# The Bilinear map that sends $(z_1,z_2,z_3)\to (\infty,0,1)$

The Bilinear map that sends $(z_1,z_2,z_3)\to (\infty,0,1)$ is given by $f(z)=\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}$

Can I extract any expression which sends the other around? I mean

$(\infty,0,1)\to (z_1,z_2,z_3)$ ?

I know one expression sending any $(z_1,z_2,z_3)\to (w_1,w_2,w_3)$ but I can not remember that expression. is there any way to construct meself from the above one which I always remember by $(23132)$

• Just compute the inverse of the map you have supplied. – mrf Sep 12 '13 at 10:42
• How to do that @mrf – Marso Sep 13 '13 at 5:50

You make the Ansatz $$f(z)={az+b \over cz+d}$$Then $f(\infty)=\frac ac=z_1, f(0)=\frac bd=z_2$ and $f(1)=\frac {a+b}{c+d}=z_3$. Now solve for $a,b,c,d$ noting that the answer is unique up to linear multiples (i.e. if $(a,b,c,d)$ a solution, then also $(\lambda a,\lambda b,\lambda c,\lambda d)$ is a solution for $\lambda\neq 0$).