Characterization of affine mappings

Proposition. A map $B:X\to Y$ between real vector spaces is affine if and only if for all $x_1,x_2\in X$ and all $\lambda\in\mathbb{R}$,

Context. This shows that “affine” is exactly the property of preserving barycentric combinations of two points (for all real weights).

Proof sketch. If $B(x)=A(x)+b$ with $A$ linear, expand both sides and use linearity of $A$. Conversely, define $b:=B(0)$ and $A(x):=B(x)-b$. The identity implies $A(\lambda x)=\lambda A(x)$ and $A(x+y)=A(x)+A(y)$, so $A$ is linear and $B(x)=A(x)+b$.