I am trying to study these notes:
https://www.dpmms.cam.ac.uk/~agk22/vb.pdf
but at page 31 I have some problems in following the construction, starting from a vector bundle $(E,B,\pi)$, of the vector bundles $\Omega^r_B(E)$ and $\Omega^r_B(\operatorname{End} E)$, which are the vector bundles of the differential $r$-forms with values in $E$ and $\operatorname{End} E$ respectively. Which are the trivializations and which is the action of the general linear group on the fibers? Where can I find a detailed construction?
My trial for $r=1$. Let $n$ be the dimension of the base and $m$ the dimension of the fiber. We suppose (but is it an assumption?) that the $U_\alpha$ of the local trivialization $\phi_{\alpha}:\pi^{-1}(U_\alpha)\rightarrow U_\alpha \times V$ are also coordinate patches. Choosing also a base for $V$ we get some coordinates on the fiber $(a_1,...,a_m)$. Now the cotangent bundle $T^*(E)$ gets the base ${dx^1,..,dx^m,da^1,..,da^m}$ and we can write locally any $\rho \in \Omega^1_B(E)$ in the form giving the coordinates:
$a_i=\sum_{a=1}^n dx^a A_{i,a}$
for some matrix $A \in R^{m \times n}$ , which depend on the point $b$ of the basis. (do I have to check something on the dependence on $b$? Or do I have to say that the topology on $\Omega^1_B(E)$ is exactly the one that makes my map continuous?) Therefore we get something like a local trivialization. From here we can also fix the structure group by looking how $A$ change from coordinate patches. This should be a matter of computation since we know the transformation rule for vectors and covectors...
Am I totally out of the way? Is this the right way to construct the bundle $\Omega^1_B(E)$ ??