Affine mapping

Let $X,Y$ be real vector spaces. A mapping $B:X\to Y$ is affine if there exist a linear mapping $A:X\to Y$ and a vector $b\in Y$ such that

Context. Affine maps preserve “straightness”: they map line segments to line segments, and therefore preserve convexity properties (see affine images/preimages of convex sets ).

Examples:

• Any translation $B(x)=x+b$ is affine (take $A=\mathrm{Id}$).
• Any linear map is affine (take $b=0$).
• In $\mathbb{R}^n$, $B(x)=Mx+b$ with a fixed matrix $M$ and vector $b$ is affine.