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.