= Solution
Use a row of local frame sections $e=(e_1,\ldots,e_m)$ and columns of coefficients: $D(eu)=e(du+Au)$. If $e'=eg$ for a smooth invertible matrix-valued function $g$, then
$$
D(e')=D(eg)=e(Ag+dg)=e'\bigl(g^{-1}Ag+g^{-1}dg\bigr).
$$
Hence the <change of frame of a vector-bundle connection> is
$$
\boxed{A'=g^{-1}Ag+g^{-1}dg.}
$$
This convention specifies both the placement of the frame matrix and the sign of the inhomogeneous term. The <vector-bundle curvature> transforms by $\Theta'=g^{-1}\Theta g$, either by applying $D^2$ to the new frame or by substituting in the formula of the next subpart.
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