Vector space/Affine-linear mapping/Example

If we consider vector spaces ${\displaystyle {}V}$ and ${\displaystyle {}W}$ over a field ${\displaystyle {}K}$ as affine spaces, then the affine mappings from ${\displaystyle {}V}$ to ${\displaystyle {}W}$ have the form

${\displaystyle \psi \colon V\longrightarrow W,v\longmapsto \varphi (v)+w,}$

where ${\displaystyle {}\varphi \colon V\rightarrow W}$ is a linear mapping, and ${\displaystyle {}w\in W}$ is a fixed vector. In particular, an affine mapping from ${\displaystyle {}K}$ to ${\displaystyle {}K}$ has the form

${\displaystyle \psi \colon K\longrightarrow K,x\longmapsto ax+b,}$

where ${\displaystyle {}a,b\in K}$.