Quadratic Mathematics | Part IB, 2003

Let UU and VV be finite-dimensional vector spaces. Suppose that bb and cc are bilinear forms on U×VU \times V and that bb is non-degenerate. Show that there exist linear endomorphisms SS of UU and TT of VV such that c(x,y)=b(S(x),y)=b(x,T(y))c(x, y)=b(S(x), y)=b(x, T(y)) for all (x,y)U×V(x, y) \in U \times V.

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