Affine space/Affine-linear mapping/Definition

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}E}$ and ${\displaystyle {}F}$ denote affine spaces over the vector spaces ${\displaystyle {}V}$ and ${\displaystyle {}W}$, respectively. A mapping

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

is called affine (or affine-linear), if there exists a linear mapping

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

such that

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

holds for all ${\displaystyle {}P\in E}$ and ${\displaystyle {}v\in V}$.