Affine space/Affine-linear mapping/Definition

Affine-linear mapping

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

$\psi \colon E\longrightarrow F$

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

$\varphi \colon V\longrightarrow W$

such that

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

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