Fundamental theorem of affine geometry

Remarks

A map with properties 1 and 2 is called semi-affine. The proof first shows that $f$ maps planes to planes and hence parallel lines to parallel lines. Parallelograms then go to parallelograms, which makes the induced map $\varphi$ on vectors well defined and additive. Comparing $f$ on the points of one line with its action on scalars defines $\sigma$ and shows that it is a field automorphism. For $k = \R$ the only field automorphism is the identity, which gives the last sentence.

The hypothesis on $k$ is needed: over $\F_2$ each line has exactly two points, so every bijection maps lines to lines. For $n \ge 3$ there are bijections of $\F_2^n$ that are not affine, for example the map that swaps two basis vectors $e_1$ and $e_2$ and fixes every other point. It is not affine, since it fixes $0$, $e_3$ and $e_1 + e_3$ but moves $e_1$. The hypothesis $n \ge 2$ is needed too: an affine line has only one line, so every bijection of it qualifies. Over $\C$, the map $(z_1, \dots, z_n) \mapsto (\bar z_1, \dots, \bar z_n)$ maps lines to lines but is not affine, so $\sigma$ cannot be dropped.

Going deeper

For a student

The theorem shows that the incidence structure of points and lines determines the affine structure of $k^n$, up to the automorphisms of $k$. It is the affine version of the fundamental theorem of projective geometry, and one can pass between the two by adding or removing a hyperplane at infinity. It shows that the collineations of a real affine space are exactly the invertible affine maps. It also explains why the automorphism group of a field enters the symmetry groups of geometries over that field.