Let $(E(M),\pi,M)$ be the frame bundle over a manifold $M$ of rank $n$. Consider a covering of $M$ by open neighborhoods $U_{\alpha}$. Let $s_{i}$ and $t_{j}$ where $i,j\in {1,...,n}$ be frames of $M$ over $U_{\alpha}$ and $U_{\beta}$, respectively. Then there is a $g_{ij}\in GL(n,\mathbb{R})$ such that $t_{j}= s_{i}.g_{ij}$.
My question: Why does connection $1-$form $\omega_{U_{\alpha}}$ over $U_{\alpha}$ with values in the Lie algebra of the group $GL(n,\mathbb{R})$ satisfy the following condition, locally?
$$\omega_{U_{\beta}}= g_{ij}^{-1}\omega_{U_{\alpha}}g_{ij}+ g_{ij}^{-1} dg_{ij}$$
Second question:Is there any book on frame bundles that can help me?
Thanks.
Aside from the somewhat long-winded text by Spivak (see volumes 2 and 3), I would recommend Lectures on Differential Geometry by one of the masters of the moving frame, Chern, along with Chen and Lam.