A cross-ratio is defined as a linear transformation $S$ which maps $(z2,z3,z4)$ to $(1,0,\infty)$ respectively. I need to prove that it is unique. Now, if i let $T$ be another linear transformation, then i can see that $ST^{-1}$ leaves $1,0, \infty$ invariant, what to do now?
where $$Sz = \frac{z-z3}{z-z4} . \frac{z2-z4}{z2-z3}$$ if possible, can anyone also tell me what would $T^{-1}$ look like