
Isometries of R2
I am reading Aigli Papantonopoulou's book Algebra: Pure and applied.
Exercise 16 of Chapter 14:Symmetries reads as follows:
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"Find all isometries of $\displaystyle \mathbb{R}^2$ that map the line y = x to the line y = 1  2x
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Could anyone please help with this exercise
Peter

Re: Isometries of R2
rather than give you the actual answer, i will give you the basic idea: how do you place one long skinny stick, on top of another? well, you can just put it directly on top, or you can "twirl it around", and then set it down, or you can "flip it over". does this give you any idea of how to proceed?

Re: Isometries of R2
I have thought about this but need some more explicit help
Peter

Re: Isometries of R2
think about the possible ways to map y = x, to the line y = 2x (these ways will be elements of O(2)). one way, is to just rotate the line y = x until it lines up with y = 2x. of course, we can rotate the line y = x the opposite direction to do this, too. since we are dealing with isometries, we need move the entire plane "along for the ride".
once we have rotated the second line onto the first, if we "flip the line over" (reflect the entire space), we also have succeeded in mapping y = x to y = 2x. of course, we could reflect first, and then apply one of our two rotations (this is easier to compute).
having mapped y = x to y = 2x, an obvious translation finishes the process.