TITLE: Is a map that preserves lines and fixes the origin necessarily linear?
QUESTION [34 upvotes]: Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$ with $\text{dim }V \ge 2$. A line is a set of the form $\{ \mathbf{u} + t\mathbf{v} : t \in \mathbb{F} \}$. A map $f: V \to W$ preserves lines if the image of every line in $V$ is a line in $W$. A map fixes the origin if $f(0) = 0$.
Is a function $f: V\to W$ that preserves lines and fixes the origin necessarily linear?

REPLY [3 votes]: You may be interested in this paper
   Affinity of a Permutation of a Finite Vector Space
It discusses the problem of how many k-flats (cosets of a k-dimentional subspace) of an n-dimensional vector space over a finite field must be preserved by a permutation to force the permutation to preserve all k-flats.  See the references for the history of this problem for other fields. And, by the way, Vilmos Totik and Wen-Xiu Ma proved (personal communication) that if f is a transformation of Euclidean n-space, n  > 1, such that for all but countably many lines L the image f(L) is a line, then the image of any line is a line, hence f is an affine transformation.