There this theorem that states the Linear Fractional Transformations map circles and straight lines to circles and straight lines. How can you prove this?
This is what the theorem states in my lecture notes: "Every Linear Fractional Transformation maps circles and straight lines in the $z$-plane on circles and straight lines in the $w$-plane (but not necessarily circles onto circles and straight lines onto straight lines)."
I've tried showing the $w=1/z$ does it by writing the equation of a circle with $z$ then with $z=1/w$ and showed that we have circles in both cases. Then you can argue that the Linear Fractional Mapping formula is a combination of rotations and translations of $w=1/z$.
However this does not seem to be enough. Any thoughts ?
Many thanks !