Affine map

Affine maps

An affine map ${\displaystyle f\colon {\mathcal {A}}\to {\mathcal {B}}}$ between two affine spaces is a map on the points that acts linearly on the vectors (that is, the vectors between points of the space). In symbols, ${\displaystyle f}$ determines a linear transformation ${\displaystyle \varphi }$ such that, for any pair of points ${\displaystyle P,Q\in {\mathcal {A}}}$: