> We all know that if L is a linear mapping from R^n to R^n, then it maps > every straight line to a straight line. But is the following proposition > true? > > 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.
-- Timothy Murphy e-mail: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland