Starting with a principal $GL\_n$ bundle, we could form the vector bundle associated to a representation $GL\_n\to GL(V)$. Could someone please explain how we get a connection on this vector bundle. 



Various definitions of Connections on bundles-2
Let $P \to M$ is a $G$-principal bundle with a connection $\omega$ on it and $R:G \to GL (V)$ is a representation, inducing a Lie algebra map $r: \mathfrak{g} \to \mathfrak{gl}(V)$. Assume that $P$ is trivial and pick a section $s$ of $P$. The section identifies sections of $P \times\_G V$ with functions $M \to V$ and $s^\* \omega$ is a $\mathfrak{g}$-valued $1$-form. The formula becomes (sections written as $V$-valued functions, $X$ a vector field)


$$
\nabla\_X v:= (dv)(X) \pm r(s^\* \omega)(X)\cdot v.
$$


This is a connection on $V$ and it does not depend on the choice of $s$ when the sign is correctly chosen (exercise). The independence on $s$ shows that you can do these things locally, so that the procedure works for nontrivial bundles as well.


