Affine space/Affine-linear mapping/Remark

A mapping

$\psi \colon E\longrightarrow F$

is affine-linear with linear part ${}\varphi$ if and only if the diagram

${\begin{matrix}V\times E&{\stackrel {+}{\longrightarrow }}&E&\\\!\!\!\!\!\varphi \times \psi \downarrow &&\downarrow \psi \!\!\!\!\!&\\W\times F&{\stackrel {+}{\longrightarrow }}&F&\!\!\!\!\!\\\end{matrix}}$

commutes. For an affine-linear mapping

$\psi \colon E\longrightarrow F,$

the linear part (assume ${}E\neq \emptyset$)

$\varphi \colon V\longrightarrow W$

is uniquely determined. This is because we must have

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

for an arbitrary point ${}P\in E$. Therefore, we denote the linear part with ${}\psi _{0}$. in particular, for two points ${}P,Q\in E$, we have

${}\psi _{0}({\overrightarrow {PQ}})={\overrightarrow {\psi (P)\psi (Q)}}\,.$