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Let $b$ be a non-degenerate bilinear form on a finite dimensional vector space $V$. Let $b'$ be any bilinear form on $V$. Show that $\exists$ $T \in L(V,V)$ such that $b'(v,w)=b(Tv,w) \, \forall v,w \in V$.

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up vote 2 down vote accepted

Let $B$ and $B'$ be matrices such that $xBy^T=b(x,y)$ and $xB'y^T=b'(x,y)$.

Since $b$ is nondegenerate, the rows of $B$ are linearly independent, so there exists a matrix $T$ such that $TB=B'$. Can you take it from here?

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Yeah I think I can, I'll post if I get stuck again. Thank you! –  Zachary Luety Mar 13 '13 at 1:36
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