Linear Algebra Done Right · Problem 3D.8

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

V

and

W

are finite-dimensional and

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

. Prove that there exist invertible

E_1 \in \mathcal{L}(V)

and

E_2 \in \mathcal{L}(W)

such that

S = E_2 T E_1

if and only if

\dim \operatorname{null} S = \dim \operatorname{null} T