Linear mappings/Wedge product/Diagonalizable/Exercise

Let $K$ be a field, and let $V$ denote a $K$-vector space. Let

$\phi :V⟶V$

be a diagonalizable $K$-linear mapping. Show that also the wedge product

$\stackrel{n}{\bigwedge }\phi :\stackrel{n}{\bigwedge }V⟶\stackrel{n}{\bigwedge }V$

is diagonalizable.

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