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Topic: Mapping from straight line to straight line
Replies: 11   Last Post: Nov 3, 2013 4:35 PM

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Timothy Murphy

Posts: 657
Registered: 12/18/07
Re: Mapping from straight line to straight line
Posted: Nov 2, 2013 8:36 AM
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emwong wrote:

> 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/
School of Mathematics, Trinity College, Dublin 2, Ireland

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