Quadratic Mathematics | Part IB, 2003

Let UU be a finite-dimensional real vector space and bb a positive definite symmetric bilinear form on U×UU \times U. Let ψ:UU\psi: U \rightarrow U be a linear map such that b(ψ(x),y)+b(x,ψ(y))=0b(\psi(x), y)+b(x, \psi(y))=0 for all xx and yy in UU. Prove that if ψ\psi is invertible, then the dimension of UU must be even. By considering the restriction of ψ\psi to its image or otherwise, prove that the rank of ψ\psi is always even.

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