Affine space/Affine-linear mapping/Remark

A mapping

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

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

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

commutes. For an affine-linear mapping

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

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

${\displaystyle \varphi \colon V\longrightarrow W}$

is uniquely determined. This is because we must have

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

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

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