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Let $p(x)= \pi A_\alpha \pi^{-1}(x) = y$, where $$A_\alpha = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\\ \end{pmatrix}$$ and $\pi:S\to C=\{z=x_1+ix_2\}\cup\infty$.

Show that $y = p(x)$ is a linear fractional transformation. I am a little confused on how to start this problem.

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    What is the function $\pi$?2011-10-02
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    Im so sorry I forgot to add that. $\pi : S ----> C$ = { z= x_{1} + i x_2} $\bigcup$ $\infty$2011-10-02
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    Should we assume $\pi$ is also a lft? If so, then all you need is that inverses of and products of lft's are lft's...2011-10-02
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    It is not stated whether or not $\pi$ is a lft or not. I think that may be the reson I am confused.2011-10-02
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    Do you have properties of [Schwarzian derivative](http://en.wikipedia.org/wiki/Schwarzian_derivative) at your disposal?2011-10-02
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    I think, from my understanding, is that I am suppossed to show that a LFT is induced by rotations of the circle. This question is so confusing.2011-10-02
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    @ Anon. I just read about the Schwarzian derv's. That is a useful trick or property. However, looking at the matrix, how would I show that the derivative of that is 0? Also, once I had this obtained, then would that show that p(x) = y is also 0 then, therefore it would be a LFT?2011-10-02
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    Marge, please tell a bit more. Is $S$, perhaps, the Riemann Sphere and $\pi$ the stereographic projection? This doesn't quite make sense given that $A$ looks like a 2D-rotation, but, pray, tell us what $S$ is? The reason I ask is that it is a standard exercise to show that rotations of the Riemann Sphere correspond to fractional linear transformations of the extended plane.2011-10-02
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    If you weren't familiar with SDs before now it would be a bad idea to put them in your homework. But they are indeed useful with their properties at your disposal (which themselves would either take hellish algebra or a high-concept view of where $S$ originally comes from - namely the local distortion of cross ratio under flow). I haven't looked at it fully but $S(p)=0$ would indeed imply $p$ is a LFT, and that might be possible to prove with $S$'s inversion and composition properties. (But you should probably do your homework separately first! Also, don't put a space after '@'; doesn't work.)2011-10-02
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    @Jyrki: Typically when I see a matrix juxtaposed with a complex variable in the context of Mobius transformations it's supposed to be the corresponding LFT, but your interpretation of a 2D rotation (along with $\pi$ being a streographic projection) seems much more plausible because the question is clearly sensible then. If Marge would give more context you could probably write an answer under that assumption.2011-10-02
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    and no matter what the topic is, [this](http://www.youtube.com/watch?v=JX3VmDgiFnY) always helps on LFTs. :)2011-10-02
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    Yes S is the Riemann Sphere and $\pi$ is the sterographic projection. I am sorry I did not state this. I am new to this site. I thank you all for you suggestions. I just need to understand them. Because I do not at the moment.2011-10-02

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