def convert_linear_mapping(cls, linear_mapping, basis_vectors):
'''
converts a standard matrix linear mapping to a given coordinate vector
linear_mapping: Matrix that inputs size basis_vectors
basis_vectors: all basis_vectors
'''
lin_width, lin_height = linear_mapping.get_size()
# convert basis vectors to a matrix
width, height = basis_vectors.get_size()
assert linear_mapping.is_square(
) and lin_width == width, "Basis Vectors and Linear Mapping should have the same shape"
matrix = Matrix(width, height)
for col_idx in range(width):
matrix.set_col(col_idx,
basis_vectors.basis_vectors[col_idx].values)
basis_vec_mat = copy.deepcopy(matrix)
# Apply Linear Mapping to basis vectors
applied = Matrix.matmul(linear_mapping, matrix)
# Solve for Linear Mapping in base_B
concat = Matrix.matrix_concat_column(basis_vec_mat, applied)
reduced = Matrix.AUG_RREF(concat, width)
# Extract Augmented Portion
extracted = Matrix(lin_width, lin_height)
count = 0
for col_idx in range(width, lin_width + width):
extracted.set_col(count, reduced.get_col(col_idx))
count += 1
return extracted
def compute_coordinate(cls, ordered_basis, x):
'''
Computes [x]_B given ordered basis and original vector x.
ordered_basis: Basis Object, ordered basis
x: vector
'''
width, height = ordered_basis.get_size()
# Construct Matrix using ordered matrix
matrix = Matrix(width + 1, height)
for col_idx in range(width):
matrix.set_col(col_idx,
ordered_basis.basis_vectors[col_idx].values)
matrix.set_col(width, x.values)
# AUG RREF to solve for coefficients
reduced = Matrix.AUG_RREF(matrix, width)
# Extract Vector Out
x_b = reduced.get_col(width)
return x_b
def in_span(self, vec_A):
'''
Computes if a given vector is already spanned by the collection
'''
if isinstance(vec_A, Matrix):
# convert to vector
vec_A = Matrix.flatten(vec_A)
assert isinstance(vec_A, Vector)
assert len(vec_A) == self.height
# Construct a Matrix
matrix = Matrix(self.width + 1, self.height)
for vector_idx in range(len(self.vectors)):
matrix.set_col(vector_idx, self.vectors[vector_idx].values)
matrix.set_col(self.width, vec_A.values)
# Row Reduce matrix
reduced = Matrix.AUG_RREF(matrix, self.width)
# All 0 values
row_indices = Matrix.zero_rows(reduced, self.width)
for row_idx in row_indices:
if reduced.get_value(row_idx, self.width) != 0:
return False
return True
def change_of_basis(cls, b_coord_basis, c_coord_basis):
'''
Computes the change of basis matrix that converts a vector in B_coord to C_coord
'''
B_width, B_height = b_coord_basis.get_size()
assert b_coord_basis.is_square() and (
B_width, B_height) == c_coord_basis.get_size(
), "Matrices must be square and identical in shape"
matrix = Matrix(B_width * 2, B_height)
for col_idx in range(B_width):
matrix.set_col(col_idx, c_coord_basis.get_col(col_idx))
count = B_width
for col_idx in range(B_width):
matrix.set_col(count, b_coord_basis.get_col(col_idx))
count += 1
reduced = Matrix.AUG_RREF(matrix, B_width)
# Extract Augmented
C_B = Matrix(B_width, B_height)
count = 0
for col_idx in range(B_width, B_width * 2):
C_B.set_col(count, reduced.get_col(col_idx))
count += 1
return C_B
