Eigenspaces/Direct sum/Dimension/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 the

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

${}\sum _{\lambda \in K}\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}\leq \dim _{K}{\left(V\right)}\,.$

Proof, Write another proof