Linear Algebra Done Right · Problem 29

Polynomials — Sheldon Axler

Mathematics — Linear Algebra · null-space-range-dual-map

Problem

Suppose

V

and

W

are finite-dimensional and

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

. (a) Prove that if

\varphi \in W'

and

\operatorname{null} T' = \operatorname{span}(\varphi)

, then

\operatorname{range} T = \operatorname{null} \varphi

. (b) Prove that if

\psi \in V'

and

\operatorname{range} T' = \operatorname{span}(\psi)

, then

\operatorname{null} T = \operatorname{null} \psi