= Solution
A rank-$k$ <complex vector bundle> is a smooth map $\pi:E\to B$ whose fibers are $k$-dimensional complex <vector spaces>, with local <diffeomorphisms> $\Phi_i:\pi^{-1}(U_i)\to U_i\times\mathbb C^k$ commuting with projection and linear on every fiber. Define <transition functions of a vector bundle> by
$$
\Phi_i\Phi_j^{-1}(b,v)=(b,g_{ij}(b)v),\qquad g_{ij}:U_i\cap U_j\to GL(k,\mathbb C).
$$
They are smooth and obey
$$
\boxed{g_{ii}=I,\qquad g_{ij}=g_{ji}^{-1},\qquad g_{ij}g_{jk}=g_{ik}}
$$
on the appropriate double/triple overlaps. The <structure group of a vector bundle> is a Lie subgroup $G\subset GL(k,\mathbb C)$ in which a chosen set of transitions takes values. It records allowed changes of fiber frame; different trivializations may exhibit different reductions of that group.
Using the same cocycle, glue the disjoint union of $U_i\times G$ by
$$
(b,h)_j\sim(b,g_{ij}(b)h)_i.
$$
The cocycle makes this an equivalence relation and makes its local charts compatible. Right multiplication $(b,h)_i\cdot a=(b,ha)_i$ is well-defined because it commutes with the left gluing <matrices>. It is free and transitive on each fiber. These charts therefore construct the associated right <principal bundle> $P\to B$ with group $G$.
For the natural line realization, take the <complex tautological line bundle> $L=\{(\ell,v):\ell\in\mathbb{CP}^1,\ v\in\ell\subset\mathbb C^2\}$. Its nonzero vectors identify with $\mathbb C^2\setminus\{0\}$ by $(\ell,v)\mapsto v$, with projection $v\mapsto[v]$. On the two standard charts, use $\zeta=z_2/z_1$ and $\eta=z_1/z_2=1/\zeta$. The local frames are $e_0=(1,\zeta)$ and $e_1=(\eta,1)$, so $e_1=e_0/\zeta$. Normalize them using the ambient Hermitian norm:
$$
u_0=\frac{(1,\zeta)}{\sqrt{1+|\zeta|^2}},\qquad
u_1=\frac{(\eta,1)}{\sqrt{1+|\eta|^2}}
=\frac{|\zeta|}{\zeta}u_0.
$$
If $v=t_0u_0=t_1u_1$, then $t_0=(|\zeta|/\zeta)t_1$. Thus the <unitary transitions of the tautological line over the projective line> are
$$
\boxed{g_{01}(\zeta)=\frac{|\zeta|}{\zeta}\in S^1,\qquad
g_{10}=g_{01}^{-1},\qquad g_{00}=g_{11}=1}.
$$
These are smooth on the overlap $\zeta\ne0$. A <Hermitian metric on a smooth complex vector bundle> generally permits precisely this unitary reduction; here the ambient norm supplies it explicitly.
The corresponding principal circle bundle consists of unit vectors in the tautological line. Sending $(\ell,v)$ to $v$ identifies its total space with $S^3\subset\mathbb C^2$, and the right circle action is $v\cdot e^{i\theta}=e^{i\theta}v$. Identify $\mathbb{CP}^1$ with $S^2$ using
$$
[z_1:z_2]\longmapsto
\frac{(2\operatorname{Re}(z_1\bar z_2),\ 2\operatorname{Im}(z_1\bar z_2),\ |z_1|^2-|z_2|^2)}{|z_1|^2+|z_2|^2}.
$$
Its affine-coordinate inverse on the chart $Z\ne-1$ is $\zeta=(X-iY)/(1+Z)$; the other chart handles the missing pole. This gives a <diffeomorphism>, and the projection becomes the <Hopf fibration>
$$
\boxed{S^3\longrightarrow S^2,\qquad
(z_1,z_2)\longmapsto(2\operatorname{Re}(z_1\bar z_2),2\operatorname{Im}(z_1\bar z_2),|z_1|^2-|z_2|^2)}.
$$
There is a distinction if the stated punctured-bundle identification is interpreted only as a projection-preserving <diffeomorphism>, rather than a fiber-linear identification. It does not determine whether the complex line is tautological or dual: for a nonzero $\phi\in L_\ell^*$, the unique $v\in\ell$ with $\phi(v)=1$ gives a smooth map $L^*\setminus0\to L\setminus0$. Locally it is inversion of a nonzero complex number. Thus <punctured line bundles do not determine their duality sign>.
For completeness, these are the only possible signs here. Complex line bundles over $S^2$ are described by the <clutching construction> with a loop in $\mathbb C^*$, reducible to $S^1$, and its integer <winding number> determines the bundle: a zero-degree ratio has a logarithm and can be removed by changes of trivializations extending over the two disks. For <winding number> $d$, the unit circle bundle has <fundamental group> $\mathbb Z/d\mathbb Z$, as the two solid-torus trivializations identify the common fiber generator and impose its $d$th power as the equatorial gluing relation. There is a fiberwise radial <deformation retraction> from the punctured <line bundle> onto its unit circle bundle. Since $\mathbb C^2\setminus0$ retracts to <simply connected> $S^3$, the hypothesis forces $|d|=1$.
Consequently the allowed unitary transitions are the displayed $g_{01}$ or its reciprocal, according to the bundle's sign. For the dual, the same underlying Hopf projection has the inverse principal action $v\cdot e^{i\theta}=e^{-i\theta}v$. \b[In either case the principal total space and projection are $S^3\to S^2$; the purely diffeomorphic hypothesis alone leaves the circle-action sign unspecified].
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