Eigenspaces/Direct sum/Dimension/Fact

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

$\phi :V⟶V$

be a linear mapping.

Then the

sum of the eigenspaces ${\mathrm{Eig}}_{\lambda }\left(\phi \right)$ is direct, and we have

$\sum _{\lambda \in K}{\dim }_K\left({\mathrm{Eig}}_{\lambda }\left(\phi \right)\right)\le {\dim }_K\left(V\right).$

Proof, Write another proof