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

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.

Then ${}\varphi$ is diagonalizable.

Write a proof