Linearity vs. takes straight lines to straight lines

[Fredrik]

Linearity vs. "takes straight lines to straight lines"

Homework Statement

Prove that if $\Lambda:\mathbb R^n\to\mathbb R^n$ is a bijection that takes straight lines to straight lines, and is such that $\Lambda(0)=0$, then $\Lambda$ is linear.

Homework Equations

Fock's theorem implies that the statement I want to prove is true. But I don't want to use that. I want to find a simpler proof of a less general theorem. It's also possible that there's a published proof of Fock's theorem that is easy to simplify then the assumptions are stronger.

Fock's theorem says (roughly) that if $\Lambda$ is a map that takes straight lines to straight lines, there's there's a 4×4 matrix A, two 4×1 matrices y,z, and a number c, such that
$$\Lambda(x)=\frac{Ax+y}{z^Tx+c}.$$ Note that if $\Lambda$ is defined on a vector space (rather than a proper subset of one), there's always an x that makes the denominator 0. So, since my $\Lambda$ has a vector space as its domain, we must have z=0, and we can redefine A and y to absorb c. But I'm also assuming that $\Lambda(0)=0$, so the theorem also says that $y=0$, which means that my $\Lambda$ is linear.

The proof of Fock's theorem is very hard. It looks like it's proved in Appendix B of this article, but the proof is so badly written that it's hard to tell. The proof is discussed in this thread. My thoughts on the first part of the proof appears in post #61.

The Attempt at a Solution

I'm nowhere near a solution. I guess the following is better than nothing, but only very slightly: Let x be arbitrary. Let v≠0 be arbitrary. Since $\Lambda$ takes straight lines to straight lines, there's a unit vector u and a function $s:\mathbb R\to\mathbb R$ such that for all t,
$$\Lambda(x+vt)=\Lambda(x)+s(t)u.$$ (The link to "post #61" above explains this in more detail). I guess I want to prove that $t\Lambda(v)=s(t)u$, but I don't see how to proceed from here.

Also, my post #50 describes a few more of my thoughts about the problem.

One of the reasons why I think it that the simpler theorem has a simpler proof is that it's an exercise in a book I own. It's exercise 1.3.1(2) on page 9 of this book (pdf link). Unfortunately, the exercise is messed up in at least two ways. It assumes that the map is surjective onto W, but it doesn't define W. And it asks the reader to prove that the map is linear, but that can't be the correct conclusion since the exercise doesn't include the assumption that the map takes 0 to 0. The exercise also doesn't assume that the map is bijective, but including that assumption can only make things easier, right?

Edit: By the way, I'm not entirely clear on whether some assumptions about smoothness or at least existence of partial derivatives must be included.

 

[Fredrik]

I got a tip that the "simple" theorem I want to prove is more or less "the fundamental theorem of affine geometry". It's quite hard to prove, so it really shouldn't be an exercise on page 9 of a book on functional analysis.