Definitions:
Extended Complex Plane $\mathbb{C}^\infty = \mathbb{C} \cup \{\infty\}$.
Stereographic projection: A mapping from a sphere in $\mathbb{R}^3$ to the extended complex plane.
Möbius Transformation: $z \mapsto \frac{a+bz}{c+dz}$ with $ad-bc \neq 0$.
The Question:
Starting with a point in $\mathbb{C}^\infty$, we map it to the sphere via inverse stereographic projection, then we apply a rotation of the sphere and we project it back to the extended complex plane by stereographic projection.
a) Show that any such mapping is a Möbius Transformation.
b) Does every Möbius transformation arise in this way? If so, give a proof. If not, then describe the subset of Möbius transformations that are obtained from a rotation of the sphere.
Intuitively I understand what is happening, when we rotate the sphere about the $z$-axis we get a rotation of the plane, rotating about another axis will cause a translation, dilatations, etc. I drew sketches of the sphere's, complex plane and different mappings, but I am not sure how I am supposed to prove it mathematically using formulas. As for the second question I also don't know where to start.