Definition (Affine map)

A Linear Map $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is called affine if for any $x,y\in\mathbb{R}^{n}$ and any $\alpha,\beta\in\mathbb{R}$ with $\alpha+\beta=1$ we have $$f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$$

Definition (Affine function)

A function $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is called affine if and only if $$f(x)=Ax+b$$for some $A\in\mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^{m}$.