Let $f: V \rightarrow V$ be a bijective map of a vector space to itself that preserves one-dimensional affine subspaces. Is $f$ already the composition of some invertible matrix and a translation? My intuition says yes, but writing down a matrix didn't work. Thanks!
Map preserving one-dimensional affine subspaces is affine
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linear-algebra
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0By "preserves" do you mean that the image of each one-dimensional affine subspace is contained in a one-dimensional affine subspace? Is the base field arbitrary? – 2012-05-24
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2In one dimension this is wrong in general, as $x \mapsto x^3$ is bijective from $\mathbb R$ to $\mathbb R$. – 2012-05-24
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1Jason Jeffers proves that this is indeed true in dimensions $n > 1$ in his article "[Lost Theorems of Geometry](http://www.jstor.org/stable/view/2695735)", American Mathematical Monthly, Vol. 107, No. 9 (Nov., 2000) (pp. 800-812) – 2012-05-24
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0@JonasMeyer: Sorry, I was a too imprecise: The image of a one-dimensional affine subspace is again a one-dimensional affine subspace. The ground field is arbitrary, but the dimension is indeed positive. Thanks martini, I will have a look into that. – 2012-05-25
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0I read Jeffers' article, and while it was very interesting, he works over the affine plane. I am not sure if this holds over arbitrary fields, or at least his proof doesn't seem to be applicable to the general case. Does anybody have any more information? I may assume that the ground field is finite, if this helps. – 2012-06-13