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