXAMPLES::

            sage: M = FreeModule(Integers(8),3)
            sage: S = FreeModule(Integers(8),3, sparse=True)
            sage: M._sparse_module() is S
            True
        """
        A = self.ambient_module().sparse_module()
        return A.span(self.basis())

    def _an_element_(self):
        """
        Returns an arbitrary element of a free module.

        EXAMPLES::

            sage: V = VectorSpace(QQ,2)
            sage: V._an_element_()
            (1, 0)
            sage: U = V.submodule([[1,0]])
            sage: U._an_element_()
            (1, 0)
            sage: W = V.submodule([])
            sage: W._an_element_()
            (0, 0)
        """
        try:
            return self.gen(0)
        except ValueError:
            return self(0)

    def some_elements(self):
        r"""
        Return some elements of this free module.

        See :class:`TestSuite` for a typical use case.

        OUTPUT:

        An iterator.

        EXAMPLES::

            sage: F = FreeModule(ZZ, 2)
            sage: tuple(F.some_elements())
            ((1, 0),
             (1, 1),
             (0, 1),
             (-1, 2),
             (-2, 3),
             ...
             (-49, 50))

            sage: F = FreeModule(QQ, 3)
            sage: tuple(F.some_elements())
            ((1, 0, 0),
             (1/2, 1/2, 1/2),
             (1/2, -1/2, 2),
             (-2, 0, 1),
             (-1, 42, 2/3),
             (-2/3, 3/2, -3/2),
             (4/5, -4/5, 5/4),
             ...
             (46/103823, -46/103823, 103823/46))

            sage: F = FreeModule(SR, 2)
            sage: tuple(F.some_elements())
            ((1, 0), (some_variable, some_variable))
        """
        from itertools import islice
        yield self.an_element()
        yield self.base().an_element() * sum(self.gens())
        some_elements_base = iter(self.base().some_elements())
        n = self.degree()
        while True:
            L = list(islice(some_elements_base, n))
            if len(L) != n:
                return
            try:
                yield self(L)
            except (TypeError, ValueError):
                pass

    def _element_constructor_(self, x, coerce=True, copy=True, check=True):
        r"""
        Create an element of this free module from ``x``.

        The ``coerce`` and ``copy`` arguments are
        passed on to the underlying element constructor. If
        ``check`` is ``True``, confirm that the
        element specified by x does in fact lie in self.

        .. note::

           In the case of an inexact base ring (i.e. RDF), we don't
           verify that the element is in the subspace, even when
           ``check=True``, to account for numerical instability
           issues.

        EXAMPLE::

            sage: M = ZZ^4
            sage: M([1,-1,0,1])  #indirect doctest
            (1, -1, 0, 1)
            sage: M(0)
            (0, 0, 0, 0)

        ::

            sage: N = M.submodule([[1,0,0,0], [0,1,1,0]])
            sage: N([1,1,1,0])
            (1, 1, 1, 0)
            sage: N((3,-2,-2,0))
            (3, -2, -2, 0)
            sage: N((0,0,0,1))
            Traceback (most recent call last):
            ...
            TypeError: element (0, 0, 0, 1) is not in free module

        Beware that using check=False can create invalid results::

            sage: N((0,0,0,1), check=False)
            (0, 0, 0, 1)
            sage: N((0,0,0,1), check=False) in N
            True
        """
        if isinstance(x, (int, long, sage.rings.integer.Integer)) and x==0:
            return self.zero_vector()
        elif isinstance(x, free_module_element.FreeModuleElement):
            if x.parent() is self:
                if copy:
                    return x.__copy__()
                else:
                    return x
            x = x.list()
        if check and self.coordinate_ring().is_exact():
            if isinstance(self, FreeModule_ambient):
                return self.element_class(self, x, coerce, copy)
            try:
                c = self.coordinates(x)
                R = self.base_ring()
                for d in c:
                    if d not in R:
                        raise ArithmeticError
            except ArithmeticError:
                raise TypeError("element {!r} is not in free module".format(x))
        return self.element_class(self, x, coerce, copy)

    def is_submodule(self, other):
        """
        Return ``True`` if ``self`` is a submodule of ``other``.

        EXAMPLES::

            sage: M = FreeModule(ZZ,3)
            sage: V = M.ambient_vector_space()
            sage: X = V.span([[1/2,1/2,0],[1/2,0,1/2]], ZZ)
            sage: Y = V.span([[1,1,1]], ZZ)
            sage: N = X + Y
            sage: M.is_submodule(X)
            False
            sage: M.is_submodule(Y)
            False
            sage: Y.is_submodule(M)
            True
            sage: N.is_submodule(M)
            False
            sage: M.is_submodule(N)
            True

            sage: M = FreeModule(ZZ,2)
            sage: M.is_submodule(M)
            True
            sage: N = M.scale(2)
            sage: N.is_submodule(M)
            True
            sage: M.is_submodule(N)
            False
            sage: N = M.scale(1/2)
            sage: N.is_submodule(M)
            False
            sage: M.is_submodule(N)
            True

        Since :meth:`basis` is not implemented in general, submodule
        testing does not work for all PID's. However, trivial cases are
        already used (and useful) for coercion, e.g. ::

            sage: QQ(1/2) * vector(ZZ['x']['y'],[1,2,3,4])
            (1/2, 1, 3/2, 2)
            sage: vector(ZZ['x']['y'],[1,2,3,4]) * QQ(1/2)
            (1/2, 1, 3/2, 2)
        """
        if not isinstance(other, FreeModule_generic):
            return False
        try:
            if self.ambient_vector_space() != other.ambient_vector_space():
                return False
            if other == other.ambient_vector_space():
                return True
        except AttributeError:
            # Not all free modules have an ambient_vector_space.
            pass
        if other.rank() < self.rank():
            return False
        R = self.base_ring()
        S = other.base_ring()
        if R != S:
            try:
                if not R.is_subring(S):
                    return False
            except NotImplementedError:
                return False
        try:
            M = other.basis_matrix().solve_left(self.basis_matrix())
        except ValueError:
            return False
        return all(x in S for x in M.list())

    def __iter__(self):
        """
        Return iterator over the elements of this free module.

        EXAMPLES::

            sage: V = VectorSpace(GF(4,'a'),2)
            sage: [x for x in V]
            [(0, 0), (a, 0), (a + 1, 0), (1, 0), (0, a), (a, a), (a + 1, a), (1, a), (0, a + 1), (a, a + 1), (a + 1, a + 1), (1, a + 1), (0, 1), (a, 1), (a + 1, 1), (1, 1)]

        ::

            sage: W = V.subspace([V([1,1])])
            sage: [x for x in W]
            [(0, 0), (a, a), (a + 1, a + 1), (1, 1)]

        TESTS::

            sage: V = VectorSpace(GF(2,'a'),2)
            sage: V.list()
            [(0, 0), (1, 0), (0, 1), (1, 1)]
        """
        G = self.gens()
        if len(G) == 0:
            yield self(0)
            return
        R     = self.base_ring()
        iters = [iter(R) for _ in range(len(G))]
        for x in iters: next(x)     # put at 0
        zero  = R(0)
        v = [zero for _ in range(len(G))]
        n = 0
        z = self(0)
        yield z
        while n < len(G):
            try:
                v[n] = next(iters[n])
                yield self.linear_combination_of_basis(v)
                n = 0
            except StopIteration:
                iters[n] = iter(R)  # reset
                next(iters[n])     # put at 0
                v[n] = zero
                n += 1

    def cardinality(self):
        r"""
        Return the cardinality of the free module.

        OUTPUT:

        Either an integer or ``+Infinity``.

        EXAMPLES::

            sage: k.<a> = FiniteField(9)
            sage: V = VectorSpace(k,3)
            sage: V.cardinality()
            729
            sage: W = V.span([[1,2,1],[0,1,1]])
            sage: W.cardinality()
            81
            sage: R = IntegerModRing(12)
            sage: M = FreeModule(R,2)
            sage: M.cardinality()
            144
            sage: (QQ^3).cardinality()
            +Infinity
        """
        return (self.base_ring().cardinality())**self.rank()

    __len__ = cardinality # for backward compatibility

    def ambient_module(self):
        """
        Return the ambient module associated to this module.

        EXAMPLES::

            sage: R.<x,y> = QQ[]
            sage: M = FreeModule(R,2)
            sage: M.ambient_module()
            Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y over Rational Field

        ::

            sage: V = FreeModule(QQ, 4).span([[1,2,3,4], [1,0,0,0]]); V
            Vector space of degree 4 and dimension 2 over Rational Field
            Basis matrix:
            [  1   0   0   0]
            [  0   1 3/2   2]
            sage: V.ambient_module()
            Vector space of dimension 4 over Rational Field
        """
        return FreeModule(self.base_ring(), self.degree())

    def basis(self):
        """
        Return the basis of this module.

        EXAMPLES::

            sage: FreeModule(Integers(12),3).basis()
            [
            (1, 0, 0),
            (0, 1, 0),
            (0, 0, 1)
            ]
        """
        raise NotImplementedError

    def gens(self):
        """
        Return a tuple of basis elements of ``self``.

        EXAMPLES::

            sage: FreeModule(Integers(12),3).gens()
            ((1, 0, 0), (0, 1, 0), (0, 0, 1))
        """
        return tuple(self.basis())

    def basis_matrix(self, ring=None):
        """
        Return the matrix whose rows are the basis for this free module.

        INPUT:

        - ``ring`` -- (default: ``self.coordinate_ring()``) a ring over
          which the matrix is defined

        EXAMPLES::

            sage: FreeModule(Integers(12),3).basis_matrix()
            [1 0 0]
            [0 1 0]
            [0 0 1]

        ::

            sage: M = FreeModule(GF(7),3).span([[2,3,4],[1,1,1]]); M
            Vector space of degree 3 and dimension 2 over Finite Field of size 7
            Basis matrix:
            [1 0 6]
            [0 1 2]
            sage: M.basis_matrix()
            [1 0 6]
            [0 1 2]

        ::

            sage: M = FreeModule(GF(7),3).span_of_basis([[2,3,4],[1,1,1]]);
            sage: M.basis_matrix()
            [2 3 4]
            [1 1 1]

        ::

            sage: M = FreeModule(QQ,2).span_of_basis([[1,-1],[1,0]]); M
            Vector space of degree 2 and dimension 2 over Rational Field
            User basis matrix:
            [ 1 -1]
            [ 1  0]
            sage: M.basis_matrix()
            [ 1 -1]
            [ 1  0]

        TESTS:

        See :trac:`3699`::

            sage: K = FreeModule(ZZ, 2000)
            sage: I = K.basis_matrix()

        See :trac:`17585`::

            sage: ((ZZ^2)*2).basis_matrix().parent()
            Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
            sage: ((ZZ^2)*2).basis_matrix(RDF).parent()
            Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

            sage: M = (ZZ^2)*(1/2)
            sage: M.basis_matrix()
            [1/2   0]
            [  0 1/2]
            sage: M.basis_matrix().parent()
            Full MatrixSpace of 2 by 2 dense matrices over Rational Field
            sage: M.basis_matrix(QQ).parent()
            Full MatrixSpace of 2 by 2 dense matrices over Rational Field
            sage: M.basis_matrix(ZZ)
            Traceback (most recent call last):
            ...
            TypeError: matrix has denominators so can't change to ZZ.
        """
        try:
            A = self.__basis_matrix
        except AttributeError:
            MAT = sage.matrix.matrix_space.MatrixSpace(self.coordinate_ring(),
                            len(self.basis()), self.degree(),
                            sparse = self.is_sparse())
            if self.is_ambient():
                A = MAT.identity_matrix()
            else:
                A = MAT(self.basis())
            A.set_immutable()
            self.__basis_matrix = A
        if ring is None or ring is A.base_ring():
            return A
        else:
            return A.change_ring(ring)

    def echelonized_basis_matrix(self):
        """
        The echelonized basis matrix (not implemented for this module).

        This example works because M is an ambient module. Submodule
        creation should exist for generic modules.

        EXAMPLES::

            sage: R = IntegerModRing(12)
            sage: S.<x,y> = R[]
            sage: M = FreeModule(S,3)
            sage: M.echelonized_basis_matrix()
            [1 0 0]
            [0 1 0]
            [0 0 1]

        TESTS::

            sage: from sage.modules.free_module import FreeModule_generic
            sage: FreeModule_generic.echelonized_basis_matrix(M)
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError

    def matrix(self):
        """
        Return the basis matrix of this module, which is the matrix whose
        rows are a basis for this module.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2)
            sage: M.matrix()
            [1 0]
            [0 1]
            sage: M.submodule([M.gen(0) + M.gen(1), M.gen(0) - 2*M.gen(1)]).matrix()
            [1 1]
            [0 3]
        """
        return self.basis_matrix()

    def direct_sum(self, other):
        """
        Return the direct sum of self and other as a free module.

        EXAMPLES::

            sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]]); V
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1/2   0  14]
            [  0   1  -3]
            sage: W = (ZZ^3).span([[1/2,4,2]]); W
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1/2   4   2]
            sage: V.direct_sum(W)
            Free module of degree 6 and rank 3 over Integer Ring
            Echelon basis matrix:
            [1/2   0  14   0   0   0]
            [  0   1  -3   0   0   0]
            [  0   0   0 1/2   4   2]
        """
        if not is_FreeModule(other):
            raise TypeError("other must be a free module")
        if other.base_ring() != self.base_ring():
            raise TypeError("base rings of self and other must be the same")
        return self.basis_matrix().block_sum(other.basis_matrix()).row_module(self.base_ring())

    def coordinates(self, v, check=True):
        r"""
        Write `v` in terms of the basis for self.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        Returns a list `c` such that if `B` is the basis
        for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2); M0,M1=M.gens()
            sage: W = M.submodule([M0 + M1, M0 - 2*M1])
            sage: W.coordinates(2*M0-M1)
            [2, -1]
        """
        return self.coordinate_vector(v, check=check).list()

    def coordinate_vector(self, v, check=True):
        """
        Return the vector whose coefficients give `v` as a linear
        combination of the basis for self.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2); M0,M1=M.gens()
            sage: W = M.submodule([M0 + M1, M0 - 2*M1])
            sage: W.coordinate_vector(2*M0 - M1)
            (2, -1)
        """
        raise NotImplementedError

    def coordinate_module(self, V):
        r"""
        Suppose V is a submodule of self (or a module commensurable with
        self), and that self is a free module over `R` of rank
        `n`. Let `\phi` be the map from self to
        `R^n` that sends the basis vectors of self in order to the
        standard basis of `R^n`. This function returns the image
        `\phi(V)`.

        .. warning::

           If there is no integer `d` such that `dV` is a
           submodule of self, then this function will give total
           nonsense.

        EXAMPLES:

        We illustrate this function with some
        `\ZZ`-submodules of `\QQ^3`.

        ::

            sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]])
            sage: W = (ZZ^3).span([[1/2,4,2]])
            sage: V.coordinate_module(W)
            Free module of degree 2 and rank 1 over Integer Ring
            User basis matrix:
            [1 4]
            sage: V.0 + 4*V.1
            (1/2, 4, 2)

        In this example, the coordinate module isn't even in
        `\ZZ^3`.

        ::

            sage: W = (ZZ^3).span([[1/4,2,1]])
            sage: V.coordinate_module(W)
            Free module of degree 2 and rank 1 over Integer Ring
            User basis matrix:
            [1/2   2]

        The following more elaborate example illustrates using this
        function to write a submodule in terms of integral cuspidal modular
        symbols::

            sage: M = ModularSymbols(54)
            sage: S = M.cuspidal_subspace()
            sage: K = S.integral_structure(); K
            Free module of degree 19 and rank 8 over Integer Ring
            Echelon basis matrix:
            [ 0  1  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
            ...
            sage: L = M[0].integral_structure(); L
            Free module of degree 19 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 0  1  1  0 -2  1 -1  1 -1 -2  2  0  0  0  0  0  0  0  0]
            [ 0  0  3  0 -3  2 -1  2 -1 -4  2 -1 -2  1  2  0  0 -1  1]
            sage: K.coordinate_module(L)
            Free module of degree 8 and rank 2 over Integer Ring
            User basis matrix:
            [ 1  1  1 -1  1 -1  0  0]
            [ 0  3  2 -1  2 -1 -1 -2]
            sage: K.coordinate_module(L).basis_matrix() * K.basis_matrix()
            [ 0  1  1  0 -2  1 -1  1 -1 -2  2  0  0  0  0  0  0  0  0]
            [ 0  0  3  0 -3  2 -1  2 -1 -4  2 -1 -2  1  2  0  0 -1  1]
        """
        if not is_FreeModule(V):
            raise ValueError("V must be a free module")
        A = self.basis_matrix()
        A = A.matrix_from_columns(A.pivots()).transpose()
        B = V.basis_matrix()
        B = B.matrix_from_columns(self.basis_matrix().pivots()).transpose()
        S = A.solve_right(B).transpose()
        return (self.base_ring()**S.ncols()).span_of_basis(S.rows())

    def degree(self):
        """
        Return the degree of this free module. This is the dimension of the
        ambient vector space in which it is embedded.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 10)
            sage: W = M.submodule([M.gen(0), 2*M.gen(3) - M.gen(0), M.gen(0) + M.gen(3)])
            sage: W.degree()
            10
            sage: W.rank()
            2
        """
        return self.__degree

    def dimension(self):
        """
        Return the dimension of this free module.

        EXAMPLES::

            sage: M = FreeModule(FiniteField(19), 100)
            sage: W = M.submodule([M.gen(50)])
            sage: W.dimension()
            1
        """
        return self.rank()

    def discriminant(self):
        """
        Return the discriminant of this free module.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 3)
            sage: M.discriminant()
            1
            sage: W = M.span([[1,2,3]])
            sage: W.discriminant()
            14
            sage: W2 = M.span([[1,2,3], [1,1,1]])
            sage: W2.discriminant()
            6
        """
        return self.gram_matrix().determinant()

    def base_field(self):
        """
        Return the base field, which is the fraction field of the base ring
        of this module.

        EXAMPLES::

            sage: FreeModule(GF(3), 2).base_field()
            Finite Field of size 3
            sage: FreeModule(ZZ, 2).base_field()
            Rational Field
            sage: FreeModule(PolynomialRing(GF(7),'x'), 2).base_field()
            Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 7
        """
        return self.base_ring().fraction_field()

    def coordinate_ring(self):
        """
        Return the ring over which the entries of the vectors are
        defined.

        This is the same as :meth:`base_ring` unless an explicit basis
        was given over the fraction field.

        EXAMPLES::

            sage: M = ZZ^2
            sage: M.coordinate_ring()
            Integer Ring

        ::

            sage: M = (ZZ^2) * (1/2)
            sage: M.base_ring()
            Integer Ring
            sage: M.coordinate_ring()
            Rational Field

        ::

            sage: R.<x> = QQ[]
            sage: L = R^2
            sage: L.coordinate_ring()
            Univariate Polynomial Ring in x over Rational Field
            sage: L.span([(x,0), (1,x)]).coordinate_ring()
            Univariate Polynomial Ring in x over Rational Field
            sage: L.span([(x,0), (1,1/x)]).coordinate_ring()
            Fraction Field of Univariate Polynomial Ring in x over Rational Field
            sage: L.span([]).coordinate_ring()
            Univariate Polynomial Ring in x over Rational Field
        """
        return self.__coordinate_ring

    def free_module(self):
        """
        Return this free module. (This is used by the
        ``FreeModule`` functor, and simply returns self.)

        EXAMPLES::

            sage: M = FreeModule(ZZ, 3)
            sage: M.free_module()
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
        """
        return self

    def gen(self, i=0):
        """
        Return the `i`-th generator for ``self``.

        Here `i` is between 0 and rank - 1, inclusive.

        INPUT:

        - `i` -- an integer (default 0)

        OUTPUT: `i`-th basis vector for ``self``.

        EXAMPLES::

            sage: n = 5
            sage: V = QQ^n
            sage: B = [V.gen(i) for i in range(n)]
            sage: B
            [(1, 0, 0, 0, 0),
            (0, 1, 0, 0, 0),
            (0, 0, 1, 0, 0),
            (0, 0, 0, 1, 0),
            (0, 0, 0, 0, 1)]
            sage: V.gens() == tuple(B)
            True

        TESTS::

            sage: (QQ^3).gen(4/3)
            Traceback (most recent call last):
            ...
            TypeError: rational is not an integer
        """
        if i < 0 or i >= self.rank():
            raise ValueError("Generator %s not defined." % i)
        return self.basis()[i]

    def gram_matrix(self):
        r"""
        Return the gram matrix associated to this free module, defined to
        be `G = B*A*B.transpose()`, where A is the inner product matrix
        (induced from the ambient space), and B the basis matrix.

        EXAMPLES::

            sage: V = VectorSpace(QQ,4)
            sage: u = V([1/2,1/2,1/2,1/2])
            sage: v = V([0,1,1,0])
            sage: w = V([0,0,1,1])
            sage: M = span([u,v,w], ZZ)
            sage: M.inner_product_matrix() == V.inner_product_matrix()
            True
            sage: L = M.submodule_with_basis([u,v,w])
            sage: L.inner_product_matrix() == M.inner_product_matrix()
            True
            sage: L.gram_matrix()
            [1 1 1]
            [1 2 1]
            [1 1 2]
        """
        if self.is_ambient():
            return sage.matrix.matrix_space.MatrixSpace(self.base_ring(), self.degree(), sparse=True)(1)
        else:
            if self._gram_matrix is None:
                B = self.basis_matrix()
                self._gram_matrix = B*B.transpose()
            return self._gram_matrix

    def has_user_basis(self):
        """
        Return ``True`` if the basis of this free module is
        specified by the user, as opposed to being the default echelon
        form.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.subspace([[2,'1/2', 1]])
            sage: W.has_user_basis()
            False
            sage: W = V.subspace_with_basis([[2,'1/2',1]])
            sage: W.has_user_basis()
            True
        """
        return False

    def inner_product_matrix(self):
        """
        Return the default identity inner product matrix associated to this
        module.

        By definition this is the inner product matrix of the ambient
        space, hence may be of degree greater than the rank of the module.

        TODO: Differentiate the image ring of the inner product from the
        base ring of the module and/or ambient space. E.g. On an integral
        module over ZZ the inner product pairing could naturally take
        values in ZZ, QQ, RR, or CC.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 3)
            sage: M.inner_product_matrix()
            [1 0 0]
            [0 1 0]
            [0 0 1]
        """
        return sage.matrix.matrix_space.MatrixSpace(self.base_ring(), self.degree(), sparse=True)(1)

    def _inner_product_is_dot_product(self):
        """
        Return whether or not the inner product on this module is induced
        by the dot product on the ambient vector space. This is used
        internally by the inner_product function for optimization.

        EXAMPLES::

            sage: FreeModule(ZZ, 3)._inner_product_is_dot_product()
            True
            sage: FreeModule(ZZ, 3, inner_product_matrix=1)._inner_product_is_dot_product()
            True
            sage: FreeModule(ZZ, 2, inner_product_matrix=[1,0,-1,0])._inner_product_is_dot_product()
            False

        ::

            sage: M = FreeModule(QQ, 3)
            sage: M2 = M.span([[1,2,3]])
            sage: M2._inner_product_is_dot_product()
            True
        """
        return True

    def is_ambient(self):
        """
        Returns False since this is not an ambient free module.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 3).span([[1,2,3]]); M
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1 2 3]
            sage: M.is_ambient()
            False
            sage: M = (ZZ^2).span([[1,0], [0,1]])
            sage: M
            Free module of degree 2 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 0]
            [0 1]
            sage: M.is_ambient()
            False
            sage: M == M.ambient_module()
            True
        """
        return False

    def is_dense(self):
        """
        Return ``True`` if the underlying representation of
        this module uses dense vectors, and False otherwise.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).is_dense()
            True
            sage: FreeModule(ZZ, 2, sparse=True).is_dense()
            False
        """
        return not self.is_sparse()

    def is_full(self):
        """
        Return ``True`` if the rank of this module equals its
        degree.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).is_full()
            True
            sage: M = FreeModule(ZZ, 2).span([[1,2]])
            sage: M.is_full()
            False
        """
        return self.rank() == self.degree()

    def is_finite(self):
        """
        Returns True if the underlying set of this free module is finite.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).is_finite()
            False
            sage: FreeModule(Integers(8), 2).is_finite()
            True
            sage: FreeModule(ZZ, 0).is_finite()
            True
        """
        return self.base_ring().is_finite() or self.rank() == 0

    def is_sparse(self):
        """
        Return ``True`` if the underlying representation of
        this module uses sparse vectors, and False otherwise.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).is_sparse()
            False
            sage: FreeModule(ZZ, 2, sparse=True).is_sparse()
            True
        """
        return self.__is_sparse

    def ngens(self):
        """
        Returns the number of basis elements of this free module.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).ngens()
            2
            sage: FreeModule(ZZ, 0).ngens()
            0
            sage: FreeModule(ZZ, 2).span([[1,1]]).ngens()
            1
        """
        try:
            return self.__ngens
        except AttributeError:
            self.__ngens = self.rank()
        return self.__ngens

    def nonembedded_free_module(self):
        """
        Returns an ambient free module that is isomorphic to this free
        module.

        Thus if this free module is of rank `n` over a ring
        `R`, then this function returns `R^n`, as an
        ambient free module.

        EXAMPLES::

            sage: FreeModule(ZZ, 2).span([[1,1]]).nonembedded_free_module()
            Ambient free module of rank 1 over the principal ideal domain Integer Ring
        """
        return FreeModule(self.base_ring(), self.rank())

    def random_element(self, prob=1.0, *args, **kwds):
        """
        Returns a random element of self.

        INPUT:

        -- ``prob`` - float. Each coefficient will be set to zero with
           probability `1-prob`. Otherwise coefficients will be chosen
           randomly from base ring (and may be zero).

        -- ``*args, **kwds`` - passed on to ``random_element()`` function
           of base ring.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2).span([[1,1]])
            sage: M.random_element()
            (-1, -1)
            sage: M.random_element()
            (2, 2)
            sage: M.random_element()
            (1, 1)

        Passes extra positional or keyword arguments through::

            sage: M.random_element(5,10)
            (9, 9)
        """
        rand = current_randstate().python_random().random
        R = self.base_ring()
        prob = float(prob)
        c = [0 if rand() > prob else R.random_element(*args, **kwds) for _ in range(self.rank())]
        return self.linear_combination_of_basis(c)

    def rank(self):
        """
        Return the rank of this free module.

        EXAMPLES::

            sage: FreeModule(Integers(6), 10000000).rank()
            10000000
            sage: FreeModule(ZZ, 2).span([[1,1], [2,2], [3,4]]).rank()
            2
        """
        return self.__rank

    def uses_ambient_inner_product(self):
        r"""
        Return ``True`` if the inner product on this module is
        the one induced by the ambient inner product.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2)
            sage: W = M.submodule([[1,2]])
            sage: W.uses_ambient_inner_product()
            True
            sage: W.inner_product_matrix()
            [1 0]
            [0 1]

        ::

            sage: W.gram_matrix()
            [5]
        """
        return self.__uses_ambient_inner_product

    def zero_vector(self):
        """
        Returns the zero vector in this free module.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2)
            sage: M.zero_vector()
            (0, 0)
            sage: M(0)
            (0, 0)
            sage: M.span([[1,1]]).zero_vector()
            (0, 0)
            sage: M.zero_submodule().zero_vector()
            (0, 0)
        """
        # Do *not* cache this -- it must be computed fresh each time, since
        # it is used by __call__ to make a new copy of the 0 element.

        return self.element_class(self, 0)

    @cached_method
    def zero(self):
        """
        Returns the zero vector in this free module.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 2)
            sage: M.zero()
            (0, 0)
            sage: M.span([[1,1]]).zero()
            (0, 0)
            sage: M.zero_submodule().zero()
            (0, 0)
            sage: M.zero_submodule().zero().is_mutable()
            False
        """
        res = self.element_class(self, 0)
        res.set_immutable()
        return res

    def are_linearly_dependent(self, vecs):
        """
        Return ``True`` if the vectors ``vecs`` are linearly dependent and
        ``False`` otherwise.

        EXAMPLES::

            sage: M = QQ^3
            sage: vecs = [M([1,2,3]), M([4,5,6])]
            sage: M.are_linearly_dependent(vecs)
            False
            sage: vecs.append(M([3,3,3]))
            sage: M.are_linearly_dependent(vecs)
            True

            sage: R.<x> = QQ[]
            sage: M = FreeModule(R, 2)
            sage: vecs = [M([x^2+1, x+1]), M([x+2, 2*x+1])]
            sage: M.are_linearly_dependent(vecs)
            False
            sage: vecs.append(M([-2*x+1, -2*x^2+1]))
            sage: M.are_linearly_dependent(vecs)
            True
        """
        from sage.matrix.constructor import matrix
        A = matrix(vecs)
        A.echelonize()
        return any(row.is_zero() for row in A.rows())

    def _magma_init_(self, magma):
        """
        EXAMPLES::

            sage: magma(QQ^9)                                   # optional - magma
            Full Vector space of degree 9 over Rational Field
            sage: (QQ^9)._magma_init_(magma)                    # optional - magma
            'RSpace(_sage_[...],9)'

        ::

            sage: magma(Integers(8)^2)                          # optional - magma
            Full RSpace of degree 2 over IntegerRing(8)
            sage: magma(FreeModule(QQ['x'], 2))                 # optional - magma
            Full RSpace of degree 2 over Univariate Polynomial Ring in x over Rational Field

        ::

            sage: A = matrix([[1,0],[0,-1]])
            sage: M = FreeModule(ZZ,2,inner_product_matrix=A); M
            Ambient free quadratic module of rank 2 over the principal ideal domain Integer Ring
            Inner product matrix:
            [ 1  0]
            [ 0 -1]
            sage: M._magma_init_(magma)                         # optional - magma
            'RSpace(_sage_[...],2,_sage_ref...)'
            sage: m = magma(M); m                               # optional - magma
            Full RSpace of degree 2 over Integer Ring
            Inner Product Matrix:
            [ 1  0]
            [ 0 -1]
            sage: m.Type()                                      # optional - magma
            ModTupRng
            sage: m.sage()                                      # optional - magma
            Ambient free quadratic module of rank 2 over the principal ideal domain Integer Ring
            Inner product matrix:
            [ 1  0]
            [ 0 -1]
            sage: m.sage() is M                               # optional - magma
            True

        Now over a field::

            sage: N = FreeModule(QQ,2,inner_product_matrix=A); N
            Ambient quadratic space of dimension 2 over Rational Field
            Inner product matrix:
            [ 1  0]
            [ 0 -1]
            sage: n = magma(N); n                                      # optional - magma
            Full Vector space of degree 2 over Rational Field
            Inner Product Matrix:
            [ 1  0]
            [ 0 -1]
            sage: n.Type()                                             # optional - magma
            ModTupFld
            sage: n.sage()                                           # optional - magma
            Ambient quadratic space of dimension 2 over Rational Field
            Inner product matrix:
            [ 1  0]
            [ 0 -1]
            sage: n.sage() is N                                      # optional - magma
            True

        How about some inexact fields::

            sage: v = vector(RR, [1, pi, 5/6])
            sage: F = v.parent()
            sage: M = magma(F); M # optional - magma
            Full Vector space of degree 3 over Real field of precision 15
            sage: M.Type() # optional - magma
            ModTupFld
            sage: m = M.sage(); m # optional - magma
            Vector space of dimension 3 over Real Field with 53 bits of precision
            sage: m is F # optional - magma
            True

        For interval fields, we can convert to Magma but there is no
        interval field in Magma so we cannot convert back::

            sage: v = vector(RealIntervalField(100), [1, pi, 0.125])
            sage: F = v.parent()
            sage: M = magma(v.parent()); M # optional - magma
            Full Vector space of degree 3 over Real field of precision 30
            sage: M.Type() # optional - magma
            ModTupFld
            sage: m = M.sage(); m # optional - magma
            Vector space of dimension 3 over Real Field with 100 bits of precision
            sage: m is F # optional - magma
            False
        """
        K = magma(self.base_ring())
        if not self._inner_product_is_dot_product():
            M = magma(self.inner_product_matrix())
            return "RSpace(%s,%s,%s)"%(K.name(), self.rank(), M._ref())
        else:
            return "RSpace(%s,%s)"%(K.name(), self.rank())

    def _macaulay2_(self, macaulay2=None):
        r"""
        EXAMPLES::

            sage: R = QQ^2
            sage: macaulay2(R) # optional - macaulay2
              2
            QQ
        """
        if macaulay2 is None:
            from sage.interfaces.macaulay2 import macaulay2
        if self._inner_product_matrix:
            raise NotImplementedError
        else:
            return macaulay2(self.base_ring())**self.rank()

class FreeModule_generic_pid(FreeModule_generic):
    """
    Base class for all free modules over a PID.
    """
    def __init__(self, base_ring, rank, degree, sparse=False, coordinate_ring=None):
        """
        Create a free module over a PID.

        EXAMPLES::

            sage: FreeModule(ZZ, 2)
            Ambient free module of rank 2 over the principal ideal domain Integer Ring
            sage: FreeModule(PolynomialRing(GF(7),'x'), 2)
            Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Finite Field of size 7
        """
        # The first check should go away once everything is categorized...
        if base_ring not in PrincipalIdealDomains():
            raise TypeError("The base_ring must be a principal ideal domain.")
        super(FreeModule_generic_pid, self).__init__(base_ring, rank, degree,
                                                     sparse, coordinate_ring)

    def scale(self, other):
        """
        Return the product of this module by the number other, which is the
        module spanned by other times each basis vector.

        EXAMPLES::

            sage: M = FreeModule(ZZ, 3)
            sage: M.scale(2)
            Free module of degree 3 and rank 3 over Integer Ring
            Echelon basis matrix:
            [2 0 0]
            [0 2 0]
            [0 0 2]

        ::

            sage: a = QQ('1/3')
            sage: M.scale(a)
            Free module of degree 3 and rank 3 over Integer Ring
            Echelon basis matrix:
            [1/3   0   0]
            [  0 1/3   0]
            [  0   0 1/3]
        """
        if other == 0:
            return self.zero_submodule()
        if other == 1 or other == -1:
            return self
        return self.span([v*other for v in self.basis()])

    def __radd__(self, other):
        """
        EXAMPLES::

            sage: int(0) + QQ^3
            Vector space of dimension 3 over Rational Field
            sage: sum([QQ^3, QQ^3])
            Vector space of degree 3 and dimension 3 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 1 0]
            [0 0 1]
        """
        if other == 0:
            return self
        else:
            raise TypeError

    def __add__(self, other):
        r"""
        Return the sum of self and other, where both self and other must be
        submodules of the ambient vector space.

        EXAMPLES:

        We add two vector spaces::

            sage: V  = VectorSpace(QQ, 3)
            sage: W  = V.subspace([V([1,1,0])])
            sage: W2 = V.subspace([V([1,-1,0])])
            sage: W + W2
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 1 0]

        We add two free `\ZZ`-modules.

        ::

            sage: M = FreeModule(ZZ, 3)
            sage: W = M.submodule([M([1,0,2])])
            sage: W2 = M.submodule([M([2,0,-4])])
            sage: W + W2
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 0 2]
            [0 0 8]

        We can also add free `\ZZ`-modules embedded
        non-integrally into an ambient space.

        ::

            sage: V = VectorSpace(QQ, 3)
            sage: W = M.span([1/2*V.0 - 1/3*V.1])

        Here the command ``M.span(...)`` creates the span of
        the indicated vectors over the base ring of `M`.

        ::

            sage: W2 = M.span([1/3*V.0 + V.1])
            sage: W + W2
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1/6  7/3    0]
            [   0 11/3    0]

        We add two modules over `\ZZ`::

            sage: A = Matrix(ZZ, 3, 3, [3, 0, -1, 0, -2, 0, 0, 0, -2])
            sage: V = (A+2).kernel()
            sage: W = (A-3).kernel()
            sage: V+W
            Free module of degree 3 and rank 3 over Integer Ring
            Echelon basis matrix:
            [5 0 0]
            [0 1 0]
            [0 0 1]

        We add a module to 0::

            sage: ZZ^3 + 0
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
        """
        if not isinstance(other, FreeModule_generic):
            if other == 0:
                return self
            raise TypeError("other (=%s) must be a free module"%other)
        if not (self.ambient_vector_space() == other.ambient_vector_space()):
            raise TypeError("ambient vector spaces must be equal")
        return self.span(self.basis() + other.basis())

    def _mul_(self, other, switch_sides=False):
        r"""
        Multiplication of the basis by ``other``.

        EXAMPLES::

            sage: A = ZZ^3
            sage: A * 3
            Free module of degree 3 and rank 3 over Integer Ring
            Echelon basis matrix:
            [3 0 0]
            [0 3 0]
            [0 0 3]

            sage: V = A.span([A([1,2,2]), A([-1,0,2])])
            sage: 2 * V
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 2  0 -4]
            [ 0  4  8]

            sage: m = matrix(3, range(9))
            sage: A * m
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 3  0 -3]
            [ 0  1  2]
            sage: m * A
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 3  0 -3]
            [ 0  1  2]

        TESTS:

        Check that :trac:`17705` is fixed::

            sage: V = GF(2)^2
            sage: W = V.subspace([[1, 0]])
            sage: x = matrix(GF(2), [[1, 1], [0, 1]])
            sage: W*x
            Vector space of degree 2 and dimension 1 over Finite Field of size 2
            Basis matrix:
            [1 1]

        """
        B = self.basis_matrix()
        B = other * B if switch_sides else B * other
        return self.span(B.rows())

    def index_in(self, other):
        """
        Return the lattice index [other:self] of self in other, as an
        element of the base field. When self is contained in other, the
        lattice index is the usual index. If the index is infinite, then
        this function returns infinity.

        EXAMPLES::

            sage: L1 = span([[1,2]], ZZ)
            sage: L2 = span([[3,6]], ZZ)
            sage: L2.index_in(L1)
            3

        Note that the free modules being compared need not be integral.

        ::

            sage: L1 = span([['1/2','1/3'], [4,5]], ZZ)
            sage: L2 = span([[1,2], [3,4]], ZZ)
            sage: L2.index_in(L1)
            12/7
            sage: L1.index_in(L2)
            7/12
            sage: L1.discriminant() / L2.discriminant()
            49/144

        The index of a lattice of infinite index is infinite.

        ::

            sage: L1 = FreeModule(ZZ, 2)
            sage: L2 = span([[1,2]], ZZ)
            sage: L2.index_in(L1)
            +Infinity
        """
        if not isinstance(other, FreeModule_generic):
            raise TypeError("other must be a free module")

        if self.ambient_vector_space() != other.ambient_vector_space():
            raise ArithmeticError("self and other must be embedded in the same ambient space.")

        if self.base_ring() != other.base_ring():
            raise NotImplementedError("lattice index only defined for modules over the same base ring.")

        if other.base_ring().is_field():
            if self == other:
                return sage.rings.integer.Integer(1)
            else:
                if self.is_subspace(other):
                    return sage.rings.infinity.infinity
            raise ArithmeticError("self must be contained in the vector space spanned by other.")

        C = [other.coordinates(b) for b in self.basis()]

        if self.rank() < other.rank():
            return sage.rings.infinity.infinity

        a = sage.matrix.matrix_space.MatrixSpace(self.base_field(), self.rank())(C).determinant()
        if sage.rings.integer_ring.is_IntegerRing(self.base_ring()):
            return a.abs()
        elif isinstance(self.base_ring, sage.rings.number_field.order.Order):
            return self.base_ring().ideal(a).norm()
        else:
            raise NotImplementedError

    def intersection(self, other):
        r"""
        Return the intersection of self and other.

        EXAMPLES:

        We intersect two submodules one of which is clearly
        contained in the other.

        ::

            sage: A = ZZ^2
            sage: M1 = A.span([[1,1]])
            sage: M2 = A.span([[3,3]])
            sage: M1.intersection(M2)
            Free module of degree 2 and rank 1 over Integer Ring
            Echelon basis matrix:
            [3 3]
            sage: M1.intersection(M2) is M2
            True

        We intersection two submodules of `\ZZ^3` of rank
        `2`, whose intersection has rank `1`.

        ::

            sage: A = ZZ^3
            sage: M1 = A.span([[1,1,1], [1,2,3]])
            sage: M2 = A.span([[2,2,2], [1,0,0]])
            sage: M1.intersection(M2)
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [2 2 2]

        We compute an intersection of two `\ZZ`-modules that
        are not submodules of `\ZZ^2`.

        ::

            sage: A = ZZ^2
            sage: M1 = A.span([[1,2]]).scale(1/6)
            sage: M2 = A.span([[1,2]]).scale(1/15)
            sage: M1.intersection(M2)
            Free module of degree 2 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1/3 2/3]

        We intersect a `\ZZ`-module with a
        `\QQ`-vector space.

        ::

            sage: A = ZZ^3
            sage: L = ZZ^3
            sage: V = QQ^3
            sage: W = L.span([[1/2,0,1/2]])
            sage: K = V.span([[1,0,1], [0,0,1]])
            sage: W.intersection(K)
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1/2   0 1/2]
            sage: K.intersection(W)
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1/2   0 1/2]

        We intersect two modules over the ring of integers of a number field::

            sage: L.<w> = NumberField(x^2 - x + 2)
            sage: OL = L.ring_of_integers()
            sage: V = L**3; W1 = V.span([[0,w/5,0], [1,0,-1/17]], OL); W2 = V.span([[0,(1-w)/5,0]], OL)
            sage: W1.intersection(W2)
            Free module of degree 3 and rank 1 over Maximal Order in Number Field in w with defining polynomial x^2 - x + 2
            Echelon basis matrix:
            [  0 2/5   0]

        """
        if not isinstance(other, FreeModule_generic):
            raise TypeError("other must be a free module")

        if self.ambient_vector_space() != other.ambient_vector_space():
            raise ArithmeticError("self and other must be embedded in the same ambient space.")

        if self.base_ring() != other.base_ring():
            if other.base_ring().is_field():
                return other.intersection(self)
            raise NotImplementedError("intersection of modules over different base rings (neither a field) is not implemented.")

        # dispense with the three easy cases
        if self == self.ambient_vector_space() or other.is_submodule(self):
            return other
        elif other == other.ambient_vector_space() or self.is_submodule(other):
            return self
        elif self.rank() == 0 or other.rank() == 0:
            if self.base_ring().is_field():
                return other.zero_submodule()
            else:
                return self.zero_submodule()

        # standard algorithm for computing intersection of general submodule
        if self.dimension() <= other.dimension():
            V1 = self; V2 = other
        else:
            V1 = other; V2 = self
        A1 = V1.basis_matrix()
        A2 = V2.basis_matrix()
        S  = A1.stack(A2)
        K  = S.integer_kernel(self.base_ring()).basis_matrix()
        n  = int(V1.dimension())
        K = K.matrix_from_columns(range(n))
        B = K*A1
        return B.row_module(self.base_ring())

    def zero_submodule(self):
        """
        Return the zero submodule of this module.

        EXAMPLES::

            sage: V = FreeModule(ZZ,2)
            sage: V.zero_submodule()
            Free module of degree 2 and rank 0 over Integer Ring
            Echelon basis matrix:
            []
        """
        return self.submodule([], check=False, already_echelonized=True)

    def denominator(self):
        """
        The denominator of the basis matrix of self (i.e. the LCM of the
        coordinate entries with respect to the basis of the ambient
        space).

        EXAMPLES::

            sage: V = QQ^3
            sage: L = V.span([[1,1/2,1/3], [-1/5,2/3,3]],ZZ)
            sage: L
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1/5 19/6 37/3]
            [   0 23/6 46/3]
            sage: L.denominator()
            30
        """
        return self.basis_matrix().denominator()

    def index_in_saturation(self):
        r"""
        Return the index of this module in its saturation, i.e., its
        intersection with `R^n`.

        EXAMPLES::

            sage: W = span([[2,4,6]], ZZ)
            sage: W.index_in_saturation()
            2
            sage: W = span([[1/2,1/3]], ZZ)
            sage: W.index_in_saturation()
            1/6
        """
        # TODO: There is probably a much faster algorithm in this case.
        return self.index_in(self.saturation())

    def saturation(self):
        r"""
        Return the saturated submodule of `R^n` that spans the same
        vector space as self.

        EXAMPLES:

        We create a 1-dimensional lattice that is obviously not
        saturated and saturate it.

        ::

            sage: L = span([[9,9,6]], ZZ); L
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [9 9 6]
            sage: L.saturation()
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [3 3 2]

        We create a lattice spanned by two vectors, and saturate.
        Computation of discriminants shows that the index of lattice in its
        saturation is `3`, which is a prime of congruence between
        the two generating vectors.

        ::

            sage: L = span([[1,2,3], [4,5,6]], ZZ)
            sage: L.saturation()
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
            sage: L.discriminant()
            54
            sage: L.saturation().discriminant()
            6

        Notice that the saturation of a non-integral lattice `L` is
        defined, but the result is integral hence does not contain
        `L`::

            sage: L = span([['1/2',1,3]], ZZ)
            sage: L.saturation()
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1 2 6]
        """
        R = self.base_ring()
        if R.is_field():
            return self
        try:
            A, _ = self.basis_matrix()._clear_denom()
            S = A.saturation().row_space()
        except AttributeError:
            # fallback in case _clear_denom isn't written
            V = self.vector_space()
            A = self.ambient_module()
            S = V.intersection(A)
        # Return exactly self if it is already saturated.
        return self if self == S else S

    def span(self, gens, base_ring=None, check=True, already_echelonized=False):
        """
        Return the R-span of the given list of gens, where R = base_ring.
        The default R is the base ring of self. Note that this span need
        not be a submodule of self, nor even of the ambient space. It must,
        however, be contained in the ambient vector space, i.e., the
        ambient space tensored with the fraction field of R.

        EXAMPLES::

            sage: V = FreeModule(ZZ,3)
            sage: W = V.submodule([V.gen(0)])
            sage: W.span([V.gen(1)])
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [0 1 0]
            sage: W.submodule([V.gen(1)])
            Traceback (most recent call last):
            ...
            ArithmeticError: Argument gens (= [(0, 1, 0)]) does not generate a submodule of self.
            sage: V.span([[1,0,0],[1/5,4,0],[6,3/4,0]])
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1/5   0   0]
            [  0 1/4   0]

        It also works with other things than integers::

            sage: R.<x>=QQ[]
            sage: L=R^1
            sage: a=L.span([(1/x,)])
            sage: a
            Free module of degree 1 and rank 1 over Univariate Polynomial Ring in x over Rational Field
            Echelon basis matrix:
            [1/x]
            sage: b=L.span([(1/x,)])
            sage: a(b.gens()[0])
            (1/x)
            sage: L2 = R^2
            sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]])
            Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field
            Echelon basis matrix:
            [x/(x^3 - 6*x^2 + 11*x - 6)  2/15*x^2 - 17/75*x - 1/75]
            [                         0 x^3 - 11/5*x^2 - 3*x + 4/5]

        Note that the ``base_ring`` can make a huge difference. We
        repeat the previous example over the fraction field of R and
        get a simpler vector space. ::

            sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]],base_ring=R.fraction_field())
            Vector space of degree 2 and dimension 2 over Fraction Field of Univariate Polynomial Ring in x over Rational Field
            Basis matrix:
            [1 0]
            [0 1]
        """
        if is_FreeModule(gens):
            gens = gens.gens()
        if base_ring is None or base_ring is self.base_ring():
            return FreeModule_submodule_pid(
                self.ambient_module(), gens, check=check, already_echelonized=already_echelonized)
        else:
            try:
                M = self.change_ring(base_ring)
            except TypeError:
                raise ValueError("Argument base_ring (= %s) is not compatible "%base_ring + \
                    "with the base field (= %s)." % self.base_field())
            try:
                return M.span(gens)
            except TypeError:
                raise ValueError("Argument gens (= %s) is not compatible "%gens + \
                    "with base_ring (= %s)."%base_ring)

    def submodule(self, gens, check=True, already_echelonized=False):
        r"""
        Create the R-submodule of the ambient vector space with given
        generators, where R is the base ring of self.

        INPUT:


        -  ``gens`` - a list of free module elements or a free
           module

        -  ``check`` - (default: True) whether or not to verify
           that the gens are in self.


        OUTPUT:


        -  ``FreeModule`` - the submodule spanned by the
           vectors in the list gens. The basis for the subspace is always put
           in reduced row echelon form.


        EXAMPLES:

        We create a submodule of `\ZZ^3`::

            sage: M = FreeModule(ZZ, 3)
            sage: B = M.basis()
            sage: W = M.submodule([B[0]+B[1], 2*B[1]-B[2]])
            sage: W
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1  1  0]
            [ 0  2 -1]

        We create a submodule of a submodule.

        ::

            sage: W.submodule([3*B[0] + 3*B[1]])
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [3 3 0]

        We try to create a submodule that isn't really a submodule,
        which results in an ``ArithmeticError`` exception::

            sage: W.submodule([B[0] - B[1]])
            Traceback (most recent call last):
            ...
            ArithmeticError: Argument gens (= [(1, -1, 0)]) does not generate a submodule of self.

        Next we create a submodule of a free module over the principal ideal
        domain `\QQ[x]`, which uses the general Hermite normal form functionality::

            sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
            sage: M = FreeModule(R, 3)
            sage: B = M.basis()
            sage: W = M.submodule([x*B[0], 2*B[1]- x*B[2]]); W
            Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field
            Echelon basis matrix:
            [ x  0  0]
            [ 0  2 -x]
            sage: W.ambient_module()
            Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
        """
        if is_FreeModule(gens):
            gens = gens.gens()
        V = self.span(gens, check=check, already_echelonized=already_echelonized)
        if check:
            if not V.is_submodule(self):
                raise ArithmeticError("Argument gens (= %s) does not generate a submodule of self."%gens)
        return V

    def span_of_basis(self, basis, base_ring=None, check=True, already_echelonized=False):
        r"""
        Return the free R-module with the given basis, where R is the base
        ring of self or user specified base_ring.

        Note that this R-module need not be a submodule of self, nor even
        of the ambient space. It must, however, be contained in the ambient
        vector space, i.e., the ambient space tensored with the fraction
        field of R.

        EXAMPLES::

            sage: M = FreeModule(ZZ,3)
            sage: W = M.span_of_basis([M([1,2,3])])

        Next we create two free `\ZZ`-modules, neither of
        which is a submodule of `W`.

        ::

            sage: W.span_of_basis([M([2,4,0])])
            Free module of degree 3 and rank 1 over Integer Ring
            User basis matrix:
            [2 4 0]

        The following module isn't in the ambient module `\ZZ^3`
        but is contained in the ambient vector space `\QQ^3`::

            sage: V = M.ambient_vector_space()
            sage: W.span_of_basis([ V([1/5,2/5,0]), V([1/7,1/7,0]) ])
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1/5 2/5   0]
            [1/7 1/7   0]

        Of course the input basis vectors must be linearly independent::

            sage: W.span_of_basis([ [1,2,0], [2,4,0] ])
            Traceback (most recent call last):
            ...
            ValueError: The given basis vectors must be linearly independent.
        """
        if is_FreeModule(basis):
            basis = basis.gens()
        if base_ring is None or base_ring == self.base_ring():
            try:
                if self.is_dense():
                    from .free_module_integer import FreeModule_submodule_with_basis_integer
                    return FreeModule_submodule_with_basis_integer(self.ambient_module(),
                                                                   basis=basis, check=check,
                                                                   already_echelonized=already_echelonized,
                                                                   lll_reduce=False)
            except TypeError:
                pass

            return FreeModule_submodule_with_basis_pid(
                self.ambient_module(), basis=basis, check=check,
                already_echelonized=already_echelonized)
        else:
            try:
                M = self.change_ring(base_ring)
            except TypeError:
                raise ValueError("Argument base_ring (= %s) is not compatible "%base_ring + \
                    "with the base ring (= %s)."%self.base_ring())
            try:
                return M.span_of_basis(basis)
            except TypeError:
                raise ValueError("Argument gens (= %s) is not compatible "%basis + \
                    "with base_ring (= %s)."%base_ring)

    def submodule_with_basis(self, basis, check=True, already_echelonized=False):
        """
        Create the R-submodule of the ambient vector space with given
        basis, where R is the base ring of self.

        INPUT:

        - ``basis`` -- a list of linearly independent vectors

        - ``check`` -- whether or not to verify that each gen is in
           the ambient vector space

        OUTPUT:

        - ``FreeModule`` -- the `R`-submodule with given basis

        EXAMPLES:

        First we create a submodule of `\\ZZ^3`::

            sage: M = FreeModule(ZZ, 3)
            sage: B = M.basis()
            sage: N = M.submodule_with_basis([B[0]+B[1], 2*B[1]-B[2]])
            sage: N
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [ 1  1  0]
            [ 0  2 -1]

        A list of vectors in the ambient vector space may fail to generate
        a submodule.

        ::

            sage: V = M.ambient_vector_space()
            sage: X = M.submodule_with_basis([ V(B[0]+B[1])/2, V(B[1]-B[2])/2])
            Traceback (most recent call last):
            ...
            ArithmeticError: The given basis does not generate a submodule of self.

        However, we can still determine the R-span of vectors in the
        ambient space, or over-ride the submodule check by setting check to
        False.

        ::

            sage: X = V.span([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], ZZ)
            sage: X
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1/2    0   1/2]
            [   0  1/2  -1/2]
            sage: Y = M.submodule([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], check=False)
            sage: X == Y
            True

        Next we try to create a submodule of a free module over the
        principal ideal domain `\QQ[x]`, using our general Hermite normal form implementation::

            sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
            sage: M = FreeModule(R, 3)
            sage: B = M.basis()
            sage: W = M.submodule_with_basis([x*B[0], 2*B[0]- x*B[2]]); W
            Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field
            User basis matrix:
            [ x  0  0]
            [ 2  0 -x]
        """
        V = self.span_of_basis(basis=basis, check=check, already_echelonized=already_echelonized)
        if check:
            if not V.is_submodule(self):
                raise ArithmeticError("The given basis does not generate a submodule of self.")
        return V

    def vector_space_span(self, gens, check=True):
        r"""
        Create the vector subspace of the ambient vector space with given
        generators.

        INPUT:


        -  ``gens`` - a list of vector in self

        -  ``check`` - whether or not to verify that each gen
           is in the ambient vector space


        OUTPUT: a vector subspace

        EXAMPLES:

        We create a `2`-dimensional subspace of `\QQ^3`.

        ::

            sage: V = VectorSpace(QQ, 3)
            sage: B = V.basis()
            sage: W = V.vector_space_span([B[0]+B[1], 2*B[1]-B[2]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [   1    0  1/2]
            [   0    1 -1/2]

        We create a subspace of a vector space over
        `\QQ(i)`.

        ::

            sage: R.<x> = QQ[]
            sage: K = NumberField(x^2 + 1, 'a'); a = K.gen()
            sage: V = VectorSpace(K, 3)
            sage: V.vector_space_span([2*V.gen(0) + 3*V.gen(2)])
            Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 1
            Basis matrix:
            [  1   0 3/2]

        We use the ``vector_space_span`` command to create a
        vector subspace of the ambient vector space of a submodule of
        `\ZZ^3`.

        ::

            sage: M = FreeModule(ZZ,3)
            sage: W = M.submodule([M([1,2,3])])
            sage: W.vector_space_span([M([2,3,4])])
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [  1 3/2   2]
        """
        if is_FreeModule(gens):
            gens = gens.gens()
        return FreeModule_submodule_field(self.ambient_vector_space(), gens, check=check)

    def vector_space_span_of_basis(self, basis, check=True):
        """
        Create the vector subspace of the ambient vector space with given
        basis.

        INPUT:

        - ``basis`` -- a list of linearly independent vectors

        - ``check`` -- whether or not to verify that each gen is in
           the ambient vector space

        OUTPUT: a vector subspace with user-specified basis

        EXAMPLES::

            sage: V = VectorSpace(QQ, 3)
            sage: B = V.basis()
            sage: W = V.vector_space_span_of_basis([B[0]+B[1], 2*B[1]-B[2]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [ 1  1  0]
            [ 0  2 -1]
        """
        return FreeModule_submodule_with_basis_field(self.ambient_vector_space(), basis, check=check)

    def quotient(self, sub, check=True):
        """
        Return the quotient of self by the given submodule sub.

        INPUT:

        -  ``sub`` - a submodule of self, or something that can
           be turned into one via self.submodule(sub).

        -  ``check`` - (default: True) whether or not to check
           that sub is a submodule.


        EXAMPLES::

            sage: A = ZZ^3; V = A.span([[1,2,3], [4,5,6]])
            sage: Q = V.quotient( [V.0 + V.1] ); Q
            Finitely generated module V/W over Integer Ring with invariants (0)
        """
        # Calling is_subspace may be way too slow and repeat work done below.
        # It will be very desirable to somehow do this step better.
        if check and (not is_FreeModule(sub) or not sub.is_submodule(self)):
            try:
                sub = self.submodule(sub)
            except (TypeError, ArithmeticError):
                raise ArithmeticError("sub must be a subspace of self")
        if self.base_ring() == sage.rings.integer_ring.ZZ:
            from .fg_pid.fgp_module import FGP_Module
            return FGP_Module(self, sub, check=False)
        else:
            raise NotImplementedError("quotients of modules over rings other than fields or ZZ is not fully implemented")

    def __truediv__(self, sub):
        """
        Return the quotient of self by the given submodule sub.

        EXAMPLES::

            sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]])
            sage: V1/W1
            Finitely generated module V/W over Integer Ring with invariants (2)
            sage: V2 = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W2 = V2.span([2*V2.0+4*V2.1, 9*V2.0+12*V2.1, 4*V2.2])
            sage: V2/W2
            Finitely generated module V/W over Integer Ring with invariants (4, 12)
        """
        return self.quotient(sub, check=True)

class FreeModule_generic_field(FreeModule_generic_pid):
    """
    Base class for all free modules over fields.
    """
    def __init__(self, base_field, dimension, degree, sparse=False):
        """
        Creates a vector space over a field.

        EXAMPLES::

            sage: FreeModule(QQ, 2)
            Vector space of dimension 2 over Rational Field
            sage: FreeModule(FiniteField(2), 7)
            Vector space of dimension 7 over Finite Field of size 2

        We test that objects of this type are initialised correctly;
        see :trac:`11166` (the failing ``repr`` is fine because this
        is an abstract base class)::

            sage: from sage.modules.free_module import FreeModule_generic_field
            sage: FreeModule_generic_field(QQ, 5, 5)
            <repr(<sage.modules.free_module.FreeModule_generic_field_with_category at 0x...>) failed: NotImplementedError>
        """
        if not isinstance(base_field, ring.Field):
            raise TypeError("The base_field (=%s) must be a field"%base_field)
        FreeModule_generic_pid.__init__(self, base_field, dimension, degree, sparse=sparse)

    def _Hom_(self, Y, category):
        r"""
        Returns a homspace whose morphisms have this vector space as domain.

        This is called by the general methods such as
        :meth:`sage.structure.parent.Parent.Hom`.

        INPUT:

        - ``Y`` - a free module (or vector space) that will
          be the codomain of the morphisms in returned homspace
        - ``category`` - the category for the homspace

        OUTPUT:

        If ``Y`` is a free module over a field, in other words, a vector space,
        then this returns a space of homomorphisms between vector spaces,
        in other words a space of linear transformations.

        If ``Y`` is a free module that is not a vector space, then
        the returned space contains homomorphisms between free modules.

        EXAMPLES::

            sage: V = QQ^2
            sage: W = QQ^3
            sage: H = V._Hom_(W, category=None)
            sage: type(H)
            <class 'sage.modules.vector_space_homspace.VectorSpaceHomspace_with_category'>
            sage: H
            Set of Morphisms (Linear Transformations) from Vector space of dimension 2 over Rational Field to Vector space of dimension 3 over Rational Field

            sage: V = QQ^2
            sage: W = ZZ^3
            sage: H = V._Hom_(W, category=None)
            sage: type(H)
            <class 'sage.modules.free_module_homspace.FreeModuleHomspace_with_category'>
            sage: H
            Set of Morphisms from Vector space of dimension 2 over Rational Field
             to Ambient free module of rank 3 over the principal ideal domain Integer Ring
             in Category of finite dimensional vector spaces with basis over
              (quotient fields and metric spaces)
        """
        if Y.base_ring().is_field():
            from . import vector_space_homspace
            return vector_space_homspace.VectorSpaceHomspace(self, Y, category)
        from . import free_module_homspace
        return free_module_homspace.FreeModuleHomspace(self, Y, category)

    def scale(self, other):
        """
        Return the product of self by the number other, which is the module
        spanned by other times each basis vector. Since self is a vector
        space this product equals self if other is nonzero, and is the zero
        vector space if other is 0.

        EXAMPLES::

            sage: V = QQ^4
            sage: V.scale(5)
            Vector space of dimension 4 over Rational Field
            sage: V.scale(0)
            Vector space of degree 4 and dimension 0 over Rational Field
            Basis matrix:
            []

        ::

            sage: W = V.span([[1,1,1,1]])
            sage: W.scale(2)
            Vector space of degree 4 and dimension 1 over Rational Field
            Basis matrix:
            [1 1 1 1]
            sage: W.scale(0)
            Vector space of degree 4 and dimension 0 over Rational Field
            Basis matrix:
            []

        ::

            sage: V = QQ^4; V
            Vector space of dimension 4 over Rational Field
            sage: V.scale(3)
            Vector space of dimension 4 over Rational Field
            sage: V.scale(0)
            Vector space of degree 4 and dimension 0 over Rational Field
            Basis matrix:
            []
        """
        if other == 0:
            return self.zero_submodule()
        return self

    def __add__(self, other):
        """
        Return the sum of self and other.

        EXAMPLES::

            sage: V = VectorSpace(QQ,3)
            sage: V0 = V.span([V.gen(0)])
            sage: V2 = V.span([V.gen(2)])
            sage: V0 + V2
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 0 1]
            sage: QQ^3 + 0
            Vector space of dimension 3 over Rational Field
        """
        if not isinstance(other, FreeModule_generic_field):
            if other == 0:
                return self
            raise TypeError("other must be a Vector Space")
        V = self.ambient_vector_space()
        if V != other.ambient_vector_space():
            raise ArithmeticError("self and other must have the same ambient space")
        return V.span(self.basis() + other.basis())

    def echelonized_basis_matrix(self):
        """
        Return basis matrix for self in row echelon form.

        EXAMPLES::

            sage: V = FreeModule(QQ, 3).span_of_basis([[1,2,3],[4,5,6]])
            sage: V.basis_matrix()
            [1 2 3]
            [4 5 6]
            sage: V.echelonized_basis_matrix()
            [ 1  0 -1]
            [ 0  1  2]
        """
        return self.basis_matrix().echelon_form()

    def intersection(self, other):
        """
        Return the intersection of self and other, which must be
        R-submodules of a common ambient vector space.

        EXAMPLES::

            sage: V  = VectorSpace(QQ,3)
            sage: W1 = V.submodule([V.gen(0), V.gen(0) + V.gen(1)])
            sage: W2 = V.submodule([V.gen(1), V.gen(2)])
            sage: W1.intersection(W2)
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [0 1 0]
            sage: W2.intersection(W1)
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [0 1 0]
            sage: V.intersection(W1)
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 1 0]
            sage: W1.intersection(V)
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [1 0 0]
            [0 1 0]
            sage: Z = V.submodule([])
            sage: W1.intersection(Z)
            Vector space of degree 3 and dimension 0 over Rational Field
            Basis matrix:
            []
        """
        if not isinstance(other, FreeModule_generic):
            raise TypeError("other must be a free module")

        if self.ambient_vector_space() != other.ambient_vector_space():
            raise ArithmeticError("self and other must have the same ambient space.")

        if self.rank() == 0 or other.rank() == 0:
            if self.base_ring().is_field():
                return other.zero_submodule()
            else:
                return self.zero_submodule()

        if self.base_ring() != other.base_ring():
            # Now other is over a ring R whose fraction field K is the base field of V = self.
            # We compute the intersection using the following algorithm:
            # 1. By explicitly computing the nullspace of the matrix whose rows
            #    are a basis for self, we obtain the matrix over a linear map
            #         phi:  K^n ----> W
            #    with kernel equal to V = self.
            # 2. Compute the kernel over R of Phi restricted to other.  Do this
            #    by clearing denominators, computing the kernel of a matrix with
            #    entries in R, then restoring denominators to the answer.
            K = self.base_ring()
            R = other.base_ring()
            B = self.basis_matrix().transpose()
            W = B.kernel()
            phi = W.basis_matrix().transpose()

            # To restrict phi to other, we multiply the basis matrix for other
            # by phi, thus computing the image of each basis vector.
            X = other.basis_matrix()
            psi = X * phi

            # Now psi is a matrix that defines an R-module morphism from other to some
            # R-module, whose kernel defines the long sought for intersection of self and other.
            L = psi.integer_kernel()

            # Finally the kernel of the intersection has basis the linear combinations of
            # the basis of other given by a basis for L.
            G = L.basis_matrix() * other.basis_matrix()
            return other.span(G.rows())

        # dispense with the three easy cases
        if self == self.ambient_vector_space():
            return other
        elif other == other.ambient_vector_space():
            return self
        elif self.dimension() == 0 or other.dimension() == 0:
            return self.zero_submodule()

        # standard algorithm for computing intersection of general subspaces
        if self.dimension() <= other.dimension():
            V1 = self; V2 = other
        else:
            V1 = other; V2 = self
        A1 = V1.basis_matrix()
        A2 = V2.basis_matrix()
        S  = A1.stack(A2)
        K  = S.kernel()
        n = int(V1.dimension())
        B = [A1.linear_combination_of_rows(v.list()[:n]) for v in K.basis()]
        return self.ambient_vector_space().submodule(B, check=False)

    def is_subspace(self, other):
        """
        True if this vector space is a subspace of other.

        EXAMPLES::

            sage: V = VectorSpace(QQ,3)
            sage: W = V.subspace([V.gen(0), V.gen(0) + V.gen(1)])
            sage: W2 = V.subspace([V.gen(1)])
            sage: W.is_subspace(V)
            True
            sage: W2.is_subspace(V)
            True
            sage: W.is_subspace(W2)
            False
            sage: W2.is_subspace(W)
            True
        """
        return self.is_submodule(other)

    def span(self, gens, base_ring=None, check=True, already_echelonized=False):
        """
        Return the K-span of the given list of gens, where K is the
        base field of self or the user-specified base_ring.  Note that
        this span is a subspace of the ambient vector space, but need
        not be a subspace of self.

        INPUT:


        -  ``gens`` - list of vectors

        -  ``check`` - bool (default: True): whether or not to
           coerce entries of gens into base field

        -  ``already_echelonized`` - bool (default: False):
           set this if you know the gens are already in echelon form


        EXAMPLES::

            sage: V = VectorSpace(GF(7), 3)
            sage: W = V.subspace([[2,3,4]]); W
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 5 2]
            sage: W.span([[1,1,1]])
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 1 1]

        TESTS::

            sage: V = FreeModule(RDF,3)
            sage: W = V.submodule([V.gen(0)])
            sage: W.span([V.gen(1)], base_ring=GF(7))
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [0 1 0]
            sage: v = V((1, pi, log(2))); v
            (1.0, 3.141592653589793, 0.6931471805599453)
            sage: W.span([v], base_ring=GF(7))
            Traceback (most recent call last):
            ...
            ValueError: Argument gens (= [(1.0, 3.141592653589793, 0.6931471805599453)]) is not compatible with base_ring (= Finite Field of size 7).
            sage: W = V.submodule([v])
            sage: W.span([V.gen(2)], base_ring=GF(7))
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [0 0 1]
        """
        if is_FreeModule(gens):
            gens = gens.gens()
        if base_ring is None or base_ring is self.base_ring():
            return FreeModule_submodule_field(
                self.ambient_module(), gens=gens, check=check, already_echelonized=already_echelonized)
        else:
            try:
                M = self.ambient_module().change_ring(base_ring)
            except TypeError:
                raise ValueError("Argument base_ring (= %s) is not compatible with the base field (= %s)." % (base_ring, self.base_field() ))
            try:
                return M.span(gens)
            except TypeError:
                raise ValueError("Argument gens (= %s) is not compatible with base_ring (= %s)." % (gens, base_ring))

    def span_of_basis(self, basis, base_ring=None, check=True, already_echelonized=False):
        r"""
        Return the free K-module with the given basis, where K is the base
        field of self or user specified base_ring.

        Note that this span is a subspace of the ambient vector space, but
        need not be a subspace of self.

        INPUT:


        -  ``basis`` - list of vectors

        -  ``check`` - bool (default: True): whether or not to
           coerce entries of gens into base field

        -  ``already_echelonized`` - bool (default: False):
           set this if you know the gens are already in echelon form


        EXAMPLES::

            sage: V = VectorSpace(GF(7), 3)
            sage: W = V.subspace([[2,3,4]]); W
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 5 2]
            sage: W.span_of_basis([[2,2,2], [3,3,0]])
            Vector space of degree 3 and dimension 2 over Finite Field of size 7
            User basis matrix:
            [2 2 2]
            [3 3 0]

        The basis vectors must be linearly independent or a
        ``ValueError`` exception is raised::

            sage: W.span_of_basis([[2,2,2], [3,3,3]])
            Traceback (most recent call last):
            ...
            ValueError: The given basis vectors must be linearly independent.
        """
        if is_FreeModule(basis):
            basis = basis.gens()
        if base_ring is None:
            return FreeModule_submodule_with_basis_field(
                self.ambient_module(), basis=basis, check=check, already_echelonized=already_echelonized)
        else:
            try:
                M = self.change_ring(base_ring)
            except TypeError:
                raise ValueError("Argument base_ring (= %s) is not compatible with the base field (= %s)." % (
                    base_ring, self.base_field() ))
            try:
                return M.span_of_basis(basis)
            except TypeError:
                raise ValueError("Argument basis (= %s) is not compatible with base_ring (= %s)." % (basis, base_ring))

    def subspace(self, gens, check=True, already_echelonized=False):
        """
        Return the subspace of self spanned by the elements of gens.

        INPUT:

        -  ``gens`` - list of vectors

        -  ``check`` - bool (default: True) verify that gens
           are all in self.

        -  ``already_echelonized`` - bool (default: False) set
           to True if you know the gens are in Echelon form.

        EXAMPLES:

        First we create a 1-dimensional vector subspace of an
        ambient `3`-dimensional space over the finite field of
        order `7`::

            sage: V = VectorSpace(GF(7), 3)
            sage: W = V.subspace([[2,3,4]]); W
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 5 2]

        Next we create an invalid subspace, but it's allowed since
        ``check=False``. This is just equivalent to computing
        the span of the element::

            sage: W.subspace([[1,1,0]], check=False)
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 1 0]

        With ``check=True`` (the default) the mistake is correctly
        detected and reported with an ``ArithmeticError`` exception::

            sage: W.subspace([[1,1,0]], check=True)
            Traceback (most recent call last):
            ...
            ArithmeticError: Argument gens (= [[1, 1, 0]]) does not generate a submodule of self.
        """
        return self.submodule(gens, check=check, already_echelonized=already_echelonized)

    def subspaces(self, dim):
        """
        Iterate over all subspaces of dimension dim.

        INPUT:

        - ``dim`` - int, dimension of subspaces to be generated

        EXAMPLE::

            sage: V = VectorSpace(GF(3), 5)
            sage: len(list(V.subspaces(0)))
            1
            sage: len(list(V.subspaces(1)))
            121
            sage: len(list(V.subspaces(2)))
            1210
            sage: len(list(V.subspaces(3)))
            1210
            sage: len(list(V.subspaces(4)))
            121
            sage: len(list(V.subspaces(5)))
            1

        ::

            sage: V = VectorSpace(GF(3), 5)
            sage: V = V.subspace([V([1,1,0,0,0]),V([0,0,1,1,0])])
            sage: list(V.subspaces(1))
            [Vector space of degree 5 and dimension 1 over Finite Field of size 3
            Basis matrix:
            [1 1 0 0 0],
             Vector space of degree 5 and dimension 1 over Finite Field of size 3
            Basis matrix:
            [1 1 1 1 0],
             Vector space of degree 5 and dimension 1 over Finite Field of size 3
            Basis matrix:
            [1 1 2 2 0],
             Vector space of degree 5 and dimension 1 over Finite Field of size 3
            Basis matrix:
            [0 0 1 1 0]]
        """
        if not self.base_ring().is_finite():
            raise RuntimeError("Base ring must be finite.")
        b = self.basis_matrix()
        from sage.matrix.echelon_matrix import reduced_echelon_matrix_iterator
        for m in reduced_echelon_matrix_iterator(self.base_ring(), dim, self.dimension(), self.is_sparse(), copy=False):
            yield self.subspace((m*b).rows())

    def subspace_with_basis(self, gens, check=True, already_echelonized=False):
        """
        Same as ``self.submodule_with_basis(...)``.

        EXAMPLES:

        We create a subspace with a user-defined basis.

        ::

            sage: V = VectorSpace(GF(7), 3)
            sage: W = V.subspace_with_basis([[2,2,2], [1,2,3]]); W
            Vector space of degree 3 and dimension 2 over Finite Field of size 7
            User basis matrix:
            [2 2 2]
            [1 2 3]

        We then create a subspace of the subspace with user-defined basis.

        ::

            sage: W1 = W.subspace_with_basis([[3,4,5]]); W1
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            User basis matrix:
            [3 4 5]

        Notice how the basis for the same subspace is different if we
        merely use the ``subspace`` command.

        ::

            sage: W2 = W.subspace([[3,4,5]]); W2
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 6 4]

        Nonetheless the two subspaces are equal (as mathematical objects)::

            sage: W1 == W2
            True
        """
        return self.submodule_with_basis(gens, check=check, already_echelonized=already_echelonized)

    def complement(self):
        """
        Return the complement of ``self`` in the
        :meth:`~sage.modules.free_module.FreeModule_ambient_field.ambient_vector_space`.

        EXAMPLES::

            sage: V = QQ^3
            sage: V.complement()
            Vector space of degree 3 and dimension 0 over Rational Field
            Basis matrix:
            []
            sage: V == V.complement().complement()
            True
            sage: W = V.span([[1, 0, 1]])
            sage: X = W.complement(); X
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  0]
            sage: X.complement() == W
            True
            sage: X + W == V
            True

        Even though we construct a subspace of a subspace, the
        orthogonal complement is still done in the ambient vector
        space `\QQ^3`::

            sage: V = QQ^3
            sage: W = V.subspace_with_basis([[1,0,1],[-1,1,0]])
            sage: X = W.subspace_with_basis([[1,0,1]])
            sage: X.complement()
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  0]

        All these complements are only done with respect to the inner
        product in the usual basis.  Over finite fields, this means
        we can get complements which are only isomorphic to a vector
        space decomposition complement. ::

            sage: F2 = GF(2,x)
            sage: V = F2^6
            sage: W = V.span([[1,1,0,0,0,0]])
            sage: W
            Vector space of degree 6 and dimension 1 over Finite Field of size 2
            Basis matrix:
            [1 1 0 0 0 0]
            sage: W.complement()
            Vector space of degree 6 and dimension 5 over Finite Field of size 2
            Basis matrix:
            [1 1 0 0 0 0]
            [0 0 1 0 0 0]
            [0 0 0 1 0 0]
            [0 0 0 0 1 0]
            [0 0 0 0 0 1]
            sage: W.intersection(W.complement())
            Vector space of degree 6 and dimension 1 over Finite Field of size 2
            Basis matrix:
            [1 1 0 0 0 0]
        """
        # Check simple cases
        if self.dimension() == 0:
            return self.ambient_vector_space()
        if self.dimension() == self.ambient_vector_space().dimension():
            return self.submodule([])
        return self.basis_matrix().right_kernel()

    def vector_space(self, base_field=None):
        """
        Return the vector space associated to self. Since self is a vector
        space this function simply returns self, unless the base field is
        different.

        EXAMPLES::

            sage: V = span([[1,2,3]],QQ); V
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
            sage: V.vector_space()
            Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 2 3]
        """
        if base_field is None:
            return self
        return self.change_ring(base_field)

    def zero_submodule(self):
        """
        Return the zero submodule of self.

        EXAMPLES::

            sage: (QQ^4).zero_submodule()
            Vector space of degree 4 and dimension 0 over Rational Field
            Basis matrix:
            []
        """
        return self.zero_subspace()

    def zero_subspace(self):
        """
        Return the zero subspace of self.

        EXAMPLES::

            sage: (QQ^4).zero_subspace()
            Vector space of degree 4 and dimension 0 over Rational Field
            Basis matrix:
            []
        """
        return self.submodule([], check=False, already_echelonized=True)

    def linear_dependence(self, vectors, zeros='left', check=True):
        r"""
        Returns a list of vectors giving relations of linear dependence for the input list of vectors.
        Can be used to check linear independence of a set of vectors.

        INPUT:

        - ``vectors`` -- A list of vectors, all from the same vector
          space.

        - ``zeros`` -- default: ``'left'`` - ``'left'`` or ``'right'``
          as a general preference for where zeros are located in the
          returned coefficients

        - ``check`` -- default: ``True`` - if ``True`` each item in
          the list ``vectors`` is checked for membership in ``self``.
          Set to ``False`` if you can be certain the vectors come from
          the vector space.

        OUTPUT:

        Returns a list of vectors.  The scalar entries of each vector provide
        the coefficients for a linear combination of the input vectors that
        will equal the zero vector in ``self``.  Furthermore, the returned list
        is linearly independent in the vector space over the same base field
        with degree equal to the length of the list ``vectors``.

        The linear independence of ``vectors`` is equivalent to the returned
        list being empty, so this provides a test - see the examples below.

        The returned vectors are always independent, and with ``zeros`` set to
        ``'left'`` they have 1's in their first non-zero entries and a qualitative
        disposition to having zeros in the low-index entries.  With ``zeros`` set
        to ``'right'`` the situation is reversed with a qualitative disposition
        for zeros in the high-index entries.

        If the vectors in ``vectors`` are made the rows of a matrix `V` and
        the returned vectors are made the rows of a matrix `R`, then the
        matrix product `RV` is a zero matrix of the proper size.  And
        `R` is a matrix of full rank.  This routine uses kernels of
        matrices to compute these relations of linear dependence,
        but handles all the conversions between sets of vectors
        and matrices. If speed is important, consider working with
        the appropriate matrices and kernels instead.

        EXAMPLES:

        We begin with two linearly independent vectors, and add three
        non-trivial linear combinations to the set.  We illustrate
        both types of output and check a selected relation of linear
        dependence. ::

            sage: v1 = vector(QQ, [2, 1, -4, 3])
            sage: v2 = vector(QQ, [1, 5, 2, -2])
            sage: V = QQ^4
            sage: V.linear_dependence([v1,v2])
            [
            <BLANKLINE>
            ]

            sage: v3 = v1 + v2
            sage: v4 = 3*v1 - 4*v2
            sage: v5 = -v1 + 2*v2
            sage: L = [v1, v2, v3, v4, v5]

            sage: relations = V.linear_dependence(L, zeros='left')
            sage: relations
            [
            (1, 0, 0, -1, -2),
            (0, 1, 0, -1/2, -3/2),
            (0, 0, 1, -3/2, -7/2)
            ]
            sage: v2 + (-1/2)*v4 + (-3/2)*v5
            (0, 0, 0, 0)

            sage: relations = V.linear_dependence(L, zeros='right')
            sage: relations
            [
            (-1, -1, 1, 0, 0),
            (-3, 4, 0, 1, 0),
            (1, -2, 0, 0, 1)
            ]
            sage: z = sum([relations[2][i]*L[i] for i in range(len(L))])
            sage: z == zero_vector(QQ, 4)
            True

        A linearly independent set returns an empty list,
        a result that can be tested. ::

            sage: v1 = vector(QQ, [0,1,-3])
            sage: v2 = vector(QQ, [4,1,0])
            sage: V = QQ^3
            sage: relations = V.linear_dependence([v1, v2]); relations
            [
            <BLANKLINE>
            ]
            sage: relations == []
            True

        Exact results result from exact fields. We start with three
        linearly independent vectors and add in two linear combinations
        to make a linearly dependent set of five vectors. ::

            sage: F = FiniteField(17)
            sage: v1 = vector(F, [1, 2, 3, 4, 5])
            sage: v2 = vector(F, [2, 4, 8, 16, 15])
            sage: v3 = vector(F, [1, 0, 0, 0, 1])
            sage: (F^5).linear_dependence([v1, v2, v3]) == []
            True
            sage: L = [v1, v2, v3, 2*v1+v2, 3*v2+6*v3]
            sage: (F^5).linear_dependence(L)
            [
            (1, 0, 16, 8, 3),
            (0, 1, 2, 0, 11)
            ]
            sage: v1 + 16*v3 + 8*(2*v1+v2) + 3*(3*v2+6*v3)
            (0, 0, 0, 0, 0)
            sage: v2 + 2*v3 + 11*(3*v2+6*v3)
            (0, 0, 0, 0, 0)
            sage: (F^5).linear_dependence(L, zeros='right')
            [
            (15, 16, 0, 1, 0),
            (0, 14, 11, 0, 1)
            ]

        TESTS:

        With ``check=True`` (the default) a mismatch between vectors
        and the vector space is caught. ::

            sage: v1 = vector(RR, [1,2,3])
            sage: v2 = vector(RR, [1,2,3,4])
            sage: (RR^3).linear_dependence([v1,v2], check=True)
            Traceback (most recent call last):
            ...
            ValueError: vector (1.00000000000000, 2.00000000000000, 3.00000000000000, 4.00000000000000) is not an element of Vector space of dimension 3 over Real Field with 53 bits of precision

        The ``zeros`` keyword is checked. ::

            sage: (QQ^3).linear_dependence([vector(QQ,[1,2,3])], zeros='bogus')
            Traceback (most recent call last):
            ...
            ValueError: 'zeros' keyword must be 'left' or 'right', not 'bogus'

        An empty input set is linearly independent, vacuously. ::

            sage: (QQ^3).linear_dependence([]) == []
            True
        """
        if check:
            for v in vectors:
                if not v in self:
                    raise ValueError('vector %s is not an element of %s' % (v, self))
        if zeros == 'left':
            basis = 'echelon'
        elif zeros == 'right':
            basis = 'pivot'
        else:
            raise ValueError("'zeros' keyword must be 'left' or 'right', not '%s'" % zeros)
        import sage.matrix.constructor
        A = sage.matrix.constructor.matrix(vectors)  # as rows, so get left kernel
        return A.left_kernel(basis=basis).basis()

    def __truediv__(self, sub):
        """
        Return the quotient of self by the given subspace sub.

        EXAMPLES::

            sage: V = RDF^3; W = V.span([[1,0,-1], [1,-1,0]])
            sage: Q = V/W; Q
            Vector space quotient V/W of dimension 1 over Real Double Field where
            V: Vector space of dimension 3 over Real Double Field
            W: Vector space of degree 3 and dimension 2 over Real Double Field
            Basis matrix:
            [ 1.0  0.0 -1.0]
            [ 0.0  1.0 -1.0]
            sage: type(Q)
            <class 'sage.modules.quotient_module.FreeModule_ambient_field_quotient_with_category'>
            sage: V([1,2,3])
            (1.0, 2.0, 3.0)
            sage: Q == V.quotient(W)
            True
            sage: Q(W.0)
            (0.0)
        """
        return self.quotient(sub, check=True)

    def quotient(self, sub, check=True):
        """
        Return the quotient of self by the given subspace sub.

        INPUT:


        -  ``sub`` - a submodule of self, or something that can
           be turned into one via self.submodule(sub).

        -  ``check`` - (default: True) whether or not to check
           that sub is a submodule.


        EXAMPLES::

            sage: A = QQ^3; V = A.span([[1,2,3], [4,5,6]])
            sage: Q = V.quotient( [V.0 + V.1] ); Q
            Vector space quotient V/W of dimension 1 over Rational Field where
            V: Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
            W: Vector space of degree 3 and dimension 1 over Rational Field
            Basis matrix:
            [1 1 1]
            sage: Q(V.0 + V.1)
            (0)

        We illustrate that the base rings must be the same::

            sage: (QQ^2)/(ZZ^2)
            Traceback (most recent call last):
            ...
            ValueError: base rings must be the same
        """
        # Calling is_submodule may be way too slow and repeat work done below.
        # It will be very desirable to somehow do this step better.
        if is_FreeModule(sub) and self.base_ring() != sub.base_ring():
            raise ValueError("base rings must be the same")
        if check and (not is_FreeModule(sub) or not sub.is_subspace(self)):
            try:
                sub = self.subspace(sub)
            except (TypeError, ArithmeticError):
                raise ArithmeticError("sub must be a subspace of self")
        A, L = self.__quotient_matrices(sub)
        from . import quotient_module
        return quotient_module.FreeModule_ambient_field_quotient(self, sub, A, L)

    def __quotient_matrices(self, sub):
        r"""
        This internal function is used by
        ``self.quotient(...)``.

        EXAMPLES::

            sage: V = QQ^3; W = V.span([[1,0,-1], [1,-1,0]])
            sage: A, L = V._FreeModule_generic_field__quotient_matrices(W)
            sage: A
            [1]
            [1]
            [1]
            sage: L
            [1 0 0]

        The quotient and lift maps are used to compute in the quotient and
        to lift::

            sage: Q = V/W
            sage: Q(W.0)
            (0)
            sage: Q.lift_map()(Q.0)
            (1, 0, 0)
            sage: Q(Q.lift_map()(Q.0))
            (1)

        An example in characteristic 5::

            sage: A = GF(5)^2; B = A.span([[1,3]]); A / B
            Vector space quotient V/W of dimension 1 over Finite Field of size 5 where
            V: Vector space of dimension 2 over Finite Field of size 5
            W: Vector space of degree 2 and dimension 1 over Finite Field of size 5
            Basis matrix:
            [1 3]
        """
        # 2. Find a basis C for a another submodule of self, so that
        #    B + C is a basis for self.
        # 3. Then the quotient map is:
        #     x |---> 'write in terms of basis for C and take the last m = #C-#B components.
        # 4. And a section of this map is:
        #     x |---> corresponding linear combination of entries of last m entries
        #    of the basis C.

        # Step 1: Find bases for spaces
        B = sub.basis_matrix()
        S = self.basis_matrix()

        n = self.dimension()
        m = n - sub.dimension()

        # Step 2: Extend basis B to a basis for self.
        # We do this by simply finding the pivot rows of the matrix
        # whose rows are a basis for sub concatenated with a basis for
        # self.
        C = B.stack(S).transpose()
        A = C.matrix_from_columns(C.pivots()).transpose()

        # Step 3: Compute quotient map
        # The quotient map is given by writing in terms of the above basis,
        # then taking the last #C columns

        # Compute the matrix D "change of basis from S to A"
        # that writes each element of the basis
        # for self in terms of the basis of rows of A, i.e.,
        # want to find D such that
        #                D * A = S
        # where D is a square n x n matrix.
        # Our algorithm is to note that D is determined if we just
        # replace both A and S by the submatrix got from their pivot
        # columns.
        P  = A.pivots()
        AA = A.matrix_from_columns(P)
        SS = S.matrix_from_columns(P)
        D  = SS * AA**(-1)

        # Compute the image of each basis vector for self under the
        # map "write an element of self in terms of the basis A" then
        # take the last n-m components.
        Q = D.matrix_from_columns(range(n - m, n))

        # Step 4. Section map
        # The lifting or section map
        Dinv = D**(-1)
        L = Dinv.matrix_from_rows(range(n - m, n))

        return Q, L

    def quotient_abstract(self, sub, check=True):
        r"""
        Returns an ambient free module isomorphic to the quotient space of
        self modulo sub, together with maps from self to the quotient, and
        a lifting map in the other direction.

        Use ``self.quotient(sub)`` to obtain the quotient
        module as an object equipped with natural maps in both directions,
        and a canonical coercion.

        INPUT:


        -  ``sub`` - a submodule of self, or something that can
           be turned into one via self.submodule(sub).

        -  ``check`` - (default: True) whether or not to check
           that sub is a submodule.


        OUTPUT:


        -  ``U`` - the quotient as an abstract *ambient* free
           module

        -  ``pi`` - projection map to the quotient

        -  ``lift`` - lifting map back from quotient


        EXAMPLES::

            sage: V = GF(19)^3
            sage: W = V.span_of_basis([ [1,2,3], [1,0,1] ])
            sage: U,pi,lift = V.quotient_abstract(W)
            sage: pi(V.2)
            (18)
            sage: pi(V.0)
            (1)
            sage: pi(V.0 + V.2)
            (0)

        Another example involving a quotient of one subspace by another.

        ::

            sage: A = matrix(QQ,4,4,[0,1,0,0, 0,0,1,0, 0,0,0,1, 0,0,0,0])
            sage: V = (A^3).kernel()
            sage: W = A.kernel()
            sage: U, pi, lift = V.quotient_abstract(W)
            sage: [pi(v) == 0 for v in W.gens()]
            [True]
            sage: [pi(lift(b)) == b for b in U.basis()]
            [True, True]
        """
        # Calling is_subspace may be way too slow and repeat work done below.
        # It will be very desirable to somehow do this step better.
        if check and (not is_FreeModule(sub) or not sub.is_subspace(self)):
            try:
                sub = self.subspace(sub)
            except (TypeError, ArithmeticError):
                raise ArithmeticError("sub must be a subspace of self")

        A, L = self.__quotient_matrices(sub)
        quomap = self.hom(A)
        quo = quomap.codomain()
        liftmap = quo.Hom(self)(L)

        return quomap.codomain(), quomap, liftmap

###############################################################################
#
# Generic ambient free modules, i.e., of the form R^n for some commutative ring R.
#
###############################################################################

class FreeModule_ambient(FreeModule_generic):
    """
    Ambient free module over a commutative ring.
    """
    def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
        """
        The free module of given rank over the given base_ring.

        INPUT:

        -  ``base_ring`` -- a commutative ring

        -  ``rank`` -- a non-negative integer

        -  ``sparse`` -- bool (default: False)

        - ``coordinate_ring`` -- a ring containing ``base_ring``
          (default: equal to ``base_ring``)

        EXAMPLES::

            sage: FreeModule(ZZ, 4)
            Ambient free module of rank 4 over the principal ideal domain Integer Ring
        """
        FreeModule_generic.__init__(self, base_ring, rank=rank,
                degree=rank, sparse=sparse, coordinate_ring=coordinate_ring)

    def __hash__(self):
        """
        The hash is obtained from the rank and the base ring.

        .. TODO::

            Make pickling so that the hash is available early enough.

        EXAMPLES::

            sage: V = QQ^7
            sage: hash(V) == hash((V.rank(), V.base_ring()))
            True
        """
        try:
            return hash((self.rank(), self.base_ring()))
        except AttributeError:
            # This is a fallback because sometimes hash is called during object
            # reconstruction (unpickle), and the above fields haven't been
            # filled in yet.
            return 0

    def _coerce_map_from_(self, M):
        """
        Return a coercion map from `M` to ``self``, or None.

        TESTS:

        Make sure :trac:`10513` is fixed (no coercion from a quotient
        vector space to an isomorphic abstract vector space)::

            sage: M = QQ^3 / [[1,2,3]]
            sage: V = QQ^2
            sage: V.coerce_map_from(M)

        """
        if isinstance(M, FreeModule_ambient):
            from sage.modules.quotient_module import FreeModule_ambient_field_quotient
            if isinstance(M, FreeModule_ambient_field_quotient):
                # No forgetful map.
                return None
            elif (self.base_ring().has_coerce_map_from(M.base_ring())
                  and self.rank() == M.rank()):
                # We could return M.hom(self.basis(), self), but the
                # complexity of this is quadratic in space and time,
                # since it constructs a matrix.
                return True
        return super(FreeModule_ambient, self)._coerce_map_from_(M)

    def _dense_module(self):
        """
        Creates a dense module with the same defining data as self.

        N.B. This function is for internal use only! See dense_module for
        use.

        EXAMPLES::

            sage: M = FreeModule(Integers(8),3)
            sage: S = FreeModule(Integers(8),3, sparse=True)
            sage: M is S._dense_module()
            True
        """
        return FreeModule(base_ring=self.base_ring(), rank = self.rank(), sparse=False)

    def _sparse_module(self):
        """
        Creates a sparse module with the same defining data as self.

        N.B. This function is for internal use only! See sparse_module for
        use.

        EXAMPLES::

            sage: M = FreeModule(Integers(8),3)
            sage: S = FreeModule(Integers(8),3, sparse=True)
            sage: M._sparse_module() is S
            True
        """
        return FreeModule(base_ring=self.base_ring(), rank = self.rank(), sparse=True)

    def echelonized_basis_matrix(self):
        """
        The echelonized basis matrix of self.

        EXAMPLES::

            sage: V = ZZ^4
            sage: W = V.submodule([ V.gen(i)-V.gen(0) for i in range(1,4) ])
            sage: W.basis_matrix()
            [ 1  0  0 -1]
            [ 0  1  0 -1]
            [ 0  0  1 -1]
            sage: W.echelonized_basis_matrix()
            [ 1  0  0 -1]
            [ 0  1  0 -1]
            [ 0  0  1 -1]
            sage: U = V.submodule_with_basis([ V.gen(i)-V.gen(0) for i in range(1,4) ])
            sage: U.basis_matrix()
            [-1  1  0  0]
            [-1  0  1  0]
            [-1  0  0  1]
            sage: U.echelonized_basis_matrix()
            [ 1  0  0 -1]
            [ 0  1  0 -1]
            [ 0  0  1 -1]
        """
        return self.basis_matrix()

    def __cmp__(self, other):
        """
        Compare the free module self with other.

        Modules are ordered by their ambient spaces, then by dimension,
        then in order by their echelon matrices. However, if
        other is a sub-module or is a quotient module then its
        comparison method is used instead of generic comparison.

        EXAMPLES:

        We compare rank three free modules over the integers and
        rationals::

            sage: QQ^3 < CC^3
            True
            sage: CC^3 < QQ^3
            False
            sage: CC^3 > QQ^3
            True

        ::

            sage: Q = QQ; Z = ZZ
            sage: Q^3 > Z^3
            True
            sage: Q^3 < Z^3
            False
            sage: Z^3 < Q^3
            True
            sage: Z^3 > Q^3
            False
            sage: Q^3 == Z^3
            False
            sage: Q^3 == Q^3
            True

        Comparison with a sub-module::

            sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
            sage: V
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
            sage: A = QQ^3
            sage: V < A
            True
            sage: A < V
            False

        Comparison with a quotient module (see :trac:`10513`)::

            sage: M = QQ^3 / [[1,2,3]]
            sage: V = QQ^2
            sage: V == M
            False
            sage: M == V
            False

        We create the module `\\ZZ^3`, and the submodule generated by
        one vector `(1,1,0)`, and check whether certain elements are
        in the submodule::

            sage: R = FreeModule(ZZ, 3)
            sage: V = R.submodule([R.gen(0) + R.gen(1)])
            sage: R.gen(0) + R.gen(1) in V
            True
            sage: R.gen(0) + 2*R.gen(1) in V
            False

            sage: w = (1/2)*(R.gen(0) + R.gen(1))
            sage: w
            (1/2, 1/2, 0)
            sage: w.parent()
            Vector space of dimension 3 over Rational Field
            sage: w in V
            False
            sage: V.coordinates(w)
            [1/2]

        """
        if self is other:
            return 0
        if not isinstance(other, FreeModule_generic):
            return cmp(type(self), type(other))
        from sage.modules.quotient_module import FreeModule_ambient_field_quotient
        if isinstance(other, FreeModule_ambient) and not isinstance(other,FreeModule_ambient_field_quotient):
            c = cmp(self.rank(), other.rank())
            if c: return c
            c = cmp(self.base_ring(), other.base_ring())
            if not c: return c
            try:
                if self.base_ring().is_subring(other.base_ring()):
                    return -1
                elif other.base_ring().is_subring(self.base_ring()):
                    return 1
            except NotImplementedError:
                pass
            return c
        else:
            # now other is not ambient or is a quotient;
            # it knows how to do the comparison.
            return -other.__cmp__(self)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: R = ZZ.quo(12)
            sage: M = R^12
            sage: M
            Ambient free module of rank 12 over Ring of integers modulo 12
            sage: print(M._repr_())
            Ambient free module of rank 12 over Ring of integers modulo 12

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: M.rename('M')
            sage: M
            M
            sage: print(M._repr_())
            Ambient free module of rank 12 over Ring of integers modulo 12

        Sparse modules print this fact.

        ::

            sage: N = FreeModule(R,12,sparse=True)
            sage: N
            Ambient sparse free module of rank 12 over Ring of integers modulo 12

        (Now clean up again.)

        ::

            sage: M.reset_name()
            sage: M
            Ambient free module of rank 12 over Ring of integers modulo 12
        """
        if self.is_sparse():
            return "Ambient sparse free module of rank %s over %s"%(self.rank(), self.base_ring())
        else:
            return "Ambient free module of rank %s over %s"%(self.rank(), self.base_ring())

    def _latex_(self):
        r"""
        Return a latex representation of this ambient free module.

        EXAMPLES::

            sage: latex(QQ^3) # indirect doctest
            \Bold{Q}^{3}

        ::

            sage: A = GF(5)^20
            sage: latex(A) # indiret doctest
            \Bold{F}_{5}^{20}

        ::

            sage: A = PolynomialRing(QQ,3,'x') ^ 20
            sage: latex(A) #indirect doctest
            (\Bold{Q}[x_{0}, x_{1}, x_{2}])^{20}
        """
        t = "%s"%latex.latex(self.base_ring())
        if t.find(" ") != -1:
            t = "(%s)"%t
        return "%s^{%s}"%(t, self.rank())

    def is_ambient(self):
        """
        Return ``True`` since this module is an ambient
        module.

        EXAMPLES::

            sage: A = QQ^5; A.is_ambient()
            True
            sage: A = (QQ^5).span([[1,2,3,4,5]]); A.is_ambient()
            False
        """
        return True

    def ambient_module(self):
        """
        Return self, since self is ambient.

        EXAMPLES::

            sage: A = QQ^5; A.ambient_module()
            Vector space of dimension 5 over Rational Field
            sage: A = ZZ^5; A.ambient_module()
            Ambient free module of rank 5 over the principal ideal domain Integer Ring
        """
        return self

    def basis(self):
        """
        Return a basis for this ambient free module.

        OUTPUT:


        -  ``Sequence`` - an immutable sequence with universe
           this ambient free module


        EXAMPLES::

            sage: A = ZZ^3; B = A.basis(); B
            [
            (1, 0, 0),
            (0, 1, 0),
            (0, 0, 1)
            ]
            sage: B.universe()
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
        """
        try:
            return self.__basis
        except  AttributeError:
            ZERO = self(0)
            one = self.coordinate_ring().one()
            w = []
            for n in range(self.rank()):
                v = ZERO.__copy__()
                v.set(n, one)
                w.append(v)
            self.__basis = basis_seq(self, w)
            return self.__basis

    def echelonized_basis(self):
        """
        Return a basis for this ambient free module in echelon form.

        EXAMPLES::

            sage: A = ZZ^3; A.echelonized_basis()
            [
            (1, 0, 0),
            (0, 1, 0),
            (0, 0, 1)
            ]
        """
        return self.basis()

    def change_ring(self, R):
        """
        Return the ambient free module over R of the same rank as self.

        EXAMPLES::

            sage: A = ZZ^3; A.change_ring(QQ)
            Vector space of dimension 3 over Rational Field
            sage: A = ZZ^3; A.change_ring(GF(5))
            Vector space of dimension 3 over Finite Field of size 5

        For ambient modules any change of rings is defined.

        ::

            sage: A = GF(5)**3; A.change_ring(QQ)
            Vector space of dimension 3 over Rational Field
        """
        if self.base_ring() is R:
            return self
        from .free_quadratic_module import is_FreeQuadraticModule
        if is_FreeQuadraticModule(self):
            return FreeModule(R, self.rank(), inner_product_matrix=self.inner_product_matrix())
        else:
            return FreeModule(R, self.rank())


    def linear_combination_of_basis(self, v):
        """
        Return the linear combination of the basis for self obtained from
        the elements of the list v.

        INPUT:

        - ``v`` - list

        EXAMPLES::

            sage: V = span([[1,2,3], [4,5,6]], ZZ)
            sage: V
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 2 3]
            [0 3 6]
            sage: V.linear_combination_of_basis([1,1])
            (1, 5, 9)

        This should raise an error if the resulting element is not in self::

            sage: W = span([[2,4]], ZZ)
            sage: W.linear_combination_of_basis([1/2])
            Traceback (most recent call last):
            ...
            TypeError: element [1, 2] is not in free module
        """
        return self(v)

    def coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the standard basis for self and
        return the resulting coefficients in a vector over the fraction
        field of the base ring.

        Returns a vector `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = Integers(16)^3
            sage: v = V.coordinate_vector([1,5,9]); v
            (1, 5, 9)
            sage: v.parent()
            Ambient free module of rank 3 over Ring of integers modulo 16
        """
        return self(v)

    def echelon_coordinate_vector(self, v, check=True):
        r"""
        Same as ``self.coordinate_vector(v)``, since self is
        an ambient free module.

        INPUT:


        -  ``v`` - vector

        -  ``check`` - bool (default: True); if True, also
           verify that v is really in self.


        OUTPUT: list

        EXAMPLES::

            sage: V = QQ^4
            sage: v = V([-1/2,1/2,-1/2,1/2])
            sage: v
            (-1/2, 1/2, -1/2, 1/2)
            sage: V.coordinate_vector(v)
            (-1/2, 1/2, -1/2, 1/2)
            sage: V.echelon_coordinate_vector(v)
            (-1/2, 1/2, -1/2, 1/2)
            sage: W = V.submodule_with_basis([[1/2,1/2,1/2,1/2],[1,0,1,0]])
            sage: W.coordinate_vector(v)
            (1, -1)
            sage: W.echelon_coordinate_vector(v)
            (-1/2, 1/2)
        """
        return self.coordinate_vector(v, check=check)

    def echelon_coordinates(self, v, check=True):
        """
        Returns the coordinate vector of v in terms of the echelon basis
        for self.

        EXAMPLES::

            sage: U = VectorSpace(QQ,3)
            sage: [ U.coordinates(v) for v in U.basis() ]
            [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
            sage: [ U.echelon_coordinates(v) for v in U.basis() ]
            [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
            sage: V = U.submodule([[1,1,0],[0,1,1]])
            sage: V
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  1]
            sage: [ V.coordinates(v) for v in V.basis() ]
            [[1, 0], [0, 1]]
            sage: [ V.echelon_coordinates(v) for v in V.basis() ]
            [[1, 0], [0, 1]]
            sage: W = U.submodule_with_basis([[1,1,0],[0,1,1]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [1 1 0]
            [0 1 1]
            sage: [ W.coordinates(v) for v in W.basis() ]
            [[1, 0], [0, 1]]
            sage: [ W.echelon_coordinates(v) for v in W.basis() ]
            [[1, 1], [0, 1]]
        """
        return self.coordinates(v, check=check)

    def random_element(self, prob=1.0, *args, **kwds):
        """
        Returns a random element of self.

        INPUT:


        - ``prob`` - float. Each coefficient will be set to zero with
           probability `1-prob`. Otherwise coefficients will be chosen
           randomly from base ring (and may be zero).

        - ``*args, **kwds`` - passed on to random_element function of base
           ring.


        EXAMPLES::

            sage: M = FreeModule(ZZ, 3)
            sage: M.random_element()
            (-1, 2, 1)
            sage: M.random_element()
            (-95, -1, -2)
            sage: M.random_element()
            (-12, 0, 0)

        Passes extra positional or keyword arguments through::

            sage: M.random_element(5,10)
            (5, 5, 5)


        ::

            sage: M = FreeModule(ZZ, 16)
            sage: M.random_element()
            (-6, 5, 0, 0, -2, 0, 1, -4, -6, 1, -1, 1, 1, -1, 1, -1)
            sage: M.random_element(prob=0.3)
            (0, 0, 0, 0, -3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -3)
        """
        rand = current_randstate().python_random().random
        R = self.base_ring()
        v = self(0)
        prob = float(prob)
        for i in range(self.rank()):
            if rand() <= prob:
                v[i] = R.random_element(*args, **kwds)
        return v

    def gen(self, i=0):
        """
        Return the `i`-th generator for ``self``.

        Here `i` is between 0 and rank - 1, inclusive.

        INPUT:

        - `i` -- an integer (default 0)

        OUTPUT: `i`-th basis vector for ``self``.

        EXAMPLES::

            sage: n = 5
            sage: V = QQ^n
            sage: B = [V.gen(i) for i in range(n)]
            sage: B
            [(1, 0, 0, 0, 0),
            (0, 1, 0, 0, 0),
            (0, 0, 1, 0, 0),
            (0, 0, 0, 1, 0),
            (0, 0, 0, 0, 1)]
            sage: V.gens() == tuple(B)
            True

        TESTS::

            sage: (QQ^3).gen(4/3)
            Traceback (most recent call last):
            ...
            TypeError: rational is not an integer

        Check that :trac:`10262` and :trac:`13304` are fixed
        (coercions involving :class:`FreeModule_ambient` used to take
        quadratic time and space in the rank of the module)::

            sage: vector([0]*50000)/1
            (0, 0, 0, ..., 0)
        """
        if i < 0 or i >= self.rank():
            raise ValueError("Generator %s not defined." % i)
        try:
            return self.__basis[i]
        except AttributeError:
            v = self(0)
            v[i] = self.base_ring().one()
            v.set_immutable()
            return v


###############################################################################
#
# Ambient free modules over an integral domain.
#
###############################################################################

class FreeModule_ambient_domain(FreeModule_ambient):
    """
    Ambient free module over an integral domain.
    """
    def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
        """
        Create the ambient free module of given rank over the given
        integral domain.

        EXAMPLES::

            sage: FreeModule(PolynomialRing(GF(5),'x'), 3)
            Ambient free module of rank 3 over the principal ideal domain
            Univariate Polynomial Ring in x over Finite Field of size 5
        """
        FreeModule_ambient.__init__(self, base_ring,
                rank, sparse, coordinate_ring)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: R = PolynomialRing(ZZ,'x')
            sage: M = FreeModule(R,7)
            sage: M
            Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
            sage: print(M._repr_())
            Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: M.rename('M')
            sage: M
            M
            sage: print(M._repr_())
            Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring

        Sparse modules print this fact.

        ::

            sage: N = FreeModule(R,7,sparse=True)
            sage: N
            Ambient sparse free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring

        (Now clean up again.)

        ::

            sage: M.reset_name()
            sage: M
            Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
        """
        if self.is_sparse():
            return "Ambient sparse free module of rank %s over the integral domain %s"%(
                self.rank(), self.base_ring())
        else:
            return "Ambient free module of rank %s over the integral domain %s"%(
                self.rank(), self.base_ring())

    def ambient_vector_space(self):
        """
        Returns the ambient vector space, which is this free module
        tensored with its fraction field.

        EXAMPLES::

            sage: M = ZZ^3;
            sage: V = M.ambient_vector_space(); V
            Vector space of dimension 3 over Rational Field

        If an inner product on the module is specified, then this
        is preserved on the ambient vector space.

        ::

            sage: N = FreeModule(ZZ,4,inner_product_matrix=1)
            sage: U = N.ambient_vector_space()
            sage: U
            Ambient quadratic space of dimension 4 over Rational Field
            Inner product matrix:
            [1 0 0 0]
            [0 1 0 0]
            [0 0 1 0]
            [0 0 0 1]
            sage: P = N.submodule_with_basis([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]])
            sage: P.gram_matrix()
            [ 2 -1  0]
            [-1  2 -1]
            [ 0 -1  2]
            sage: U == N.ambient_vector_space()
            True
            sage: U == V
            False
        """
        try:
            return self.__ambient_vector_space
        except AttributeError:
            self.__ambient_vector_space = FreeModule(self.base_field(), self.rank(), sparse=self.is_sparse())
            return self.__ambient_vector_space

    def coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the standard basis for self and
        return the resulting coefficients in a vector over the fraction
        field of the base ring.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        Returns a vector `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = ZZ^3
            sage: v = V.coordinate_vector([1,5,9]); v
            (1, 5, 9)
            sage: v.parent()
            Vector space of dimension 3 over Rational Field
        """
        # Calling the element constructor directly, since the
        # usual call method indirectly relies on coordinate_vector,
        # hence, an infinite recursion would result
        try:
            out = self.ambient_vector_space()._element_constructor_(v)
        except TypeError:
            raise ArithmeticError("Error transforming the given vector into the ambient vector space")
        if check and out not in self:
            raise ArithmeticError("The given vector does not belong to this free module")
        return out

    def vector_space(self, base_field=None):
        """
        Returns the vector space obtained from self by tensoring with the
        fraction field of the base ring and extending to the field.

        EXAMPLES::

            sage: M = ZZ^3;  M.vector_space()
            Vector space of dimension 3 over Rational Field
        """
        if base_field is None:
            R = self.base_ring()
            return self.change_ring(R.fraction_field())
        else:
            return self.change_ring(base_field)

###############################################################################
#
# Ambient free modules over a principal ideal domain.
#
###############################################################################

class FreeModule_ambient_pid(FreeModule_generic_pid, FreeModule_ambient_domain):
    """
    Ambient free module over a principal ideal domain.
    """
    def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
        """
        Create the ambient free module of given rank over the given
        principal ideal domain.

        INPUT:

        - ``base_ring`` -- a principal ideal domain

        - ``rank`` -- a non-negative integer

        - ``sparse`` -- bool (default: False)

        - ``coordinate_ring`` -- a ring containing ``base_ring``
          (default: equal to ``base_ring``)

        EXAMPLES::

            sage: ZZ^3
            Ambient free module of rank 3 over the principal ideal domain Integer Ring

        We create the same module with coordinates in ``QQ``::

            sage: from sage.modules.free_module import FreeModule_ambient_pid
            sage: M = FreeModule_ambient_pid(ZZ, 3, coordinate_ring=QQ)
            sage: M
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
            sage: v = M.basis()[0]; v
            (1, 0, 0)
            sage: type(v)
            <type 'sage.modules.vector_rational_dense.Vector_rational_dense'>
        """
        FreeModule_ambient_domain.__init__(self, base_ring=base_ring,
                rank=rank, sparse=sparse, coordinate_ring=coordinate_ring)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: M = FreeModule(ZZ,7)
            sage: M
            Ambient free module of rank 7 over the principal ideal domain Integer Ring
            sage: print(M._repr_())
            Ambient free module of rank 7 over the principal ideal domain Integer Ring

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: M.rename('M')
            sage: M
            M
            sage: print(M._repr_())
            Ambient free module of rank 7 over the principal ideal domain Integer Ring

        Sparse modules print this fact.

        ::

            sage: N = FreeModule(ZZ,7,sparse=True)
            sage: N
            Ambient sparse free module of rank 7 over the principal ideal domain Integer Ring

        (Now clean up again.)

        ::

            sage: M.reset_name()
            sage: M
            Ambient free module of rank 7 over the principal ideal domain Integer Ring
        """
        if self.is_sparse():
            return "Ambient sparse free module of rank %s over the principal ideal domain %s"%(
                self.rank(), self.base_ring())
        else:
            return "Ambient free module of rank %s over the principal ideal domain %s"%(
                self.rank(), self.base_ring())


###############################################################################
#
# Ambient free modules over a field (i.e., a vector space).
#
###############################################################################

class FreeModule_ambient_field(FreeModule_generic_field, FreeModule_ambient_pid):
    """

    """
    def __init__(self, base_field, dimension, sparse=False):
        """
        Create the ambient vector space of given dimension over the given
        field.

        INPUT:


        -  ``base_field`` - a field

        -  ``dimension`` - a non-negative integer

        -  ``sparse`` - bool (default: False)


        EXAMPLES::

            sage: QQ^3
            Vector space of dimension 3 over Rational Field
        """
        FreeModule_ambient_pid.__init__(self, base_field, dimension, sparse=sparse)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: V = FreeModule(QQ,7)
            sage: V
            Vector space of dimension 7 over Rational Field
            sage: print(V._repr_())
            Vector space of dimension 7 over Rational Field

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: V.rename('V')
            sage: V
            V
            sage: print(V._repr_())
            Vector space of dimension 7 over Rational Field

        Sparse modules print this fact.

        ::

            sage: U = FreeModule(QQ,7,sparse=True)
            sage: U
            Sparse vector space of dimension 7 over Rational Field

        (Now clean up again.)

        ::

            sage: V.reset_name()
            sage: V
            Vector space of dimension 7 over Rational Field
        """
        if self.is_sparse():
            return "Sparse vector space of dimension %s over %s"%(self.dimension(), self.base_ring())
        else:
            return "Vector space of dimension %s over %s"%(self.dimension(), self.base_ring())

    def ambient_vector_space(self):
        """
        Returns self as the ambient vector space.

        EXAMPLES::

            sage: M = QQ^3
            sage: M.ambient_vector_space()
            Vector space of dimension 3 over Rational Field
        """
        return self

    def base_field(self):
        """
        Returns the base field of this vector space.

        EXAMPLES::

            sage: M = QQ^3
            sage: M.base_field()
            Rational Field
        """
        return self.base_ring()

    def _element_constructor_(self, e, *args, **kwds):
        """
        Create an element of this vector space.

        EXAMPLE::

            sage: k.<a> = GF(3^4)
            sage: VS = k.vector_space()
            sage: VS(a)
            (0, 1, 0, 0)
        """
        try:
            k = e.parent()
            if finite_field.is_FiniteField(k) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
                return self(e._vector_())
        except AttributeError:
            pass
        return FreeModule_generic_field._element_constructor_(self, e, *args, **kwds)

###############################################################################
#
# R-Submodule of K^n where K is the fraction field of a principal ideal domain $R$.
#
###############################################################################

class FreeModule_submodule_with_basis_pid(FreeModule_generic_pid):
    r"""
    Construct a submodule of a free module over PID with a distinguished basis.

    INPUT:

    - ``ambient`` -- ambient free module over a principal ideal domain `R`,
      i.e. `R^n`;

    - ``basis`` -- list of elements of `K^n`, where `K` is the fraction field
      of `R`. These elements must be linearly independent and will be used as
      the default basis of the constructed submodule;

    - ``check`` -- (default: ``True``) if ``False``, correctness of the input
      will not be checked and type conversion may be omitted, use with care;

    - ``echelonize`` -- (default:``False``) if ``True``, ``basis`` will be
      echelonized and the result will be used as the default basis of the
      constructed submodule;

    - `` echelonized_basis`` -- (default: ``None``) if not ``None``, must be
      the echelonized basis spanning the same submodule as ``basis``;

    - ``already_echelonized`` -- (default: ``False``) if ``True``, ``basis``
      must be already given in the echelonized form.

    OUTPUT:

    - `R`-submodule of `K^n` with the user-specified ``basis``.

    EXAMPLES::

        sage: M = ZZ^3
        sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W
        Free module of degree 3 and rank 2 over Integer Ring
        User basis matrix:
        [1 2 3]
        [4 5 6]

    Now we create a submodule of the ambient vector space, rather than
    ``M`` itself::

        sage: W = M.span_of_basis([[1,2,3/2],[4,5,6]]); W
        Free module of degree 3 and rank 2 over Integer Ring
        User basis matrix:
        [  1   2 3/2]
        [  4   5   6]
    """

    def __init__(self, ambient, basis, check=True,
        echelonize=False, echelonized_basis=None, already_echelonized=False):
        r"""
        See :class:`FreeModule_submodule_with_basis_pid` for documentation.

        TESTS::

            sage: M = ZZ^3
            sage: W = M.span_of_basis([[1,2,3],[4,5,6]])
            sage: TestSuite(W).run()

        We test that the issue at :trac:`9502` is solved::

            sage: parent(W.basis()[0])
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1 2 3]
            [4 5 6]
            sage: parent(W.echelonized_basis()[0])
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1 2 3]
            [4 5 6]

        Now we test that the issue introduced at :trac:`9502` and reported at
        :trac:`10250` is solved as well::

            sage: V = (QQ^2).span_of_basis([[1,1]])
            sage: w = sqrt(2) * V([1,1])
            sage: 3 * w
            (3*sqrt(2), 3*sqrt(2))
        """
        if not isinstance(ambient, FreeModule_ambient_pid):
            raise TypeError("ambient (=%s) must be ambient." % ambient)
        self.__ambient_module = ambient
        R = ambient.base_ring()
        R_coord = R

        # Convert all basis elements to the ambient module
        try:
            basis = [ambient(x) for x in basis]
        except TypeError:
            # That failed, try the ambient vector space instead
            V = ambient.ambient_vector_space()
            R_coord = V.base_ring()
            try:
                basis = [V(x) for x in basis]
            except TypeError:
                raise TypeError("each element of basis must be in "
                                "the ambient vector space")

        if echelonize and not already_echelonized:
            basis = self._echelonized_basis(ambient, basis)

        # The following is WRONG - we should call __init__ of
        # FreeModule_generic_pid. However, it leads to a bunch of errors.
        FreeModule_generic.__init__(self, base_ring=R, coordinate_ring=R_coord,
                                    rank=len(basis), degree=ambient.degree(),
                                    sparse=ambient.is_sparse())
        C = self.element_class
        w = [C(self, x.list(), coerce=False, copy=False) for x in basis]
        self.__basis = basis_seq(self, w)

        if echelonize or already_echelonized:
            self.__echelonized_basis = self.__basis
        else:
            if echelonized_basis is None:
                echelonized_basis = self._echelonized_basis(ambient, basis)
            w = [C(self, x.list(), coerce=False, copy=True)
                 for x in echelonized_basis]
            self.__echelonized_basis = basis_seq(self, w)
        if check and len(basis) != len(self.__echelonized_basis):
            raise ValueError("The given basis vectors must be linearly "
                             "independent.")

    def __hash__(self):
        """
        The hash is given by the basis.

        EXAMPLES::

            sage: M = ZZ^3
            sage: W = M.span_of_basis([[1,2,3],[4,5,6]])
            sage: hash(W) == hash(W.basis())
            True
        """
        return hash(self.__basis)

    def construction(self):
        """
        Returns the functorial construction of self, namely, the subspace
        of the ambient module spanned by the given basis.

        EXAMPLE::

            sage: M = ZZ^3
            sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1 2 3]
            [4 5 6]
            sage: c, V = W.construction()
            sage: c(V) == W
            True
        """
        from sage.categories.pushout import SubspaceFunctor
        return SubspaceFunctor(self.basis()), self.ambient_module()

    def echelonized_basis_matrix(self):
        """
        Return basis matrix for self in row echelon form.

        EXAMPLES::

            sage: V = FreeModule(ZZ, 3).span_of_basis([[1,2,3],[4,5,6]])
            sage: V.basis_matrix()
            [1 2 3]
            [4 5 6]
            sage: V.echelonized_basis_matrix()
            [1 2 3]
            [0 3 6]
        """
        try:
            return self.__echelonized_basis_matrix
        except AttributeError:
            pass
        self._echelonized_basis(self.ambient_module(), self.__basis)
        return self.__echelonized_basis_matrix

    def _echelonized_basis(self, ambient, basis):
        """
        Given the ambient space and a basis, construct and cache the
        echelonized basis matrix and returns its rows.

        N.B. This function is for internal use only!

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[0,2,1]])
            sage: N._echelonized_basis(M,N.basis())
            [(1, 1, 0), (0, 2, 1)]
            sage: V = QQ^3
            sage: W = V.submodule_with_basis([[1,1,0],[0,2,1]])
            sage: W._echelonized_basis(V,W.basis())
            [(1, 0, -1/2), (0, 1, 1/2)]
            sage: V = SR^3
            sage: W = V.submodule_with_basis([[1,0,1]])
            sage: W._echelonized_basis(V,W.basis())
            [(1, 0, 1)]
        """
        # Return the first rank rows (i.e., the nonzero rows).
        d = self._denominator(basis)
        MAT = sage.matrix.matrix_space.MatrixSpace(
            ambient.base_ring(), len(basis), ambient.degree(), sparse = ambient.is_sparse())
        if d != 1:
            basis = [x*d for x in basis]
        A = MAT(basis)
        E = A.echelon_form()
        if d != 1:
            E = E.matrix_over_field()*(~d)   # divide out denominator
        r = E.rank()
        if r < E.nrows():
            E = E.matrix_from_rows(range(r))
        self.__echelonized_basis_matrix = E
        return E.rows()

    def __cmp__(self, other):
        r"""
        Compare the free module self with other.

        Modules are ordered by their ambient spaces, then by dimension,
        then in order by their echelon matrices.

        .. note::

           Use :meth:`is_submodule` to determine if one
           module is a submodule of another.

        EXAMPLES:

        First we compare two equal vector spaces.

        ::

            sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
            sage: W = span([[5,6,7], [8,9,10]], QQ)
            sage: V == W
            True

        Next we compare a one dimensional space to the two dimensional
        space defined above.

        ::

            sage: M = span([[5,6,7]], QQ)
            sage: V == M
            False
            sage: M < V
            True
            sage: V < M
            False

        We compare a `\ZZ`-module to the one-dimensional
        space above.

        ::

            sage: V = span([[5,6,7]], ZZ).scale(1/11);  V
            Free module of degree 3 and rank 1 over Integer Ring
            Echelon basis matrix:
            [5/11 6/11 7/11]
            sage: V < M
            True
            sage: M < V
            False

        We test that :trac:`5525` is fixed::

            sage: A = (QQ^1).span([[1/3]],ZZ); B = (QQ^1).span([[1]],ZZ);
            sage: A.intersection(B)
            Free module of degree 1 and rank 1 over Integer Ring
            Echelon basis matrix:
            [1]

        """
        if self is other:
            return 0
        if not isinstance(other, FreeModule_generic):
            return cmp(type(self), type(other))
        c = cmp(self.ambient_vector_space(), other.ambient_vector_space())
        if c: return c
        c = cmp(self.dimension(), other.dimension())
        if c: return c
        c = cmp(self.base_ring(), other.base_ring())
        if c: return c
        # We use self.echelonized_basis_matrix() == other.echelonized_basis_matrix()
        # with the matrix to avoid a circular reference.
        return cmp(self.echelonized_basis_matrix(), other.echelonized_basis_matrix())

    def _denominator(self, B):
        """
        The LCM of the denominators of the given list B.

        N.B.: This function is for internal use only!

        EXAMPLES::

            sage: V = QQ^3
            sage: L = V.span([[1,1/2,1/3], [-1/5,2/3,3]],ZZ)
            sage: L
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [ 1/5 19/6 37/3]
            [   0 23/6 46/3]
            sage: L._denominator(L.echelonized_basis_matrix().list())
            30

        """
        if len(B) == 0:
            return 1
        d = B[0].denominator()
        from sage.arith.all import lcm
        for x in B[1:]:
            d = lcm(d,x.denominator())
        return d

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: L = ZZ^8
            sage: E = L.submodule_with_basis([ L.gen(i) - L.gen(0) for i in range(1,8) ])
            sage: E # indirect doctest
            Free module of degree 8 and rank 7 over Integer Ring
            User basis matrix:
            [-1  1  0  0  0  0  0  0]
            [-1  0  1  0  0  0  0  0]
            [-1  0  0  1  0  0  0  0]
            [-1  0  0  0  1  0  0  0]
            [-1  0  0  0  0  1  0  0]
            [-1  0  0  0  0  0  1  0]
            [-1  0  0  0  0  0  0  1]

        ::

            sage: M = FreeModule(ZZ,8,sparse=True)
            sage: N = M.submodule_with_basis([ M.gen(i) - M.gen(0) for i in range(1,8) ])
            sage: N # indirect doctest
            Sparse free module of degree 8 and rank 7 over Integer Ring
            User basis matrix:
            [-1  1  0  0  0  0  0  0]
            [-1  0  1  0  0  0  0  0]
            [-1  0  0  1  0  0  0  0]
            [-1  0  0  0  1  0  0  0]
            [-1  0  0  0  0  1  0  0]
            [-1  0  0  0  0  0  1  0]
            [-1  0  0  0  0  0  0  1]
        """
        if self.is_sparse():
            s = "Sparse free module of degree %s and rank %s over %s\n"%(
                self.degree(), self.rank(), self.base_ring()) + \
                "User basis matrix:\n%s"%self.basis_matrix()
        else:
            s = "Free module of degree %s and rank %s over %s\n"%(
                self.degree(), self.rank(), self.base_ring()) + \
                "User basis matrix:\n%s"%self.basis_matrix()
        return s

    def _latex_(self):
        r"""
        Return latex representation of this free module.

        EXAMPLES::

            sage: A = ZZ^3
            sage: M = A.span_of_basis([[1,2,3],[4,5,6]])
            sage: M._latex_()
            '\\mathrm{RowSpan}_{\\Bold{Z}}\\left(\\begin{array}{rrr}\n1 & 2 & 3 \\\\\n4 & 5 & 6\n\\end{array}\\right)'
        """
        return "\\mathrm{RowSpan}_{%s}%s"%(latex.latex(self.base_ring()), latex.latex(self.basis_matrix()))

    def ambient_module(self):
        """
        Return the ambient module related to the `R`-module self,
        which was used when creating this module, and is of the form
        `R^n`. Note that self need not be contained in the ambient
        module, though self will be contained in the ambient vector space.

        EXAMPLES::

            sage: A = ZZ^3
            sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]])
            sage: M
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [  1   2 3/7]
            [  4   5   6]
            sage: M.ambient_module()
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
            sage: M.is_submodule(M.ambient_module())
            False
        """
        return self.__ambient_module

    def echelon_coordinates(self, v, check=True):
        r"""
        Write `v` in terms of the echelonized basis for self.

        INPUT:


        -  ``v`` - vector

        -  ``check`` - bool (default: True); if True, also
           verify that v is really in self.


        OUTPUT: list

        Returns a list `c` such that if `B` is the basis
        for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: A = ZZ^3
            sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]])
            sage: M.coordinates([8,10,12])
            [0, 2]
            sage: M.echelon_coordinates([8,10,12])
            [8, -2]
            sage: B = M.echelonized_basis(); B
            [
            (1, 2, 3/7),
            (0, 3, -30/7)
            ]
            sage: 8*B[0] - 2*B[1]
            (8, 10, 12)

        We do an example with a sparse vector space::

            sage: V = VectorSpace(QQ,5, sparse=True)
            sage: W = V.subspace_with_basis([[0,1,2,0,0], [0,-1,0,0,-1/2]])
            sage: W.echelonized_basis()
            [
            (0, 1, 0, 0, 1/2),
            (0, 0, 1, 0, -1/4)
            ]
            sage: W.echelon_coordinates([0,0,2,0,-1/2])
            [0, 2]
        """
        if not isinstance(v, free_module_element.FreeModuleElement):
            v = self.ambient_vector_space()(v)
        elif v.degree() != self.degree():
            raise ArithmeticError("vector is not in free module")
        # Find coordinates of v with respect to rref basis.
        E = self.echelonized_basis_matrix()
        P = E.pivots()
        w = v.list_from_positions(P)
        # Next use the transformation matrix from the rref basis
        # to the echelon basis.
        T = self._rref_to_echelon_matrix()
        x = T.linear_combination_of_rows(w).list(copy=False)
        if not check:
            return x
        if v.parent() is self:
            return x
        lc = E.linear_combination_of_rows(x)
        if list(lc) != list(v):
            raise ArithmeticError("vector is not in free module")
        return x

    def user_to_echelon_matrix(self):
        """
        Return matrix that transforms a vector written with respect to the
        user basis of self to one written with respect to the echelon
        basis. The matrix acts from the right, as is usual in Sage.

        EXAMPLES::

            sage: A = ZZ^3
            sage: M = A.span_of_basis([[1,2,3],[4,5,6]])
            sage: M.echelonized_basis()
            [
            (1, 2, 3),
            (0, 3, 6)
            ]
            sage: M.user_to_echelon_matrix()
            [ 1  0]
            [ 4 -1]

        The vector `v=(5,7,9)` in `M` is `(1,1)`
        with respect to the user basis. Multiplying the above matrix on the
        right by this vector yields `(5,-1)`, which has components
        the coordinates of `v` with respect to the echelon basis.

        ::

            sage: v0,v1 = M.basis(); v = v0+v1
            sage: e0,e1 = M.echelonized_basis()
            sage: v
            (5, 7, 9)
            sage: 5*e0 + (-1)*e1
            (5, 7, 9)
        """
        try:
            return self.__user_to_echelon_matrix
        except AttributeError:
            if self.base_ring().is_field():
                self.__user_to_echelon_matrix = self._user_to_rref_matrix()
            else:
                rows = sum([self.echelon_coordinates(b,check=False) for b in self.basis()], [])
                M = sage.matrix.matrix_space.MatrixSpace(self.base_ring().fraction_field(),
                                             self.dimension(),
                                             sparse = self.is_sparse())
                self.__user_to_echelon_matrix = M(rows)
        return self.__user_to_echelon_matrix


    def echelon_to_user_matrix(self):
        """
        Return matrix that transforms the echelon basis to the user basis
        of self. This is a matrix `A` such that if `v` is a
        vector written with respect to the echelon basis for self then
        `vA` is that vector written with respect to the user basis
        of self.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
            sage: W.echelonized_basis()
            [
            (1, 0, -1),
            (0, 1, 2)
            ]
            sage: A = W.echelon_to_user_matrix(); A
            [-5/3  2/3]
            [ 4/3 -1/3]

        The vector `(1,1,1)` has coordinates `v=(1,1)` with
        respect to the echelonized basis for self. Multiplying `vA`
        we find the coordinates of this vector with respect to the user
        basis.

        ::

            sage: v = vector(QQ, [1,1]); v
            (1, 1)
            sage: v * A
            (-1/3, 1/3)
            sage: u0, u1 = W.basis()
            sage: (-u0 + u1)/3
            (1, 1, 1)
        """
        try:
            return self.__echelon_to_user_matrix
        except AttributeError:
            self.__echelon_to_user_matrix = ~self.user_to_echelon_matrix()
            return self.__echelon_to_user_matrix

    def _user_to_rref_matrix(self):
        """
        Returns a transformation matrix from the user specified basis to row
        reduced echelon form, for this module over a PID.

        Note: For internal use only! See user_to_echelon_matrix.

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[0,1,1]])
            sage: T = N.user_to_echelon_matrix(); T # indirect doctest
            [1 1]
            [0 1]
            sage: N.basis_matrix()
            [1 1 0]
            [0 1 1]
            sage: N.echelonized_basis_matrix()
            [ 1  0 -1]
            [ 0  1  1]
            sage: T * N.echelonized_basis_matrix() == N.basis_matrix()
            True
        """
        try:
            return self.__user_to_rref_matrix
        except AttributeError:
            A = self.basis_matrix()
            P = self.echelonized_basis_matrix().pivots()
            T = A.matrix_from_columns(P)
            self.__user_to_rref_matrix = T
        return self.__user_to_rref_matrix

    def _rref_to_user_matrix(self):
        """
        Returns a transformation matrix from row reduced echelon form to
        the user specified basis, for this module over a PID.

        Note: For internal use only! See user_to_echelon_matrix.

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[0,1,1]])
            sage: U = N.echelon_to_user_matrix(); U # indirect doctest
            [ 1 -1]
            [ 0  1]
            sage: N.echelonized_basis_matrix()
            [ 1  0 -1]
            [ 0  1  1]
            sage: N.basis_matrix()
            [1 1 0]
            [0 1 1]
            sage: U * N.basis_matrix() == N.echelonized_basis_matrix()
            True
        """
        try:
            return self.__rref_to_user_matrix
        except AttributeError:
            self.__rref_to_user_matrix = ~self._user_to_rref_matrix()
            return self.__rref_to_user_matrix

    def _echelon_to_rref_matrix(self):
        """
        Returns a transformation matrix from the some matrix to the row
        reduced echelon form for this module over a PID.

        Note: For internal use only! and not used!

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[1,1,2]])
            sage: N
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1 1 0]
            [1 1 2]
            sage: T = N._echelon_to_rref_matrix(); T
            [1 0]
            [0 2]
            sage: type(T)
            <type 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
            sage: U = N._rref_to_echelon_matrix(); U
            [  1   0]
            [  0 1/2]
            sage: type(U)
            <type 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
        """
        try:
            return self.__echelon_to_rref_matrix
        except AttributeError:
            A = self.echelonized_basis_matrix()
            T = A.matrix_from_columns(A.pivots())
            self.__echelon_to_rref_matrix = T
        return self.__echelon_to_rref_matrix

    def _rref_to_echelon_matrix(self):
        """
        Returns a transformation matrix from row reduced echelon form to
        some matrix for this module over a PID.

        Note: For internal use only!

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[1,1,2]])
            sage: N
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1 1 0]
            [1 1 2]
            sage: T = N._echelon_to_rref_matrix(); T
            [1 0]
            [0 2]
            sage: type(T)
            <type 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
            sage: U = N._rref_to_echelon_matrix(); U
            [  1   0]
            [  0 1/2]
            sage: type(U)
            <type 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
        """
        try:
            return self.__rref_to_echelon_matrix
        except AttributeError:
            self.__rref_to_echelon_matrix = ~self._echelon_to_rref_matrix()
            return self.__rref_to_echelon_matrix

    def vector_space(self, base_field=None):
        """
        Return the vector space associated to this free module via tensor
        product with the fraction field of the base ring.

        EXAMPLES::

            sage: A = ZZ^3; A
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
            sage: A.vector_space()
            Vector space of dimension 3 over Rational Field
            sage: M = A.span_of_basis([['1/3',2,'3/7'],[4,5,6]]); M
            Free module of degree 3 and rank 2 over Integer Ring
            User basis matrix:
            [1/3   2 3/7]
            [  4   5   6]
            sage: M.vector_space()
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [1/3   2 3/7]
            [  4   5   6]
        """
        if base_field is None:
            K = self.base_ring().fraction_field()
            V = self.ambient_vector_space()
            return V.submodule_with_basis(self.basis())
        return self.change_ring(base_field)

    def ambient_vector_space(self):
        """
        Return the ambient vector space in which this free module is
        embedded.

        EXAMPLES::

            sage: M = ZZ^3;  M.ambient_vector_space()
            Vector space of dimension 3 over Rational Field

        ::

            sage: N = M.span_of_basis([[1,2,'1/5']])
            sage: N
            Free module of degree 3 and rank 1 over Integer Ring
            User basis matrix:
            [  1   2 1/5]
            sage: M.ambient_vector_space()
            Vector space of dimension 3 over Rational Field
            sage: M.ambient_vector_space() is N.ambient_vector_space()
            True

        If an inner product on the module is specified, then this
        is preserved on the ambient vector space.

        ::

            sage: M = FreeModule(ZZ,4,inner_product_matrix=1)
            sage: V = M.ambient_vector_space()
            sage: V
            Ambient quadratic space of dimension 4 over Rational Field
            Inner product matrix:
            [1 0 0 0]
            [0 1 0 0]
            [0 0 1 0]
            [0 0 0 1]
            sage: N = M.submodule([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]])
            sage: N.gram_matrix()
            [2 1 1]
            [1 2 1]
            [1 1 2]
            sage: V == N.ambient_vector_space()
            True
        """
        return self.ambient_module().ambient_vector_space()

    def basis(self):
        """
        Return the user basis for this free module.

        EXAMPLES::

            sage: V = ZZ^3
            sage: V.basis()
            [
            (1, 0, 0),
            (0, 1, 0),
            (0, 0, 1)
            ]
            sage: M = V.span_of_basis([['1/8',2,1]])
            sage: M.basis()
            [
            (1/8, 2, 1)
            ]
        """
        return self.__basis

    def change_ring(self, R):
        """
        Return the free module over `R` obtained by coercing each
        element of the basis of ``self`` into a vector over the
        fraction field of `R`, then taking the resulting `R`-module.

        INPUT:

        -  ``R`` - a principal ideal domain

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.subspace([[2, 1/2, 1]])
            sage: W.change_ring(GF(7))
            Vector space of degree 3 and dimension 1 over Finite Field of size 7
            Basis matrix:
            [1 2 4]

        ::

            sage: M = (ZZ^2) * (1/2)
            sage: N = M.change_ring(QQ)
            sage: N
            Vector space of degree 2 and dimension 2 over Rational Field
            Basis matrix:
            [1 0]
            [0 1]
            sage: N = M.change_ring(QQ['x'])
            sage: N
            Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field
            Echelon basis matrix:
            [1/2   0]
            [  0 1/2]
            sage: N.coordinate_ring()
            Univariate Polynomial Ring in x over Rational Field

        The ring must be a principal ideal domain::

            sage: M.change_ring(ZZ['x'])
            Traceback (most recent call last):
            ...
            TypeError: the new ring Univariate Polynomial Ring in x over Integer Ring should be a principal ideal domain
        """
        if self.base_ring() is R:
            return self
        if R not in PrincipalIdealDomains():
            raise TypeError("the new ring %r should be a principal ideal domain" % R)

        K = R.fraction_field()
        V = VectorSpace(K, self.degree())
        B = [V(b) for b in self.basis()]
        M = self.ambient_module().change_ring(R)
        if self.has_user_basis():
            return M.span_of_basis(B)
        else:
            return M.span(B)

    def coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the user basis for self.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        Returns a vector `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = ZZ^3
            sage: M = V.span_of_basis([['1/8',2,1]])
            sage: M.coordinate_vector([1,16,8])
            (8)
        """
        # First find the coordinates of v wrt echelon basis.
        w = self.echelon_coordinate_vector(v, check=check)
        # Next use transformation matrix from echelon basis to
        # user basis.
        T = self.echelon_to_user_matrix()
        return T.linear_combination_of_rows(w)

    def echelonized_basis(self):
        """
        Return the basis for self in echelon form.

        EXAMPLES::

            sage: V = ZZ^3
            sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]])
            sage: M.basis()
            [
            (1/2, 3, 1),
            (0, 1/6, 0)
            ]
            sage: B = M.echelonized_basis(); B
            [
            (1/2, 0, 1),
            (0, 1/6, 0)
            ]
            sage: V.span(B) == M
            True
        """
        return self.__echelonized_basis

    def echelon_coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the echelonized basis for self.

        INPUT:


        -  ``v`` - vector

        -  ``check`` - bool (default: True); if True, also
           verify that v is really in self.

        Returns a list `c` such that if `B` is the echelonized basis
        for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = ZZ^3
            sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]])
            sage: B = M.echelonized_basis(); B
            [
            (1/2, 0, 1),
            (0, 1/6, 0)
            ]
            sage: M.echelon_coordinate_vector(['1/2', 3, 1])
            (1, 18)
        """
        return FreeModule(self.base_ring().fraction_field(), self.rank())(self.echelon_coordinates(v, check=check))

    def has_user_basis(self):
        """
        Return ``True`` if the basis of this free module is
        specified by the user, as opposed to being the default echelon
        form.

        EXAMPLES::

            sage: V = ZZ^3; V.has_user_basis()
            False
            sage: M = V.span_of_basis([[1,3,1]]); M.has_user_basis()
            True
            sage: M = V.span([[1,3,1]]); M.has_user_basis()
            False
        """
        return True

    def linear_combination_of_basis(self, v):
        """
        Return the linear combination of the basis for self obtained from
        the coordinates of v.

        INPUT:

        - ``v`` - list

        EXAMPLES::

            sage: V = span([[1,2,3], [4,5,6]], ZZ); V
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 2 3]
            [0 3 6]
            sage: V.linear_combination_of_basis([1,1])
            (1, 5, 9)

        This should raise an error if the resulting element is not in self::

            sage: W = (QQ**2).span([[2, 0], [0, 8]], ZZ)
            sage: W.linear_combination_of_basis([1, -1/2])
            Traceback (most recent call last):
            ...
            TypeError: element [2, -4] is not in free module
        """
        R = self.base_ring()
        check = (not R.is_field()) and any([a not in R for a in list(v)])
        return self(self.basis_matrix().linear_combination_of_rows(v),
                    check=check, copy=False, coerce=False)


class FreeModule_submodule_pid(FreeModule_submodule_with_basis_pid):
    """
    An `R`-submodule of `K^n` where `K` is the
    fraction field of a principal ideal domain `R`.

    EXAMPLES::

        sage: M = ZZ^3
        sage: W = M.span_of_basis([[1,2,3],[4,5,19]]); W
        Free module of degree 3 and rank 2 over Integer Ring
        User basis matrix:
        [ 1  2  3]
        [ 4  5 19]

    Generic tests, including saving and loading submodules and elements::

        sage: TestSuite(W).run()
        sage: v = W.0 + W.1
        sage: TestSuite(v).run()
    """
    def __init__(self, ambient, gens, check=True, already_echelonized=False):
        """
        Create an embedded free module over a PID.

        EXAMPLES::

            sage: V = ZZ^3
            sage: W = V.span([[1,2,3],[4,5,6]])
            sage: W
            Free module of degree 3 and rank 2 over Integer Ring
            Echelon basis matrix:
            [1 2 3]
            [0 3 6]
        """
        FreeModule_submodule_with_basis_pid.__init__(self, ambient, basis=gens,
            echelonize=True, already_echelonized=already_echelonized)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: M = ZZ^8
            sage: L = M.submodule([ M.gen(i) - M.gen(0) for i in range(1,8) ])
            sage: L # indirect doctest
            Free module of degree 8 and rank 7 over Integer Ring
            Echelon basis matrix:
            [ 1  0  0  0  0  0  0 -1]
            [ 0  1  0  0  0  0  0 -1]
            [ 0  0  1  0  0  0  0 -1]
            [ 0  0  0  1  0  0  0 -1]
            [ 0  0  0  0  1  0  0 -1]
            [ 0  0  0  0  0  1  0 -1]
            [ 0  0  0  0  0  0  1 -1]
        """
        if self.is_sparse():
            s = "Sparse free module of degree %s and rank %s over %s\n"%(
                self.degree(), self.rank(), self.base_ring()) + \
                "Echelon basis matrix:\n%s"%self.basis_matrix()
        else:
            s = "Free module of degree %s and rank %s over %s\n"%(
                self.degree(), self.rank(), self.base_ring()) + \
                "Echelon basis matrix:\n%s"%self.basis_matrix()
        return s

    def coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the user basis for self.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        Returns a list `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = ZZ^3
            sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
            sage: W.coordinate_vector([1,5,9])
            (5, -1)
        """
        return self.echelon_coordinate_vector(v, check=check)

    def has_user_basis(self):
        r"""
        Return ``True`` if the basis of this free module is
        specified by the user, as opposed to being the default echelon
        form.

        EXAMPLES::

            sage: A = ZZ^3; A
            Ambient free module of rank 3 over the principal ideal domain Integer Ring
            sage: A.has_user_basis()
            False
            sage: W = A.span_of_basis([[2,'1/2',1]])
            sage: W.has_user_basis()
            True
            sage: W = A.span([[2,'1/2',1]])
            sage: W.has_user_basis()
            False
        """
        return False


class FreeModule_submodule_with_basis_field(FreeModule_generic_field, FreeModule_submodule_with_basis_pid):
    """
    An embedded vector subspace with a distinguished user basis.

    EXAMPLES::

        sage: M = QQ^3; W = M.submodule_with_basis([[1,2,3], [4,5,19]]); W
        Vector space of degree 3 and dimension 2 over Rational Field
        User basis matrix:
        [ 1  2  3]
        [ 4  5 19]

    Since this is an embedded vector subspace with a distinguished user
    basis possibly different than the echelonized basis, the
    echelon_coordinates() and user coordinates() do not agree::

        sage: V = QQ^3

    ::

        sage: W = V.submodule_with_basis([[1,2,3], [4,5,6]])
        sage: W
        Vector space of degree 3 and dimension 2 over Rational Field
        User basis matrix:
        [1 2 3]
        [4 5 6]

    ::

        sage: v = V([1,5,9])
        sage: W.echelon_coordinates(v)
        [1, 5]
        sage: vector(QQ, W.echelon_coordinates(v)) * W.echelonized_basis_matrix()
        (1, 5, 9)

    ::

        sage: v = V([1,5,9])
        sage: W.coordinates(v)
        [5, -1]
        sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
        (1, 5, 9)

    Generic tests, including saving and loading submodules and elements::

        sage: TestSuite(W).run()

        sage: K.<x> = FractionField(PolynomialRing(QQ,'x'))
        sage: M = K^3; W = M.span_of_basis([[1,1,x]])
        sage: TestSuite(W).run()
    """
    def __init__(self, ambient, basis, check=True,
        echelonize=False, echelonized_basis=None, already_echelonized=False):
        """
        Create a vector space with given basis.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [1 2 3]
            [4 5 6]
        """
        FreeModule_submodule_with_basis_pid.__init__(
            self, ambient, basis=basis, check=check, echelonize=echelonize,
            echelonized_basis=echelonized_basis, already_echelonized=already_echelonized)

    def _repr_(self):
        """
        The printing representation of self.

        EXAMPLES::

            sage: V = VectorSpace(QQ,5)
            sage: U = V.submodule([ V.gen(i) - V.gen(0) for i in range(1,5) ])
            sage: U # indirect doctest
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]
            sage: print(U._repr_())
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: U.rename('U')
            sage: U
            U
            sage: print(U._repr_())
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        Sparse vector spaces print this fact.

        ::

            sage: VV = VectorSpace(QQ,5,sparse=True)
            sage: UU = VV.submodule([ VV.gen(i) - VV.gen(0) for i in range(1,5) ])
            sage: UU # indirect doctest
            Sparse vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        (Now clean up again.)

        ::

            sage: U.reset_name()
            sage: U
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]
        """
        if self.is_sparse():
            return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                    self.degree(), self.dimension(), self.base_field()) + \
                    "User basis matrix:\n%s"%self.basis_matrix()
        else:
            return "Vector space of degree %s and dimension %s over %s\n"%(
                    self.degree(), self.dimension(), self.base_field()) + \
                    "User basis matrix:\n%s"%self.basis_matrix()

    def _denominator(self, B):
        """
        Given a list (of field elements) returns 1 as the common
        denominator.

        N.B.: This function is for internal use only!

        EXAMPLES::

            sage: U = QQ^3
            sage: U
            Vector space of dimension 3 over Rational Field
            sage: U.denominator()
            1
            sage: V = U.span([[1,1/2,1/3], [-1/5,2/3,3]])
            sage: V
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [   1    0 -5/3]
            [   0    1    4]
            sage: W = U.submodule_with_basis([[1,1/2,1/3], [-1/5,2/3,3]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            User basis matrix:
            [   1  1/2  1/3]
            [-1/5  2/3    3]
            sage: W._denominator(W.echelonized_basis_matrix().list())
            1
        """
        return 1

    def _echelonized_basis(self, ambient, basis):
        """
        Given the ambient space and a basis, construct and cache the
        echelonized basis matrix and returns its rows.

        N.B. This function is for internal use only!

        EXAMPLES::

            sage: M = ZZ^3
            sage: N = M.submodule_with_basis([[1,1,0],[0,2,1]])
            sage: N._echelonized_basis(M,N.basis())
            [(1, 1, 0), (0, 2, 1)]
            sage: V = QQ^3
            sage: W = V.submodule_with_basis([[1,1,0],[0,2,1]])
            sage: W._echelonized_basis(V,W.basis())
            [(1, 0, -1/2), (0, 1, 1/2)]
        """
        MAT = sage.matrix.matrix_space.MatrixSpace(
            base_ring=ambient.base_ring(),
            nrows=len(basis), ncols=ambient.degree(),
            sparse=ambient.is_sparse())
        A = MAT(basis)
        E = A.echelon_form()
        # Return the first rank rows (i.e., the nonzero rows).
        return E.rows()[:E.rank()]

    def is_ambient(self):
        """
        Return False since this is not an ambient module.

        EXAMPLES::

            sage: V = QQ^3
            sage: V.is_ambient()
            True
            sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
            sage: W.is_ambient()
            False
        """
        return False


class FreeModule_submodule_field(FreeModule_submodule_with_basis_field):
    """
    An embedded vector subspace with echelonized basis.

    EXAMPLES:

    Since this is an embedded vector subspace with echelonized basis,
    the echelon_coordinates() and user coordinates() agree::

        sage: V = QQ^3
        sage: W = V.span([[1,2,3],[4,5,6]])
        sage: W
        Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [ 1  0 -1]
        [ 0  1  2]

    ::

        sage: v = V([1,5,9])
        sage: W.echelon_coordinates(v)
        [1, 5]
        sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix()
        (1, 5, 9)
        sage: v = V([1,5,9])
        sage: W.coordinates(v)
        [1, 5]
        sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
        (1, 5, 9)
    """
    def __init__(self, ambient, gens, check=True, already_echelonized=False):
        """
        Create an embedded vector subspace with echelonized basis.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.span([[1,2,3],[4,5,6]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
        """
        if is_FreeModule(gens):
            gens = gens.gens()
        FreeModule_submodule_with_basis_field.__init__(self, ambient, basis=gens, check=check,
            echelonize=not already_echelonized, already_echelonized=already_echelonized)

    def _repr_(self):
        """
        The default printing representation of self.

        EXAMPLES::

            sage: V = VectorSpace(QQ,5)
            sage: U = V.submodule([ V.gen(i) - V.gen(0) for i in range(1,5) ])
            sage: U # indirect doctest
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]
            sage: print(U._repr_())
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        The system representation can be overwritten, but leaves _repr_
        unmodified.

        ::

            sage: U.rename('U')
            sage: U
            U
            sage: print(U._repr_())
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        Sparse vector spaces print this fact.

        ::

            sage: VV = VectorSpace(QQ,5,sparse=True)
            sage: UU = VV.submodule([ VV.gen(i) - VV.gen(0) for i in range(1,5) ])
            sage: UU # indirect doctest
            Sparse vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]

        (Now clean up again.)

        ::

            sage: U.reset_name()
            sage: U
            Vector space of degree 5 and dimension 4 over Rational Field
            Basis matrix:
            [ 1  0  0  0 -1]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]
        """
        if self.is_sparse():
            return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                self.degree(), self.dimension(), self.base_field()) + \
                "Basis matrix:\n%s"%self.basis_matrix()
        else:
            return "Vector space of degree %s and dimension %s over %s\n"%(
                self.degree(), self.dimension(), self.base_field()) + \
                "Basis matrix:\n%s"%self.basis_matrix()

    def echelon_coordinates(self, v, check=True):
        """
        Write `v` in terms of the echelonized basis of self.

        INPUT:


        -  ``v`` - vector

        -  ``check`` - bool (default: True); if True, also
           verify that v is really in self.


        OUTPUT: list

        Returns a list `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.span([[1,2,3],[4,5,6]])
            sage: W
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]

        ::

            sage: v = V([1,5,9])
            sage: W.echelon_coordinates(v)
            [1, 5]
            sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix()
            (1, 5, 9)
        """
        if not isinstance(v, free_module_element.FreeModuleElement):
            v = self.ambient_vector_space()(v)
        if v.degree() != self.degree():
            raise ArithmeticError("v (=%s) is not in self"%v)
        E = self.echelonized_basis_matrix()
        P = E.pivots()
        if len(P) == 0:
            if check and v != 0:
                raise ArithmeticError("vector is not in free module")
            return []
        w = v.list_from_positions(P)
        if not check:
            # It's really really easy.
            return w
        if v.parent() is self:   # obvious that v is really in here.
            return w
        # the "linear_combination_of_rows" call dominates the runtime
        # of this function, in the check==False case when the parent
        # of v is not self.
        lc = E.linear_combination_of_rows(w)
        if lc.list() != v.list():
            raise ArithmeticError("vector is not in free module")
        return w

    def coordinate_vector(self, v, check=True):
        """
        Write `v` in terms of the user basis for self.

        INPUT:

        - ``v`` -- vector

        - ``check`` -- bool (default: True); if True, also verify that
           `v` is really in self.

        OUTPUT: list

        Returns a list `c` such that if `B` is the basis for self, then

        .. MATH::

            \\sum c_i B_i = v.

        If `v` is not in self, raise an ``ArithmeticError`` exception.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.span([[1,2,3],[4,5,6]]); W
            Vector space of degree 3 and dimension 2 over Rational Field
            Basis matrix:
            [ 1  0 -1]
            [ 0  1  2]
            sage: v = V([1,5,9])
            sage: W.coordinate_vector(v)
            (1, 5)
            sage: W.coordinates(v)
            [1, 5]
            sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
            (1, 5, 9)

        ::

            sage: V = VectorSpace(QQ,5, sparse=True)
            sage: W = V.subspace([[0,1,2,0,0], [0,-1,0,0,-1/2]])
            sage: W.coordinate_vector([0,0,2,0,-1/2])
            (0, 2)
        """
        return self.echelon_coordinate_vector(v, check=check)

    def has_user_basis(self):
        """
        Return ``True`` if the basis of this free module is
        specified by the user, as opposed to being the default echelon
        form.

        EXAMPLES::

            sage: V = QQ^3
            sage: W = V.subspace([[2,'1/2', 1]])
            sage: W.has_user_basis()
            False
            sage: W = V.subspace_with_basis([[2,'1/2',1]])
            sage: W.has_user_basis()
            True
        """
        return False

def basis_seq(V, vecs):
    """
    This converts a list vecs of vectors in V to an Sequence of
    immutable vectors.

    Should it? I.e. in most other parts of the system the return type
    of basis or generators is a tuple.

    EXAMPLES::

        sage: V = VectorSpace(QQ,2)
        sage: B = V.gens()
        sage: B
        ((1, 0), (0, 1))
        sage: v = B[0]
        sage: v[0] = 0 # immutable
        Traceback (most recent call last):
        ...
        ValueError: vector is immutable; please change a copy instead (use copy())
        sage: sage.modules.free_module.basis_seq(V, V.gens())
        [
        (1, 0),
        (0, 1)
        ]
    """
    for z in vecs:
        z.set_immutable()
    return Sequence(vecs, universe=V, check = False, immutable=True, cr=True)


class RealDoubleVectorSpace_class(FreeModule_ambient_field):
    def __init__(self,n):
        FreeModule_ambient_field.__init__(self,sage.rings.real_double.RDF,n)

    def coordinates(self,v):
        return v

class ComplexDoubleVectorSpace_class(FreeModule_ambient_field):
    def __init__(self,n):
        FreeModule_ambient_field.__init__(self,sage.rings.complex_double.CDF,n)

    def coordinates(self,v):
        return v



######################################################

def element_class(R, is_sparse):
    """
    The class of the vectors (elements of a free module) with base ring
    R and boolean is_sparse.

    EXAMPLES::

        sage: FF = FiniteField(2)
        sage: P = PolynomialRing(FF,'x')
        sage: sage.modules.free_module.element_class(QQ, is_sparse=True)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: sage.modules.free_module.element_class(QQ, is_sparse=False)
        <type 'sage.modules.vector_rational_dense.Vector_rational_dense'>
        sage: sage.modules.free_module.element_class(ZZ, is_sparse=True)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: sage.modules.free_module.element_class(ZZ, is_sparse=False)
        <type 'sage.modules.vector_integer_dense.Vector_integer_dense'>
        sage: sage.modules.free_module.element_class(FF, is_sparse=True)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: sage.modules.free_module.element_class(FF, is_sparse=False)
        <type 'sage.modules.vector_mod2_dense.Vector_mod2_dense'>
        sage: sage.modules.free_module.element_class(GF(7), is_sparse=False)
        <type 'sage.modules.vector_modn_dense.Vector_modn_dense'>
        sage: sage.modules.free_module.element_class(P, is_sparse=True)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: sage.modules.free_module.element_class(P, is_sparse=False)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_dense'>
    """
    import sage.modules.vector_real_double_dense
    import sage.modules.vector_complex_double_dense

    if sage.rings.integer_ring.is_IntegerRing(R) and not is_sparse:
        from .vector_integer_dense import Vector_integer_dense
        return Vector_integer_dense
    elif sage.rings.rational_field.is_RationalField(R) and not is_sparse:
        from .vector_rational_dense import Vector_rational_dense
        return Vector_rational_dense
    elif sage.rings.finite_rings.integer_mod_ring.is_IntegerModRing(R) and not is_sparse:
        from .vector_mod2_dense import Vector_mod2_dense
        if R.order() == 2:
            return Vector_mod2_dense
        from .vector_modn_dense import Vector_modn_dense, MAX_MODULUS
        if R.order() < MAX_MODULUS:
            return Vector_modn_dense
        else:
            return free_module_element.FreeModuleElement_generic_dense
    elif sage.rings.real_double.is_RealDoubleField(R) and not is_sparse:
        return sage.modules.vector_real_double_dense.Vector_real_double_dense
    elif sage.rings.complex_double.is_ComplexDoubleField(R) and not is_sparse:
        return sage.modules.vector_complex_double_dense.Vector_complex_double_dense
    elif sage.symbolic.ring.is_SymbolicExpressionRing(R) and not is_sparse:
        import sage.modules.vector_symbolic_dense
        return sage.modules.vector_symbolic_dense.Vector_symbolic_dense
    elif sage.symbolic.callable.is_CallableSymbolicExpressionRing(R) and not is_sparse:
        import sage.modules.vector_callable_symbolic_dense
        return sage.modules.vector_callable_symbolic_dense.Vector_callable_symbolic_dense
    else:
        if is_sparse:
            return free_module_element.FreeModuleElement_generic_sparse
        else:
            return free_module_element.FreeModuleElement_generic_dense
    raise NotImplementedError