Linear Algebra Done Right · Problem 3C.6

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-linear-maps

Problem

Suppose

v_1, \ldots, v_m

is a basis of

V

and

W

is finite-dimensional. Suppose

T \in \mathcal{L}(V, W)

. Prove that there exists a basis

w_1, \ldots, w_n

W

such that all entries in the first column of

\mathcal{M}(T)

[with respect to the bases

v_1, \ldots, v_m

and

w_1, \ldots, w_n

] are 0 except for possibly a 1 in the first row, first column. In this exercise, unlike Exercise 5, you are given the basis of

V

instead of being able to choose a basis of

V