  We consider several characterizations of $\mathbb R$-linear mappings. In
particular, we give a characterization of linear mappings whose range is $\geq$
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.
