On geometrical characterizations of \mathbb R-linear mappings

2020/08/04 by Fuchino, Sakaé
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2008.02156

Abstract

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

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