Affine-linear mapping/Fixed point/Linear/Fact

Let ${}K$ be a field and let ${}E$ be an affine space over the vector space ${}V$. Let ${}P\in E$.

Then the affine-linear mappings

$\psi \colon E\longrightarrow E$

having ${}P$ as a fixed point correspond to the linear mappings

$\varphi \colon V\longrightarrow V.$

Proof, Write another proof