Linear Algebra Done Right · Problem 3E.19

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · quotient-spaces

Problem

Suppose

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

and

U

is a subspace of

V

. Let

\pi

denote the quotient map from

V

onto

V/U

. Prove that there exists

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

such that

T = S \circ \pi

if and only if

U \subseteq \operatorname{null} T