Linear Algebra Done Right · Problem 3D.5

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

V

is finite-dimensional,

U

is a subspace of

V

, and

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

. Prove that there exists an invertible linear map

T

from

V

to itself such that

Tu = Su

for every

u \in U

if and only if

S

is injective.