4.6 change of basis
If V = ℝ2, B = {b_1,b_2}, and C = {c_1,c_2}, then row reduction of [c_1 c_2 b_1 b_2] to [I P] produces a matrix P that satisfies [x]_B = P[x]_C for all x in V.
The statement is false. Matrix P satisfies [x]_C = P[x]_B for all x in V.
Let B and C be bases for a vector space V. The columns of the change-of-coordinates matrix PC←B are B-coordinate vectors of the vectors in C.
The statement is false. The columns of the matrix PC←B are the C-coordinate vectors of the vectors in B.
If V = ℝn and C is the standard basis for V, then PC←B is the same as the change-of-coordinates matrix PB that satisfies x=PB[x]_B for all x in V.
The statement is true. If C is the standard basis for ℝn, then b_iC=b_i for 1 ≤ i ≤ n, and PB = b_1 ... b_n.
The columns of PC←B are linearly independent.
The statement is true. The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set B.
