Affine space/Over vector space/Definition

An affine space over a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$ is a set ${\displaystyle {}E}$, together with a mapping

${\displaystyle V\times E\longrightarrow E,(v,P)\longmapsto P+v,}$

which satisfies the following three conditions:

1. ${\displaystyle {}P+0=P}$ for all ${\displaystyle {}P\in E}$,
2. ${\displaystyle {}(P+v)+w=P+(v+w)}$ for all ${\displaystyle {}v,w\in V}$ and ${\displaystyle {}P\in E}$,
3. For two points ${\displaystyle {}P,Q\in E}$, there exists exactly one vector ${\displaystyle {}v\in V}$ such that ${\displaystyle {}Q=P+v}$.