On geometrical characterizations of $\mathbb{R}$-linear mappings

We consider several characterizations of $\mathbb{R}$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\ge$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point) by the mappings. This characterization and its affine version generalize the Fundamental Theorem of Affine Geometry. While the algebraic characterization of $\mathbb{R}$-linear mappings as additive functions depend on the axiom of set theory, our results are provable in (the modern version of) Zermelo’s axiom system without Axiom of Choice.