Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties:


1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine $\mathbb{R}$-lines


2) If $r$ and $s$ are parallel straight affine $\mathbb{R}$-lines, then so are $\alpha(r)$ and $\alpha(s)$.


**Question:** does $\alpha$ have to be $\mathbb{R}$-linear?


This (perhaps not exactly research level...) question came to me when I tried to answer the following (also elementary - Edit: as the comment by Misha points out, it was not elementary indeed! Just classical from nineteenth Century) question: does a set-theoretical bijection $\alpha:\mathbb{R}^n\to\mathbb{R}^n$ that sends lines to lines and preserves parallellism have to be $\mathbb{R}$-affine? When $n=1$ the answer is vacuously no, as there is only one line and every set-theoretical bijection would do. But when $n\geq 2$, assuming $\alpha(0)=0$, one can prove${}^\*$ that $\alpha$ is actually $\mathbb{Q}$-linear, and then -if one assumes continuity- also that $\alpha$ is $\mathbb{R}$-linear. But what if we don't assume continuity?


${}^\*$: First let's prove $\alpha$ is additive, that is $\alpha(u+v)=\alpha(u)+\alpha(v)$. If $u$ and $v$ are not multiples of each other, then the nondegenerate parallelogram corresponding to the points $0,u,v,u+v$ is sent to the nondegenerate parallelogram $0,\alpha(u),\alpha(v),\alpha(u+v)$, hence $\alpha(u+v)$ is the vector sum of $\alpha(u)$ and $\alpha(v)$. If $u,v$ are multiples of each other, write $u=u\_1+u\_2$ and $v=v\_1+v\_2$ generically enough that none of the following pairs $(w,w')$ of points is such that $w,w'$ are multiples of each other: $(u\_1+v\_1, u\_2+v\_2)$, $(u\_1,v\_1)$, $(u\_2,v\_2)$, $(u\_1,u\_2)$, $(v\_1,v\_2)$. Now, 


$$\alpha(u+v)=\alpha(u\_1+u\_2+v\_1+v\_2)=\alpha(u\_1+v\_1)+\alpha(u\_2+v\_2)=$$


$$\alpha(u\_1)+\alpha(v\_1)+\alpha(u\_2)+\alpha(v\_2)=\alpha(u\_1+u\_2)+\alpha(v\_1+v\_2)=\alpha(u)+\alpha(v).$$


So $\alpha$ is additive. For every $u$, by additivity $\alpha(u)=n \alpha (\frac{1}{n} u)$, so for every $u$, $\alpha(\frac{1}{n} u)=\frac{1}{n} \alpha(u)$. The above implies $\mathbb{Q}$-linearity.



A little question on certain parallel-lines-preserving maps
The [fundamental theorem of projective geometry](http://www.wikipedia.org/wiki/Collineation) proven by von Staudt around 1850 (at least, this is when he published the proof) states:


Let $n\ge 2$, $F$ be a field and $f: FP^n\to FP^n$ is a *collineation*, i.e., a bijection preserving collinearity of points. Then $f$ is a composition of an automorphism of the field $F$ and of an element of $PGL(n+1, F)$. 


In the setting of the question of Qfwfq, the bijection induces a collineation of ${\mathbb R}P^n$, so it is a real-linear map, as ${\mathbb R}$ has no automorphisms. (The point is that every automorphism of real numbers has to be continuous as one can encode inequalities into equations, as $x^2\ge 0$ for every real number $x$.) 


What's even more interesting is *how* von Staudt proved his theorem: He noticed that (once coordinate lines in the plane are fixed), one can *encode algebraic operations in $F$ into point-line projective configurations* (convert algebra into geometry). Von Staudt's theorem and its proof had many generalizations and applications. For instance, Tits' rigidity theorem for spherical buildings is a generalization of the von Staudt's result (Margulis' super-rigidity theorem in turn, is based on Tits theorem in measurable setting). Von Staudt's proof also has lead to 
[Mnev's Universality Theorem](http://www.wikipedia.org/Mnev's_universality_theorem), 
to universality results for character varieties [here](http://www.math.ucdavis.edu/~kapovich/EPR/ihes.pdf), and [here](https://mathoverflow.net/questions/123198/can-the-sl-2-character-variety-of-a-three-manifold-be-nonreduced/123259#123259), Ravi Vakil's Murphy's law in algebraic geometry 
[here](http://math.stanford.edu/~vakil/files/Mjul0705.pdf) and so on. 


