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

Proof

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

$\psi \colon E\longrightarrow E$

with this linear part. Because of

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

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

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