Linear Algebra Done Right · Problem 3D.23

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · change-of-basis

Problem

Suppose that

u_1, \ldots, u_n

and

v_1, \ldots, v_n

are bases of

V

. Let

T \in \mathcal{L}(V)

be such that

Tv_k = u_k

for each

k = 1, \ldots, n

. Prove that

\mathcal{M}(T, (v_1, \ldots, v_n)) = \mathcal{M}(I, (u_1, \ldots, u_n), (v_1, \ldots, v_n)).