I'm working through Lee's "Riemannian Manifolds: an Introduction to Curvature" by myself, and I'm a bit stuck on problem 4-5:
Let $\nabla$ be a connection on $M$, let $\{E_i\}$ be a local frame on some open subset $U\subset M$, and let $\{\phi^i\}$ be the dual coframe. Show that there is a uniquely determined matrix of $1$-forms $\omega_i^j$ on $U$, called the connection $1$-forms for this frame, such that
\begin{equation} \nabla_XE_i=\omega_i^j(X)E_j \end{equation} for all $X\in TM$.
I know my understanding of differential forms is lacking, but I don't even understand where they come into this problem... Are we just saying that, since the covariant derivative on a section produces a new section, there must be a linear transformation between them, and this linear transformation depends on the direction $X$?
It is exactly what you think, since $E_i$ is a local frame $\{E_i(x)\}$ is a basis of $T_xM$ for every $x\in U$; $\nabla_XE_i$ is an element of $TM_{\mid U}$, there exits $\omega_i(x)$ such that $\nabla_XE_i(x)=\sum_j\omega_i^j(x)E_i(x)$.