Linear Algebra Done Right · Problem 6A.2

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose S∈L(V)S \in \mathcal{L}(V). Define ⟨⋅,⋅⟩1\langle \cdot, \cdot \rangle_1 by ⟨u,v⟩1=⟨Su,Sv⟩\langle u, v \rangle_1 = \langle Su, Sv \rangle for all u,v∈Vu, v \in V. Show that ⟨⋅,⋅⟩1\langle \cdot, \cdot \rangle_1 is an inner product on VV if and only if SS is injective.

Related theory

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