2.II.15C

Define the dual $V^{*}$ of a vector space $V$. Given a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ of $V$ define its dual and show it is a basis of $V^{*}$. For a linear transformation $\alpha: V \rightarrow W$ define the dual $\alpha^{*}: W^{*} \rightarrow V^{*}$.

Explain (with proof) how the matrix representing $\alpha: V \rightarrow W$ with respect to given bases of $V$ and $W$ relates to the matrix representing $\alpha^{*}: W^{*} \rightarrow V^{*}$ with respect to the corresponding dual bases of $V^{*}$ and $W^{*}$.

Prove that $\alpha$ and $\alpha^{*}$ have the same rank.

Suppose that $\alpha$ is an invertible endomorphism. Prove that $\left(\alpha^{*}\right)^{-1}=\left(\alpha^{-1}\right)^{*}$.

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