Eigenvector/Dual space/Dual basis/Exercise

Let

$\varphi \colon V\longrightarrow V$

be an endomorphism on a finite-dimensional ${}K$-vector space ${}V$, and let ${}v\in V$ be an eigenvector for ${}\varphi$ with eigenvalue ${}\lambda \in K$. Let

${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$

be the dual mapping of ${}\varphi$. We consider bases of ${}V$ of the form ${}v,u_{1},\ldots ,u_{r}$ with the dual basis ${}v^{*},u_{1}^{*},\ldots ,u_{r}^{*}$. Give examples of the following behavior.

a) ${}v^{*}$ is an eigenvector of ${}{\varphi }^{*}$ with the eigenvalue ${}\lambda$ independent of ${}u_{1},\ldots ,u_{r}$.

b) ${}v^{*}$ is an eigenvector of ${}{\varphi }^{*}$ with the eigenvalue ${}\lambda$ with respect to some basis ${}v,u_{1},\ldots ,u_{r}$, but not with respect to another basis ${}v,w_{1},\ldots ,w_{r}$.

c) ${}v^{*}$ is for no basis ${}v,u_{1},\ldots ,u_{r}$ an eigenvector of ${}{\varphi }^{*}$.

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