Linear mapping/Diagonalizable/Direct sum of eigenspaces/Fact

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

${\displaystyle \varphi \colon V\longrightarrow V}$

be a linear mapping.

Then ${\displaystyle {}\varphi }$ is

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

eigenspaces.