I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$:
$$ f[\lambda x+ (1-\lambda) y]=\lambda f(x)+(1-\lambda)f(y), \text{ for all } \lambda\in\mathbb{R}, $$
admit always a representation as $f(x)=g(x)+c$ where $g:V\rightarrow W$ is a linear map and $c\in W$.