Find the bilinear transformation which maps the points $2, i, -2$ into the points $1, i, -1$
This is what I did but I couldn't find a get through
$S(z)=\frac{az+c}{cz+d}$ So, $S(2)=1$ , $S(i)=i$ and $S(-2)=-1$
By solving the first part and second part I got $b=2c$ and it is not helping a lot to find the Transformation
I don't have Cross-Ratio in hand. The approach must be using Bilinear Transformation. How to solve this question then?
Let $z_1=2 , z_2=i, z_3=-2, w_1=1, w_2=i, w_3=-1$. Substitute these numbers into this equation $$ \frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)} = \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} $$ and then solve for $w$ in terms of $z$.