Let $\displaystyle f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear map. Let $\displaystyle O$ and $\displaystyle O'$ be two points on the Euclidean space. Define $\displaystyle \varphi:{\varepsilon}^n\rightarrow{\varepsilon}^n$ as:
$\displaystyle \varphi(O)=O'$
$\displaystyle \varphi(P)=O'+f(OP)$ ($\displaystyle OP$ is a vector from point O to P)

Show that $\displaystyle \varphi$ is an affine map.