>> If L is a bijective continuous mapping from R^n to R^n that maps every >> straight line to straight line and L(0)=0, then L is linear. (P) >> >> If P is true, how to prove it? If P is false, what is a counter example? > > I believe it is true if n>=2. > (It is not true if n=1.) > Roughly speaking, choose coordinates on a line so each point P > is defined by x in R, say P = P(x). > Then I think you can find constructions in 2 dimensions, using just lines, > to define P(x+y) and P(xy). > It follows that your bijection defines an automorphism of R as a field. > It is easy to show that if this is continuous it is linear. > I think the result follows from this.
The argument can be put more simply as follows. It is easy to construct the mid-point C of two points A,B using only straight lines. (Eg take any point P not on the line AB, and take any line l parallel to AB not going through P or A. Suppose the line cuts AP,BP at E,F. Let EB,FA meet in X. Then PX cuts AB in C. This is the standard construction of 4 points with cross-ratio -1, the 4th point in this case being the point where AB meets the line at infinity.) By the same argument, given A,B we can construct the point D such that B is the mid-point of AD, etc.
It follows that a bijective map sending straight lines into straight lines must preserve mid-points. It is evident that this will give a dense set of points on the line AB which must be mapped into corresponding points on any line A'B'.
-- Timothy Murphy e-mail: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland