題組內容

1. (20%) True or False. In each case, answer T if the statement is always true and F otherwise. In the case of a false statement, explain why or give an example to show that the statement is not always true.

(e) Let {u1, u2}, {v1, v2}, and {w1, w2} be bases for \( \mathbb{R}^2 \). If X is the transition matrix corresponding to a change of basis from {u1, u2} to {v1, v2} and Y is the transition matrix corresponding to a change of basis from {v1, v2} to {w1, w2}, then Z = XY is the transition matrix corresponding to a change of basis from {u1, u2} to {w1, w2}.