Endomorphism/Characteristic polynomial/Different linear factors/Diagonalizable/Fact/Proof/Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Suppose that the characteristic polynomial ${}\chi _{\varphi }$ factors into different linear factors. Show that ${}\varphi$ is diagonalizable.

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