Linear Algebra Done Right · Problem 3C.5

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-linear-maps

Problem

Suppose

V

and

W

are finite-dimensional and

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

. Prove that there exist a basis of

V

and a basis of

W

such that with respect to these bases, all entries of

\mathcal{M}(T)

are 0 except that the entries in row

k

, column

k

, equal 1 if

1 \leq k \leq \dim \operatorname{range} T