Quadratic Mathematics | Part IB, 2001

Let VV be a finite-dimensional vector space over a field kk. Describe a bijective correspondence between the set of bilinear forms on VV, and the set of linear maps of VV to its dual space VV^{*}. If ϕ1,ϕ2\phi_{1}, \phi_{2} are non-degenerate bilinear forms on VV, prove that there exists an isomorphism α:VV\alpha: V \rightarrow V such that ϕ2(u,v)=ϕ1(u,αv)\phi_{2}(u, v)=\phi_{1}(u, \alpha v) for all u,vVu, v \in V. If furthermore both ϕ1,ϕ2\phi_{1}, \phi_{2} are symmetric, show that α\alpha is self-adjoint (i.e. equals its adjoint) with respect to ϕ1\phi_{1}.

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