## Help with proof that phi(P) is an affine transformation

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

Show that $\varphi$ is an affine map.