Basis preserving maps of linear spaces
type: Item
instance of: scholarly article
title: Basis preserving maps of linear spaces
zbMATH Open document ID: 0820.51001
DOI: 10.1007/BF01195135
author: Alexander Kreuzer
author: Wen-ling Huang
published in: Archiv der Mathematik
publication date: 1995-07-03T00:00:00Z
review text: The authors consider a linear space \((P, {\mathfrak L})\) which fulfilles the exchange condition. Let \(f : P \to P\) be a map for which \(f(B)\) is a basis of \(P\) if and only if \(B\) is a basis of \(P\). Then \(f\) is injective and maps lines on lines. In particular, if \((P, {\mathfrak L})\) is an affine or projective space over the real numbers, then \(f\) is an automorphism. If \(f : P \to P\) is a surjection of an arbitrary affine or projective space \((P, {\mathfrak L})\) which maps any basis to a basis, then \(f\) is an automorphism.
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51D10
zbMATH DE Number: 769540
zbMATH Keywords: characterizations of geometrical mappings
zbMATH Keywords: isomorphisms
zbMATH Keywords: exchange spaces
reviewed by: Alexander P. Kreuzer
MaRDI profile type: Publication
cites work: A Contribution to Chronogeometry
cites work: On Isometries of Euclidean Spaces
cites work: Q5662517
cites work: Zur Einbettung von Inzidenzräumen und angeordneten Räumen. (On the embedding of incidence spaces and ordered spaces)
full work available at URL: https://doi.org/10.1007/bf01195135
OpenAlex ID: W2086965653
description: scientific article; zbMATH DE number 769540