Affine-linear mapping/Fixed point/Linear/Fact/Proof

The assignment is given by ${\displaystyle {}\psi \mapsto \psi _{0}}$. We have to show that for every linear mapping ${\displaystyle {}\varphi \colon V\rightarrow V}$, there is a unique affine-linear mapping

${\displaystyle \psi \colon E\longrightarrow E}$

with this linear part. Because of

${\displaystyle {}\psi (Q)=\psi (P)+\varphi ({\overrightarrow {PQ}})=P+\varphi ({\overrightarrow {PQ}})\,,}$

there can exist at most one such an affine-linear mapping, and, by this rule, we can define such a mapping.