2.I.8B

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

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