Linear Algebra Done Right · Problem 25

Polynomials — Sheldon Axler

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

Problem

Suppose VV is finite-dimensional and φ1,…,φm∈V′\varphi_1, \ldots, \varphi_m \in V'. Define a linear map Γ ⁣:V→Fm\Gamma\colon V \to \mathbf{F}^m by Γ(v)=(φ1(v),…,φm(v))\Gamma(v) = (\varphi_1(v), \ldots, \varphi_m(v)). (a) Prove that φ1,…,φm\varphi_1, \ldots, \varphi_m spans V′V' if and only if Γ\Gamma is injective. (b) Prove that φ1,…,φm\varphi_1, \ldots, \varphi_m is linearly independent if and only if Γ\Gamma is surjective.

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