Linear mapping/Diagonalizable/Direct sum of eigenspaces/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.

Then ${}\varphi$ is

diagonalizable if and only if ${}V$ is the direct sum of the

eigenspaces.

Proof, Write another proof