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 $U(1)$.