Quadratic Mathematics | Part IB, 2002

Explain what is meant by a sesquilinear form on a complex vector space VV. If ϕ\phi and ψ\psi are two such forms, and ϕ(v,v)=ψ(v,v)\phi(v, v)=\psi(v, v) for all vVv \in V, prove that ϕ(v,w)=ψ(v,w)\phi(v, w)=\psi(v, w) for all v,wVv, w \in V. Deduce that if α:VV\alpha: V \rightarrow V is a linear map satisfying ϕ(α(v),α(v))=ϕ(v,v)\phi(\alpha(v), \alpha(v))=\phi(v, v) for all vVv \in V, then ϕ(α(v),α(w))=ϕ(v,w)\phi(\alpha(v), \alpha(w))=\phi(v, w) for all v,wVv, w \in V.

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