On the Fundamental Theorem of Affine Geometry
Canadian mathematical bulletin, Tome 5 (1962) no. 1, pp. 67-69
Voir la notice de l'article provenant de la source Cambridge University Press
The fundamental theorem of affine geometry is an easy corollary of the corresponding projective theorem 2.26 in Artin's Geometric Algebra. However, a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278] and his axioms 1 and 2 may be of some interest.Lipman's [desarguian] affine geometry G determined a left linear vector space L={a, b,...} over a skew field F. We wish to construct 1−1 transformations γ of G onto itself such that γ and γ-1 map straight lines onto straight lines preserving parallelism. Designate any point 0 as the origin of G. Multiplying γ with a suitable translation, we may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of L onto itself which preserves linear dependence. Since Γ-1 will have the same properties, Γ must also preserve linear independence.
Scherk, P. On the Fundamental Theorem of Affine Geometry. Canadian mathematical bulletin, Tome 5 (1962) no. 1, pp. 67-69. doi: 10.4153/CMB-1962-011-1
@article{10_4153_CMB_1962_011_1,
author = {Scherk, P.},
title = {On the {Fundamental} {Theorem} of {Affine} {Geometry}},
journal = {Canadian mathematical bulletin},
pages = {67--69},
year = {1962},
volume = {5},
number = {1},
doi = {10.4153/CMB-1962-011-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1962-011-1/}
}
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