Understanding a linear (Ehresmann) connection on a vector bundle I am trying to read this mathematical introduction to Gauge theory and got stuck on the following:
Suppose $\pi : E \rightarrow M$ is a fibre bundle with fibre $F$ and a connection represented by the vertical and horizontal projections $\pi_p^v$ and $\pi_p^h$ for all $p \in E$.
Now specializing to vector bundles the article states that a connection on $E$ "is linear if $\pi_p^h$ varies affinely with $p$ in a fixed fiber $F_{\pi(p)}$, and the canonical zero section of $E$ is horizontal."
I seem to be unable to wrap my head around what that is supposed to mean.
Unfortunately I have not even found a reference on when a section is supposed to be horizontal!
My guess goes like this: Let $\sigma$ be a section of $E$. Then call $\sigma$ horizontal whenever  
\begin{equation}
\pi_{\sigma(x)}^v \left ( \mathrm{Im} \; \mathrm{d} \sigma(x) \right ) = \{0\} \qquad \forall x \in M
\end{equation}
The first part is troubling me even more, because it makes me feel I have to interpret the expression
\begin{equation}
\pi_p^h - \pi_q^h
\end{equation}
as the result of a linear map $R$ acting on $(p - q)$ for arbitrary $p$ and $q$ in the same fibre. But I do not see how this could be possible without somehow identifying $T_p E$ and $T_q E$. And what would the codomain of $R$ be?
 A: All the notations come from the paper linked in your question, especially p.27 & p.28
You are right about what is an horizontal section. Locally it means that $s$ is a horizontal section if for all $X\in\mathfrak{X}(U)$ we get for all $x\in U$ 
$$\pi^{v}_{s(x)}(\text{d}s_x(X_{x}))=0$$
If you look at the canonical zero section $0_U:x\to (x,0)$ we get $(\text{d}0_U)_{x}:\xi\to (\xi,0)$, so $\Gamma(x,0)=0$.
If you look now at the affine condition, it means an affine dependency of $\pi^{h}_p$ for $p\in\pi(p)=x$. In the paper linked, a way of identifying all the distinct $TE_p$ locally on $U$ with $p=(x,f)$ is given, by identifying $TE_{(x,f)}$ with $T_xU\times F$, for all distinct $f$, that is to say for all distinct $p$ in the same fiber.
In that case, $\Gamma(x,f)$ is identified with a linear map $T_xU\to F$, and $\pi^h_{(x,f)}(\xi,y)=(\xi,-\Gamma(x,f)\xi)$ must depend the affine way of $f$, and since $\Gamma(x,0)=0$, must depend linearly of $f$, which means exactly $$\Gamma(x)\in\text{Hom}(F,\text{Hom}(T_xU,F))\cong\text{Hom}(T_xU,F^*\otimes F)$$
