Affine space/Affine-linear mapping/Remark

A mapping

$\psi :E⟶F$

is affine-linear with linear part $\phi$ if and only if the diagram

$\begin{array}{cccc}V\times E& \stackrel{+}{⟶}& E& \\ \phi \times \psi \downarrow & & \downarrow \psi & \\ W\times F& \stackrel{+}{⟶}& F& \end{array}$

commutes. For an affine-linear mapping

$\psi :E⟶F,$

the linear part (assume $E\ne \mathrm{\varnothing }$)

$\phi :V⟶W$

is uniquely determined. This is because we must have

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

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

${\psi }_0(\overrightarrow{PQ})=\overrightarrow{\psi (P)\psi (Q)}.$