Linear Fractional Transformations 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. 
 A: I base my answer on the guess that $\pi$ is the stereographic projection from the Riemann sphere $S$ to the extended complex plane $\mathbf{C}\cup\{\infty\}$. I'm assuming that the equator of the sphere is on the complex plane. Align the 3-axes as follows: let the $x$-axis coincide with the real axis, $y$-axis with the imaginary axis, and $z$-axis stick out of the complex plane. I am further assuming that the exercise is about showing that a rotation about any of these 3 axes corresponds to a LFT of the complex plane. Here's a plan written in terms of extended hints:
Exercise #1: Check that a rotation about the $z$-axes yields an LFT. Well, 
this rotation by the angle $\phi$, call it $R(\phi)$ amounts to multiplication by $e^{i\phi}$.
Exercise #2: Let $\psi$ be a 90 degree rotation about the $y$-axis. In the 3D-coordinates this is $\psi(x,y,z)=(-z,y,x)$. Verify that the mapping $\pi\circ\psi\circ\pi^{-1}$ is a LFT.
Exercise #3: Show that an arbitrary rotation about the $x$-axis corresponds to a LFT. Hint: The LFTs form a group, right? What can you say about the 3D-mapping $\psi\circ R(\phi)\circ \psi^{-1}$?
Exercise #4: The same as in exercise #2, but this time we rotate about the $x$-axis, so instead of $\psi$ we look at $\tau(x,y,z)=(x,-z,y)$. Alternatively you can use a little bit of geometric thinking and combine ideas from exercises #1 and #3.
Exercise #5: The same as in exercise #4, but use $\tau$ instead of $\psi$ to show that an arbitrary rotation about the $y$-axis corresponds to a LFT.
If the question was only about those '2D' rotations then you are done. If you want to show the same for all the rotations of $S$ about an arbitrary axis, then I give you one more ...
Hint #6: Have you heard of Euler angles?
