Linear Algebra Done Right · Problem 3D.10

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

V

and

W

are finite-dimensional and

U

is a subspace of

V

. Let

\mathcal{E} = \{T \in \mathcal{L}(V, W) : U \subseteq \operatorname{null} T\}.

(a) Show that

\mathcal{E}

is a subspace of

\mathcal{L}(V, W)

. (b) Find a formula for

\dim \mathcal{E}

in terms of

\dim V

\dim W

, and

\dim U

. Hint: Define

\Phi \colon \mathcal{L}(V, W) \to \mathcal{L}(U, W)

\Phi(T) = T|_U

. What is

\operatorname{null} \Phi

? What is

\operatorname{range} \Phi