Question

prove that the Möbius transformation is a sequence of 4 more simple transformations involving 2 translations, inversion/reflection,dilatation/rotation

Answer 1

\[
f_1(z)= c z +d\\
f_2(z)=\frac 1 z\\
f_3(z) =\frac {bc -ad} {c}z\\
f_4(z) = z+ \frac a c\\
f_4\circ f_3\circ f_2\circ f_1(z) = \frac { a z + b}{c z + d}
\]

Answer 2

i know this, but I need to prove it

Answer 3

All you have to do is to compute the last line, which is pretty easy to do.