Linear Algebra Done Right · Problem 3C.2

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-linear-maps

Problem

Suppose

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

, where

V

and

W

are finite-dimensional and nonzero. Prove that

\dim \operatorname{range} T = 1

if and only if there exist a basis of

V

and a basis of

W

such that with respect to these bases, all entries of

\mathcal{M}(T)

equal 1.