Linear Algebra Done Right · Problem 24

Polynomials — Sheldon Axler

Mathematics — Linear Algebra · dual-space-spanning-independence

Problem

Suppose

V

is finite-dimensional and

v_1, \ldots, v_m \in V

. Define a linear map

\Gamma\colon V' \to \mathbf{F}^m

\Gamma(\varphi) = (\varphi(v_1), \ldots, \varphi(v_m))

. (a) Prove that

v_1, \ldots, v_m

spans

V

if and only if

\Gamma

is injective. (b) Prove that

v_1, \ldots, v_m

is linearly independent if and only if

\Gamma

is surjective.