I've been trying to understand principal bundles, and to that end have been looking at the bundle $$ \pi: SU(2) \to \mathbb{CP}^1,~~~ (a_{ij}) \mapsto [a_{11},a_{21}], $$ with fibre $U(1)$. I assumed that the bundle would be trivial over the standard nbds $U_1,U_2 \subset \mathbb{C}$, but can't seem to identify the local trivializations. Now $$ \pi^{1}(U_1) = \{\left( \array{a &  \overline{b}\\\ b & \overline{a}} \right)~ a \neq 0\}, ~~~ \pi^{1}(U_2) = \{\left( \array{a &  \overline{b}\\\ b & \overline{a}} \right)~ b \neq 0\}, $$ and any trivialization $\alpha_1:\pi^{1}(U_i) \to U_i \times U(1)$, will map $$ \alpha_1:\left( \array{a &  \overline{b}\\\ b & \overline{a}} \right) \mapsto ([a,b],h_{a,b}^1), $$ for some $h_{a,b}^1$. Defining $h^1_{a,b} = arg(a) = \frac{a}{a}$, and similarly $h^2$, works, but then the transition functions are not in $U(1)$.
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You have two charts $U_0=\Bbb C$ and $U_\infty=(\Bbb C\setminus 0)\cup \infty$. The transition function on the intersection $\Bbb C\setminus 0$ is $g(z)=z/z$. 

