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.