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

TL;DR

This paper provides new geometric characterizations of real linear mappings, including those with multi-dimensional ranges, extending the Fundamental Theorem of Affine Geometry, and proves these results within Zermelo's set theory without the Axiom of Choice.

Contribution

It introduces geometric characterizations of real linear mappings based on line preservation and contraction, generalizing classical affine geometry results.

Findings

Characterization of linear mappings with range ≥ 2D via line preservation.

Extension of the Fundamental Theorem of Affine Geometry.

Results are provable without the Axiom of Choice.

Abstract

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.