I am having trouble to understand how one may write down the connection in terms of local trivialization of the vector bundle.
Assume $\pi: E\rightarrow X$ is a vector bundle with rank $n$. A connection on $E$ is a map $A:TE\rightarrow \pi^{*}E$ such that $A$ is a splitting of the sequence $0\rightarrow \pi^{*}E\rightarrow TE\rightarrow \pi^{*}TX\rightarrow 0$ and commutes with multiplication by scalars. Here the map from $\pi^{*}E$ to $TE$ is defined by $(e,v)\rightarrow (e,\frac{d}{dt}|_{t=0}(e+tv))$
The author (Cliff Taubes) asserts that if we let $x_{i}$ be the $n$ corresponding trivialization functions $E_{U}\rightarrow U\times \mathbb{C}^{n}$ such that we have $x(e)=(\pi(e),(x^{1}(e)...x^{n}(e))$ Then the connection $A$ takes value in $\mathbb{C}^{n}$ (viewed as a one form on $E$ with values in $\pi^{*}E$). So far I can follow since $\pi^{*}E$ has extra $n$ dimensions. But I feel at loss with the following assertion because I do not know how to construct an inverse to form a splitting map:
Let $A^{a}$ be the coordinates, then we can write $A^{a}=dx^{a}+A^{ab}x^{b}$ Here $A^{ab}$ is a 1-form pulled back from $U$. We can think of $A$ as an End($\mathbb{C}^{n}$)valued 1-form on $U$.
My questions are:
1): How do I get this formula? How should I interpret it in terms of the exact sequence?
2): How can I show any connection $A$ can written in this form, and any $A^{ab}$ defined on $U$ is suffice to define $A$?
3): There is a marked difference between this definition and the definition on wikipedia, for example in wikipedia connection is defined to be a map from sections to sections. What is the reason for the discrepancy in the language?