LinearSystemSolver

A linear algebra lib build through Linear Algebra Refresher Course on Udacity.

It provides functionality to:

1. Solve linear system
2. Common manipulations on vector, line, plane and hyperplane

Usage

Manipulation on Vector

# Vector constructor receives an Iterable
# create 3-dimension vectors
v = Vector([1, 2, 3])

# create n-dimension vector
n = 3
w = Vector(range(n))

# dimension and coordinates
v.dimension    # 3
v.coordinates  # [Decimal('1'), Decimal('2'), Decimal('3')]

# plus and minus
v_plus_w = v.plus(w)
v_minus_w = v.minus(w)

# times scalar
v_times_3 = v.times_scalar(3)

# check parallel,orthogonal and zero (optional parameter tolerance)
v.is_parallel_to(w)
v.is_orthogonal_to(w, tolerance=1e-8)
v.is_zero(tolerance=1e-10)

# magnitude and normalized vector
v.magnitude()
v.normalized()  # returns a vector

# dot product
v.dot(w)

# angle (optional parameter in_degrees, default False)
v.angle_with(w)                   # returns in radians
v.angle_with(w, in_degrees=True)  # returns in degrees

# cross product (returns a vector)
v.cross(w)

# area of triangle and parallelogram made up by two vectors
v.area_of_triangle_with(w)
v.area_of_parallelogram_with(w)

Solve Linear System

# Linear System is made up of equations
# Each equation is represented by a hyperplane

# Each hyperplane is made up of a normal vector and a constant term
p1 = Hyperplane(normal_vector=Vector(['0', '1', '1']), constant_term='1')   # y + z = 1
p2 = Hyperplane(normal_vector=Vector(['1', '-1', '1']), constant_term='2')  # x - y + z = 2
p3 = Hyperplane(normal_vector=Vector(['1', '2', '-5']), constant_term='3')  # 1 + 2y - 5z = 3

# Linear system receives an Iterable of Hyperplanes
system = LinearSystem([p1, p2, p3])

# get solution
solution = system.compute_solution()
print(solution)
# output:
# x_1 = 2.556
# x_2 = 0.778
# x_3 = 0.222

# if there is no solution
p1 = Hyperplane(normal_vector=Vector(['1', '1', '1']), constant_term='1')  # x + y + z = 1
p2 = Hyperplane(normal_vector=Vector(['1', '1', '1']), constant_term='2')  # x + y + z = 2
system = LinearSystem([p1, p2])
solution = system.compute_solution()
print(solution)
# output:
# No solutions

# if there is infinite solutions
p1 = Hyperplane(normal_vector=Vector(['0', '1', '1']), constant_term='1')   # y + z = 1
p2 = Hyperplane(normal_vector=Vector(['1', '-1', '1']), constant_term='2')  # x - y + z = 2
system = LinearSystem([p1, p2])
solution = system.compute_solution()
print(solution)
# output:
# x_1 = 3.0 + -2.0 t_1
# x_2 = 1.0 + -1.0 t_1
# x_3 = 0.0 + 1.0 t_1

# solution is a Parametrization Object
# Goal of Parametrization is to represent infinite solutions
# For linear system:
#   y + z = 1
#   x - y + z = 2
# Reduce to reduced row-echelon form:
#   x = 3 - 2z
#   y = 1 - z
#   z = 0 + z (make up to 3 dimension)
# Solution is a line : [3, 1, 0] + z * [-2, -1, 1]
# x,y are pivot variables and z is the free variable
# Parametrization is made up of a basepoint and a list of direction vectors representing free variables
solution.basepoint          # Vector(Decimal(3), Decimal(1), Decimal(0))
solution.direction_vectors  # [Vector(Decimal(-2), Decimal(-1), Decimal(1))]

Line and Plane

Line are Plane are just hyperplane with fixed dimension, somehow redundant, see more in documentation.

Documentation

See docstrings in each file.

Test

All tests are under test folder and written in unit testing framework unittest, see more in its documentation.