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 $U(1)$.



Transition Functions of the Principal Bundle $SU(2) \to \mathbb{CP}^1$
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 $g(z)=z/|z|$. 


