Why is the Cross-ratio $ \lambda (z_1 , \dots , z_4 ) $ the image of $ z_4 $ under $ T $ satisfying some conditions?

Question

Could someone please explain to me why is the cross-ratio : $$ \lambda ( z_1 , \dots , z_4 ) = \dfrac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_3)(z_2 - z_4)} $$ the image of $ z_4 $ under the ( unique ) linear map $ T \ : \ \mathbb{P}^1 \to \mathbb{P}^1 $ such that : $ \begin{cases} T(z_1) = 1 \\ T(z_2) = \infty \\ T(z_3) = 0 \end{cases} $ ?

Thanks in advance for your help.

Answer 1

Fix $z_1, z_2, z_3 \in \mathbb{CP}^1$. Consider the map $T : \mathbb{CP}^1 \to \mathbb{CP}^1$ given by

$$T(z) = \frac{(z_1 - z_2)(z_3 - z)}{(z_1 - z_3)(z_2 - z)}.$$

Then $T(z_1) = 1$, $T(z_2) = \infty$, and $T(z_3) = 0$. Now note that

$$T(z_4) = \frac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_3)(z_2 - z_4)} = \lambda(z_1, z_2, z_3, z_4).$$