I was reading this question [link text](https://mathoverflow.net/questions/5772/principal-bundles-representations-and-vector-bundles)
and can't seem to see why, if $\pi: P \to B$ is a principle $G$-bundle and $$\rho:G \to GL\_n(\mathbb{C})$$ is a representation of $G$, then the total space $P \times\_{\rho} \mathbb{C}^n$ is locally trivial.



Local Triviality of an Associated Bundle
A principal G-bundle $\pi: P \to B$ is locally equivalent to a product. Depending on who you ask, this is either part of the definition, or a short lemma. It means that there is a cover of B by open sets U, together with bundle isomorphisms $\alpha\_U: \pi^{-1}U \to G \times U$ that are both G-equivariant, and induce transition sections in G: see [Wikipedia](http://en.wikipedia.org/wiki/Principal_bundle#Trivializations_and_cross_sections).


The associated bundle construction is a quotient of $P \times \mathbb{C}^n$ by an equivalence relation given by the actions of G on the left and right factors. Since $\alpha\_U$ is G-equivariant, the map $\alpha\_u \times id: \pi^{-1}U \times \mathbb{C}^n \to G \times U \times \mathbb{C}^n$ is also a G-equivariant bundle map, so you get bundle isomorphisms on the quotient bundles: $\alpha\_u \times\_\rho id: \pi^{-1}U \times\_\rho \mathbb{C}^n \to U \times \mathbb{C}^n$.


