Linear Mathematics | Part IB, 2001

Define the dual VV^{*} of a vector space VV. Given a basis {v1,,vn}\left\{v_{1}, \ldots, v_{n}\right\} of VV define its dual and show it is a basis of VV^{*}. For a linear transformation α:VW\alpha: V \rightarrow W define the dual α:WV\alpha^{*}: W^{*} \rightarrow V^{*}.

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

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

Suppose that α\alpha is an invertible endomorphism. Prove that (α)1=(α1)\left(\alpha^{*}\right)^{-1}=\left(\alpha^{-1}\right)^{*}.

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