Doubt on how to prove proposition about bundles I've started studying bundles and fiber bundles and to get some practice I've tried to prove the following proposition: "Every vector bundle $(E,B,\pi,F,G)$ is associated to a given principal bundle $(E',B',\pi',G',G')$."
I've been told that this proposition is true, but as a first time trying to deal with this kind of thing I'm a little confused yet. Can someone give me just a little hint on how to start this proof? I don't want a full answer, because I want to do it by myself, I just want a hint on how to start.
Thanks very much in advance!
 A: Suppose you have a trivialization of $\pi: E \to B$ (with fiber $F$), so that you have a covering $U_\alpha$ of $B$ and a bundle isomorphism $\phi_\alpha$ between $\pi^{-1}(U_\alpha) \to U_\alpha$ and $p_\alpha =\text{proj}: U_\alpha \times F \to U_\alpha$; the isomorphism should respect of course the $G$-action on fibers. On intersections $U_\alpha \cap U_\beta$ you can then put these together to construct a bundle isomorphism from $(U_\alpha \cap U_\beta) \times F$ to itself, lying over the identity on $U_\alpha \cap U_\beta$. This in turn is given by a map $U_\alpha \cap U_\beta \to \text{Aut}(F)$ that factors through the action $G \to \text{Aut}(F)$. (Cf. Wikipedia on local trivialization; normally we require the action on $F$ to be faithful.) 
So the structure of the bundle can be described in terms of a family of maps $\phi_{\alpha \beta}: U_\alpha \cap U_\beta \to G$, called &Ccaron;ech cocycle data; i.e., the original bundle is retrieved by taking all the trivial bundles $p_\alpha: U_\alpha \times F \to U$, and gluing them together by identifying the bundle restrictions of $p_\alpha, p_\beta$ over $U_\alpha \cap U_\beta$ along the isomorphism induced by $\phi_{\alpha \beta}$. 
The point is that you can use the same cocycle data to create a principal bundle, replacing the action $G \to \text{Aut}(F)$ by the canonical action $G \to \text{Aut}(G)$ (say as left $G$-spaces). There is a nice description of the original bundle in terms of this principal bundle $E' \to B$, which can roughly be expressed as $E \cong E' \otimes_G F$, where the tensor product is meant to indicate identifications $(e g, f) \sim (e, g f)$ (NB: there is a left $G$-bundle map $E' \times G \to E'$, parallel to the fact that the right $G$-action of $G$ on itself is a left $G$-space map by virtue of the associative law). 
More details can be found in Wikipedia (there is an article "associated bundle"), but this should be a start. Or did you want something more? 
