Notes.md

1.1 Systems of Linear Equations

A linear equation in the variables x₁, ..., x_n is an equation that can be written in the form a₁x₁ + a₂x₂ + ... + a_nx_n = b where b and the coefficients a₁, ..., a_n are real or complex numbers. The subscript n may be any positive integer.

A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables - say, x₁, ..., x_n.

A solution of the system is a list (s₁, s₂, ..., s_n) of numbers that makes each equation a true statement when the values s₁, ..., s_n are substituted for x₁, ..., x_n, respectively.

The set of all possible solutions is called the solution set of the linear system. Two linear systems are called equivalent if they have the same solution set.

A system of linear equations has

1. no solution, or
2. exactly one solution, or
3. infinitely many solutions.

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system with the coefficients of each variable aligned in the columns, the matrix is called the coefficient matrix (or matrix of coefficients) of the system. An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

The size of a matrix tells how many rows and columns it has. An m x n matrix is a rectangular array of numbers with m rows and n columns.

Elementary Row Operations

1. (Replacement) Replace one row by the sum of itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Two Fundamental Questions about a Linear System

1. Is the system consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?

1.2 Row Reduction and Echelon Forms

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeros.

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.

Any nonzero matrix may be row reduced (that is, transformed by elementary row operations) into more than one matrix in echelon form, using different sequences of row operations.

Uniqueness of the Reduced Echelon Form
Each matrix is row equivalent to one and only one reduced echelon matrix.

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

A pivot is a nonzero number in a pivot position that is used as needed to create zeros via row operations.

The variables corresponding to the pivot columns in the matrix are called basic variables. The other variables are called free variables.

Existence and Uniqueness Theorem
A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column - that is, if and only if an echelon form of the augmented matrix has no row of the form [0 ... 0 b] with b nonzero. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

1.3 Vector Equations

A matrix with only one column is called a column vector, or simply a vector. Two vectors in R2 are equal if and only if their corresponding entries are equal.

Given two vectors u and v, their sum is the vector u+v obtained by adding corresponding entries of u and v.

Given a vector u and a real number c, the scalar multiple of u by c is the vector cu obtained by multiplying each entry in u by c.

If n is a positive integer, Rⁿ (read "r-n") denotes the collection of all lists of n real numbers, usually written as n x 1 column matrices.

The vector whose entries are all zero is called the zero vector and is denoted by 0.

Algebraic Properties of Rⁿ
For all u, v, w, in Rⁿ and all scalars c and d:
(i) u + v = v + u
(ii) (u + v) + w = u + (v + w)
(iii) u + 0 = 0 + u = u
(iv) u + (-u) = -u + u = 0 where -u denotes (-1)u
(v) c(u + v) = cu + cv
(vi) (c + d)u = cu + du
(vii) c(du) = (cd)u
(viii) 1u = u

Given vectors v₁, v₂, ..., v_p in Rⁿ and given scalars c₁, c₂, ..., c_p, the vector y defined by y = c₁v₁ + ... c_pv_p is called a linear combination of v₁, ..., v_p with weights c₁, ..., c_p.

A vector equation x₁a₁ + x₂a₂ + ... + x_na_n = b has the same solution set as the linear system whose augmented matrix is [a₁ a₂ ... a_n b]. In particular, b can be generated by a linear combination of a₁, ..., a_n if and only if there exists a solution to the linear system corresponding to the matrix.

If v₁, ..., v_p are in Rⁿ, then the set of all linear combinations of v₁, ..., v_p are denoted by span{v₁, ..., v_p} and is called the subset of Rⁿ spanned (or generated) by v₁, ..., v_p. That is, span{v₁, ..., v_p} is the collection of all vectors that can be written in the form c₁v₁ + c₂v₂ + ... + c_pv_p with c₁, ..., c_p scalars.

1.4 The Matrix Equation Ax = b

If A is an m x n matrix, with columns a₁, ..., a_n, and if x is in Rⁿ, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights.

If A is an m x n matrix, with columns a₁, ..., a_n, and if b is in R^m, the matrix equation Ax = b has the same solution set as the vector equation x₁a₁ + x₂a₂ + ... + x_na_n = b which, in turn, has the same solution set as the system of linear equations whose augmented matrix is [ a₁ a₂ ... a_n b ].

The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.

Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false.
a. For each b in R^m, the equation Ax = b has a solution.
b. Each b in R^m is a linear combination of the columns of A.
c. The columns of A span R^m.
d. A has a pivot position in every row.

If A is an m x n matrix, u and v are vectors in Rⁿ, and c is a scalar, then
a. A(u + v) = Au + Av.
b. A(cu) = c(Au).

1.5 Solution Sets of Linear Systems

A system of linear equations is said to be homogenous if it can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in R^m. Such a system Ax = 0 always has at least one solution, namely, x = 0 (the zero vector in Rⁿ). This zero solution is usually called the trivial solution. For a given equation Ax = 0, the important question is whether there exists a nontrivial solution, that is, a nonzero vector x that satisfies Ax = 0.

The homogenous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

Whenever a solution set is described explicitly with vectors, we say that the solution is in parametric vector form.

Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + v_h, where v_h is any solution of the homogenous equation Ax = 0.

1.7 Linear Independence

An indexed set of vectors {v₁, ..., v_p} in Rⁿ is said to be linearly independent if the vector equation x₁v₁ + x₂v₂ + ... + x_pv_p = 0 has only the trivial solution. The set {v₁, ..., v_p} is said to be linearly dependent if there exist weights c₁, ..., c_p, not all zero, such that c₁v₁ + c₂v₂ + ... + c_pv_p = 0.

The columns of a matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution.

A set of two vectors {v₁, v₂} is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other.

Characterization of Linearly Dependent Sets
An indexed set S = {v₁, ..., v_p} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v₁ is not equal to 0, then some v_j (with j > 1) is a linear combination of the preceding vectors, v₁, ..., v_j-1.

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v₁, ..., v_p} in Rⁿ is linearly dependent if p > n.

If a set S = {v₁, ..., v_p} in Rⁿ contains the zero vector, then the set is linearly dependent.

1.8 Introduction to Linear Transformations

A transformation(or function or mapping) T from Rⁿ to R^m is a rule that assigns to each vector x in Rⁿ a vector T(x) in R^m.

The set Rⁿ is called the domain of T, and R^m is called the codomain of T.

For x in Rⁿ, the vector T(x) in R^m is called the image of x(under the action of T).

The set of all images T(x) is called the range of T.

A transformation(or mapping) T is linear if:
(i) T(u + v) = T(u) + T(v) for all u, v in the domain of T.
(ii) T(cu) = cT(u) for all scalars c and all u in the domain of T.

If T is a linear transformation, then T(0) = 0 and T(cu + dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.

Repeated action of the above property produces a useful generalization:
T(c₁v₁ + ... + c_pv_p) = c₁T(v₁) + ... + c_pT(v_p)
_{Note: This is sometimes referred to as a superposition principle.}

1.9 The Matrix of a Linear Transformation

Let T : Rⁿ → R^m be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in Rⁿ.
In fact, A is the m x n matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rⁿ:
A = [T(e₁) ... T(e_n) ]. This matrix A is called the standard matrix for the linear transformation T.
_{Note: The term linear transformation focuses on a property of a mapping, while matrix transformation describes how such a mapping is implemented.}

A mapping T: Rⁿ → R^m is said to be onto R^m if each b in R^m is the image of at least one x in Rⁿ.

A mapping T: Rⁿ → R^m is said to be one-to-one R^m if each b in R^m is the image of at most one x in Rⁿ.

Let T: Rⁿ → R^m be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

Let T: Rⁿ → R^m be a linear transformation, and let A be the standard matrix for T. Then:
a. T maps Rⁿ onto R^m if and only if the columns of A span R^m.
b. T is one-to-one if and only if the columns of A are linearly independent.

2.1 Matrix Operations

If A is an m x n matrix - that is, a matrix with m rows and n columns - then the scalar entry in the ith row and the jth column of A is denoted by a_ij and is called the (i, j)-entry of A.

Each column of A is a list of m real numbers, which identifies a vector in R^m. Often, these columns are denoted by a₁, ..., a_n, and matrix A is written as A = [ a₁ a₂ ... a_n ]. Observe that the number a_ij is the ith entry (from the top) of the jth column vector a_j.

The diagonal entries in an m x n matrix A = [ a_ij ] are a₁₁, a₂₂, a₃₃, ..., and they form the main diagonal of A.

A diagonal matrix is a square n x n matrix whose nondiagonal entries are zero.

An m x n matrix whose entries are all zero is a zero matrix and is written as 0.

We say that two matrices are equal if they have the same size (i.e., the same number of rows and the same number of columns) and if their corresponding columns are equal, which amounts to saying that their corresponding entries are equal.

If A and B are m x n matrices, then the sum A + B is the m x n matrix whose columns are the sums of the corresponding columns in A and B. Since vector addition of the columns is done entrywise, each entry in A + B is the sum of the corresponding entries in A and B.
_{Note: The sum A + B is defined only when A and B are the same size.}

If r is a scalar and A is a matrix, then the scalar multiple rA is the matrix whose columns are r times the corresponding columns in A.

Let A, B, and C be matrices of the same size, and let r and s be scalars.
a. A + B = B + A.
b. (A + B) + C = A + (B + C).
c. A + 0 = A.
d. r(A + B) = rA + rB.
e. (r + s)A = rA + sA.
f. r(sA) = (rs)A.

If A is an m x n matrix, and if B is an n x p matrix with columns b₁, ..., b_p, then the product AB is the m x p matrix whose columns are Ab₁, ..., Ab_p. That is, AB = A[ b₁ b₂ ... b_p ] = [ Ab₁ Ab₂ ... Ab_p ].
_{Note: AB has the same number of rows as A and the same number of columns as B.}

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.

Row Column Rule for Computing AB
If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. If (AB)_ij denotes the (i, j)-entry in AB, and if A is an m x n matrix, then (AB)_ij = a_i1b_1j + a_i2b_2j + ... + a_inb_nj.

Let row_i(A) denote the ith row of a matrix A. Then row_i(AB) = row_i(A) · B.

Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are defined.
a. A(BC) = (AB)C. (associative law of multiplication)
b. A(B + C) = AB + AC. (left distributive law)
c. (B + C)A = BA + CA. (right distributive law)
d. r(AB) = (rA)B = A(rB). (for any scalar r)
e. I_mA = A = AI_n. (identity for matrix multiplication)

Warnings

1. In general, AB is not equal to BA.
2. The cancellation laws do not hold for matrix multiplication. That is, if AB = AC, then it is not true in general that B = C.
3. If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.

If A is an n x n matrix and if k is a positive integer, then A^k denotes the product of k copies of A.

If A is nonzero and if x is in Rⁿ, then A^kx is the result of left-multiplying x by A repeatedly k times. if k = 0, then A⁰x should be x itself. Thus A⁰ is interpreted as the identity matrix.

Given an m x n matrix A, the transpose of A is the n x m matrix, denoted by A^T, whose columns are formed from the corresponding rows of A.

Let A and B denote matrices whose sizes are appropriate for the following sums and products.
a. (A^T)^T = A.
b. (A + B)^T = A^T + B^T.
c. For any scalar r, (rA)^T = rA^T.
d. (AB)^T = B^TA^T.
_{Note: The transpose of a product of matrices equals the product of their transposes in the reverse order.}

2.2 The Inverse of Matrices

An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = I and AC = I where I = I_n, the n x n identity matrix. This unique inverse is denoted by A^-1, so that A^-1A = I and AA^-1 = I.

A matrix that is not invertible is sometimes called a singular matrix, and an invertible matrix is called a nonsingular matrix.

Let A be a 2 x 2 matrix with elements a, b, c, and d. The determinant of A, denoted by det A, is the quantity ad - bc. Matrix A is invertible if and only if det A is not equal to 0.

If A is an invertible n x n matrix, then for each b in Rⁿ, the equation Ax = b has the unique solution x = A^-1b.

If A is an invertible matrix, then A^-1 is invertible and (A^-1)^-1 = A.

If A and B are n x n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, (AB)^-1 = B^-1A^-1.

If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. That is, (A^T)^-1 = (A^-1)^T.

An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.

If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as EA, where the m x n matrix E is created by performing the same row operation on I_m.

Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.

An n x n matrix A is invertible if and only if A is row equivalent to I_n, and in this case, any sequence of elementary row operations that reduces A to I_n also transforms I_n into A^-1.

2.3 Characterizations of Invertible Matrices

Let A and B be square matrices. If AB = I, then A and B are both invertible, with B = A^-1 and A = B⁰¹.

Let T : Rⁿ -> Rⁿ be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x) = A^-1x is the unique function satisfying the equations S(T(x)) = x for all x in Rⁿ and T(S(x)) = x for all x in Rⁿ.

2.8 Subspaces of Rⁿ

A subspace of Rⁿ is any set H in Rⁿ that has three properties:
a. The zero vector is in H.
b. For each u and v in H, the sum u+v is in H.
c. For each u in H and scalar c, the vector cu is in H.

The column space of a matrix A is the set Col A of all linear combinations of the columns of A.

The null space of a matrix A is the set Nul A of all solutions of the homogeneous equation Ax = 0.

The null space of an m x n matrix A is a subspace of Rⁿ. Equivalently, the set of all solutions of a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rⁿ.

A basis for a subspace H of Rⁿ is a linearly independent set in H that spans H.

The pivot columns of a matrix A form a basis for the column space of A.

2.9 Dimension and Rank

Suppose the set B = {b₁, ..., b_p} is a basis for a subspace H. For each x in H, the coordinates of x relative to the basis B are weights c₁, ..., c_p such that x = c₁b₁ + ... + c_pb_p, and the vector in R^p [x]_B = [c₁, ..., c_p] is called the coordinate vector of x (relative to B) or the B-coordinate vector of x.

The dimension of a nonzero subspace H, denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero subspace {0} is defined to be zero.

The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.

The Rank Theorem
If a matrix A has n columns, then rank A + dim Nul A = n.

The Basis Theorem
Let H be a p-dimensional subspace of Rⁿ. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H that spans H is automatically a basis for H.

3.1 Introduction to Determinants

For n >= 2, the determinant of an n x n matrix A = [a_ij] is the sum of n terms of the form +-a_1j det A_1j, with plus and minus signs alternating, where the entries a₁₁, a₁₂, ..., a_1n are from the first rows of A. In symbols, det A = a₁₁ det A₁₁ - a₁₂ det A₁₂ + ... + (-1)⁽¹⁺ⁿ⁾ a_1n det A_1n = sum from j = 1 to n of (-1)^(1+j) a_1j det A_1j.

Given A = [a_ij], the (i, j) - cofactor of A is the number C_ij given by C_ij = (-1)^(i+j) det A_ij. The determinant of an n x n matrix A can be computed by a cofactor expansion across any row or down any column. The expansion across the ith row is det A = a_i1C_i1 + a_i2C_i2 + ... + a_inC_in. The expansion across the jth column is det A = a_1jC_1j + a_2jC_2j + ... + a_njC_nj.

If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.

3.2 Properties of Determinants

Row Operations
Let A be a square matrix.
a. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
b. If two rows of A are interchanged to produce B, then det B = -det A.
c. If one row of A is multiplied by k to produce B, then det B = k x det A.

A square matrix A is invertible if and only if det A is not equal to 0.

If A is an n x n matrix, then det A^T = det A.

If A and B are n x n matrices, then det(AB) = det(A) x det(B).

5.1 Eigenvectors and Eigenvalues

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = λx for some scalar λ. A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ.

The eigenvalues of a triangular matrix are the entries on its main diagonal.

If v₁, ..., v_r are eigenvectors that correspond to distinct eigenvalues λ₁, ..., λ_r of an n x n matrix A, then the set {v₁, ..., v_r} is linearly independent.

5.2 The Characteristic Equation

Let A be an n x n matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling), and let r be the number of such row interchanges. Then the determinant of A, written as det A, is (-1)^r times the product of the diagonal entries u₁₁, ..., u_nn. If A is invertible, then the determinant is nonzero because the diagonal entries are all pivots.

Properties of Determinants
Let A and B be n x n matrices.
a. A is invertible if and only if det A is not 0.
b. det AB = (det A)(det B).
c. det A^T = det A.
d. If A is triangular, then det A is the product of the entries on the main diagonal of A.
e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor.

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the characteristic equation det(A-λI) = 0.

If A is an n x n matrix, then det(A-λI) is a polynomial of degree n called the characteristic polynomial of A. In general, the multiplicity of an eigenvalue λ is its multiplicity as a root of the characteristic equation.

If A and B are n x n matrices, then A is similar to B if there is an invertible matrix P such that P^-1AP = B, or, equivalently, A = PBP^-1. Changing A into P^-1AP is called a similarity transformation.

If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues(with the same multiplicities).

5.3 Diagonalization

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, if A = PDP^-1 for some invertible matrix P and some diagonal matrix D.

The Diagonalization Theorem
An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A = PDP^-1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A the correspond to, respectively, to the eigenvectors in P. In otherwords, A is diagonalizable if and only if there are enough eigenvectors to form a basis for Rⁿ. We call such a basis an eigenvector basis of Rⁿ.

An n x n matrix with n distinct eigenvalues is diagonalizable.

Let A be an n x n matrix whose distinct eigenvalues are λ₁, ..., λ_p.
a. For 1 <= k <= p, the dimensions of the eigenspace for λ_k is less than or equal to the multiplicity of he eigenvalue λ_k.
b. The matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n, and this happens if and only if (i) the characteristic polynomial factors completely into linear factors and (ii) the dimension of the eigenspace for each λ_k equals the multiplicity of λ_k.
c. If A is diagonalizable and B_k is a basis for the eigenspace corresponding to λ_k for each k, then the total collection of vectors in the set B₁, ..., B_p forms an eigenvector basis for Rⁿ.

4.1 Vector Spaces and Subspaces

A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to 10 axioms (or rules) listed below. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d.

1. The sum of u and v, denoted by u+v, is in V.
2. u + v = v + u.
3. (u + v) + w = u + (v + w).
4. There is a zero vector 0 in V such that u + 0 = u.
5. For each u in V, there is a vector -u in V such that u + (-u) = 0.
6. The scalar multiple of u by c, denoted by cu, is in V.
7. c(u + v) = cu + cv.
8. (c + d)u = cu + du.
9. c(du) = (cd)u.
10. 1u = u.

A subspace of a vector space V is a subset H of V that has three properties:
a. The zero vector of V is in H.
b. H is closed under vector addition.
c. H is closed under scalar multiplication.

If v₁, ..., v_p are in a vector space V, then span{v₁, ..., v_p} is a subspace of V.

4.2 Null Spaces, Column Spaces, and Linear Transformations

The null space of an m x n matrix A, written as Nul A, is the set of all solutions of the homogeneous equation Ax = 0. In set notation, Nul A = {x: x in Rⁿ and Ax = 0}. The null space is a subspace of Rⁿ. Equivalently, the set of all solutions of a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rⁿ.

The column space of an m x n matrix A, written as Col A, is the set of all linear combinations of the columns of A. If A = [a₁ ... a_n], then Col A = span{a₁, ..., a_n}. The column spaces is a subspace of R^m. The column space is all of R^m if and only if the equation Ax = b has a solution for each b in R^m.

A linear transformation T from a vector space V to a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that (i) T(u + v) = T(u) + T(v) for all u, v, in V, and (ii) T(cu) = cT(u) for all u in V and all scalars c.

The kernel (or null space) of such a T is the set of all u in V such that T(u) = 0.

The range of T is the set of all vectors in W of the form T(x) for some x in V.

4.3 Linearly Independent Sets; Bases

An indexed set {v₁, ..., v_p} of two or more vectors, with v₁ not equal to 0 is lnearly dependent if and only if some v_j (with j > 1) is a linear combination of the preceding vectors, v₁, ..., v_j-1.

Let H be a subspace of vector space V. An indexed set of vectors B = {b₁, ..., b_p} in V is a basis for H if (i) B is a linearly independent set, and (ii) the subspace spanned by B coincides with H; that is, H = span{b₁, ..., b_p}.

The Spanning Set Theorem
Let S = {v₁, ..., v_p} be a set in V, and let H = span{v₁, ..., v_p}.
a. If one of the vectors in S - say, v_k - is a linear combination of the remaining vectors in S, then the set formed from S by removing v_k still spans H.
b. IF H not equal to {0}, some subset of S is a basis for H.

The pivot columns of a matrix A form a basis for Col A.

4.4 Coordinate Systems

The Unique Representation Theorem
Let B = {b₁, ..., b_n} be a basis for a vector space V. Then for each x in V, there exists a unique set of scalars c₁, ..., c_n such that x = c₁b₁ + ... + c_nb_n. The coordinates of x relative to the basis B (or the B-coordinates of x) are the weights c₁, ..., c_n.

Let B = {b₁, ..., b_n} be a basis for a vector space V. Then the coordinate mapping x -> [x]_B is a one-to-one linear transformation from V onto R_n.

In general, a one-to-one linear transformation from a vector space V to a vector space W is called an isomorphism from V to W.

Overarching

The Invertible Matrix Theorem
Let A be a n x n matrix. Then, the following statements are equivalent.

a. A is an invertible matrix.
b. A is row equivalent to the n x n identity matrix.
c. A has n pivot positions.
d. The equation Ax = 0 has only the trivial solution.
e. The columns of A form a linearly independent set.
f. The linear transformation x -> Ax is one-to-one.
g. The equation Ax = b has at least one solution for each b in Rⁿ.
h. The columns of A span Rⁿ.
i. The linear transformation x -> Ax maps Rⁿ onto Rⁿ.
j. There is an n x n matrix C such that CA = I.
k. There is an n x n matrix D such that AD = I.
l. A^T is an invertible matrix.
m. The columns of A form a basis of Rⁿ.
n. Col A = Rⁿ.
o. dim Col A = n.
p. rank A = n.
q. Nul A = {0}.
r. dim Nul A = 0.
s. The number 0 is not an eigenvalue of A.
t. The determinant of A is not 0.