Clutching construction

Remarks

If $f$ and $g$ are homotopic, then $E_f \cong E_g$. Every complex vector bundle of rank $n$ over $S^k$ is isomorphic to some $E_f$. Since $GL_n(\C)$ is connected, this gives a bijection between homotopy classes of maps $S^{k-1} \to GL_n(\C)$ and isomorphism classes of rank $n$ complex vector bundles over $S^k$. The same construction with $GL_n(\R)$ gives real vector bundles; then base points and orientations need more care, as $GL_n(\R)$ has two components.

Example: for $k = 2$ and $n = 1$, the maps $f_m(z) = z^m$ from the unit circle to $\C^\times = GL_1(\C)$ represent all homotopy classes. So the line bundles over $S^2$ are the $E_{f_m}$, $m \in \Z$, classified by the degree $m$. The tautological line bundle over $\C P^1 = S^2$ is $E_{f_m}$ for $m = 1$ or $m = -1$, depending on orientation conventions.

Non-example: the constant map $f = 0$ into the $n \times n$ matrices is continuous, but its values are not invertible, so it is not a map into $GL_n(\C)$. The gluing then collapses fibres over the equator, and the quotient is not a vector bundle.