Basis preserving maps of linear spaces
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.
characterizations of geometrical mappings
scientific article; zbMATH DE number 769540