Charts for the complex projective line .

Question

Let $\mathbb{C}P^1$ be the complex projective line quotient of $\mathbb C^2 \setminus \{0\}$ by the equivalence relation :
$$(z_1,z_2) \sim \;(z_1',z_2') \iff \exists \rho \in \mathbb C^*, \quad (z_1',z_2')=\rho (z_1,z_2) $$ For $i=1,2$, let $U_i=\{[z_1,z_2] \in \mathbb{C}P^1 : z_i \neq0\}$ and $\phi_i : U_i \to \mathbb C$ defined by : $$\phi_i([z_1,z_2])=\frac{z_2}{z_1} \text{ and } \phi_2([z_1,z_2])= \frac{z_1}{z_2}$$ I want to find $\phi_i ^{-1} $ for $i=1,2.$ I didn't know how to find it. Thanks in advance.

Answer 1

Verify that $\phi_1^{-1}(z)=[1,z]$, and similarly for $\phi_2^{-1}$.