Linear Algebra Done Right · Problem 31

Polynomials — Sheldon Axler

Mathematics — Linear Algebra · annihilator-inclusion-map

Problem

Suppose

U

is a subspace of

V

. Let

i\colon U \to V

be the inclusion map defined by

i(u) = u

. Thus

i' \in \mathcal{L}(V', U')

. (a) Show that

\operatorname{null} i' = U^0

. (b) Prove that if

V

is finite-dimensional, then

\operatorname{range} i' = U'

. (c) Prove that if

V

is finite-dimensional, then

\tilde{i}'

is an isomorphism from

V'/U^0

onto

U'

. The isomorphism in (c) is natural in that it does not depend on a choice of basis in either vector space.