Linear Algebra Done Right · Problem 3D.9

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

V

is finite-dimensional and

T \colon V \to W

is a surjective linear map of

V

onto

W

. Prove that there is a subspace

U

V

such that

T|_U

is an isomorphism of

U

onto

W

. Here

T|_U

means the function

T

restricted to

U

. Thus

T|_U

is the function whose domain is

U

, with

T|_U

defined by

T|_U(u) = Tu

for every

u \in U