Wedge product/Eigenvalues/Exercise

Let ${}V$ be a vector space over the field ${}K$, and let

$\varphi \colon V\longrightarrow V$

be an endomorphism. Let

$\bigwedge ^{m}\varphi \colon \bigwedge ^{m}V\longrightarrow \bigwedge ^{m}V$

be the ${}m$-th wedge product of ${}\varphi$. Let ${}v_{1},\ldots ,v_{m}$ be linearly independent eigenvectors of ${}\varphi$ with eigenvalues ${}a_{1},\ldots ,a_{m}$. Show that ${}a_{1}\cdots a_{m}$ is an eigenvalue of ${}\bigwedge ^{m}\varphi$.

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