I am trying to solve the following problem:
I have to admit I am completly stuck and don't even know where to start. The notes I am reading introduce $E_{1}\oplus E_{1}$ rather briefly. I had a look at Lee's books and I found a short discussion about the direct sum of two vector bundles. I think I am fine with $E_{1}\oplus E_{1}$ being a vector bundle and that a connection on $E_{1}\oplus E_{1}$ can be defined by $\nabla:=\nabla_{1}\oplus\nabla_{2}$. What puzzles me is the matrix representation of a connection. We never talked about anything like a correspondence between connections and matrices. Furthermore forms are introduced in the next chapter. Yet two entries are forms. I guess, since they are 1-forms, they are "only" linear maps.
I would really appreciate some tips or source where I can look it up or that explains this "correspondence" so that I can solve the problem myself.
Many thanks in advance!
If $\nabla$ and $\nabla'$ are connections on $E$, then their difference $\nabla - \nabla'$ is an "endomorphism-valued $1$-form" $B \in \Omega^1(M,\mathrm{End}(E))$. If $E=E_1 \oplus E_2$ then we can define the connection $\nabla^1$ by
$$ \nabla^1_X(s_1) = \mathrm{pr}_1 \nabla_X(s_1) $$
for a (local) section $s_1$ of $E_1$, where $\mathrm{pr}_1 \colon E \to E_1$ is the projection defined by the direct sum. You can do the same for $\nabla^2$. Then $\nabla-\nabla^1-\nabla^2$ is an endomorphism-valued $1$-form on $E$, and taking a look at the definition you see that
$$ \tilde{A}_X(s_1) = \mathrm{pr}_2 \nabla_X(s_1) .$$