def test_columnspace_second():
M = Matrix([[1, 2, 0, 2, 5], [-2, -5, 1, -1, -8], [0, -3, 3, 4, 1],
[3, 6, 0, -7, 2]])
# now check the vectors
basis = M.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
#check by columnspace definition
a, b, c, d, e = symbols('a b c d e')
X = Matrix([a, b, c, d, e])
for i in range(len(basis)):
eq = M * X - basis[i]
assert len(solve(eq, X)) != 0
#check if rank-nullity theorem holds
assert M.rank() == len(basis)
assert len(M.nullspace()) + len(M.columnspace()) == M.cols
vec_in_space = proj(random_vector, orth_comp) + proj(random_vector, S[1])
#test = Matrix([Matrix(S), vec_in_space])
#print np.array(test)
#print np.array(test.rref()[0])
print 'New vector inside S: ' + str(vec_in_space)
# Create a vector not in span(A)
vec_not_in_space = np.random.rand(1, 4)[0]
#test = Matrix([Matrix(S), vec_not_in_space])
#print np.array(test)
#print np.array(test.rref()[0])
print 'New vector not in S: ' + str(vec_not_in_space)
# (2)
# Calculate dim(S), i.e. the number of vectors in its basis
dim = len(A.columnspace())
print 'Dim(S): ' + str(dim)
# (3)
asArray = np.array(A).astype(np.float)
thing = orth(asArray)
vec1 = []
vec2 = []
for i in range(len(thing)):
vec1.append(thing[i][0])
vec2.append(thing[i][1])
vec1 = np.array(vec1)
vec1 = np.array(vec1).astype(np.float)
ortho_basis = np.array((vec1, vec2))
