= Solution
A <connection on a vector bundle> is a linear operator $\nabla:\Gamma(E)\to\Omega^1(M;E)$ satisfying $\nabla(fs)=df\otimes s+f\nabla s$. Choose coordinates $x^i$ and a <local frame> $e_1,\ldots,e_m$, and define the <connection matrix> by $\nabla e_b=\sum_a A^a{}_b\otimes e_a$, with $A^a{}_b=\sum_iA^a{}_{b i}dx^i$. For a section $s=\sum_a s^ae_a$, the coefficient column satisfies
$$
\boxed{d_As=ds+As,\qquad
(\nabla_i s)^a=\partial_i s^a+\sum_b A^a{}_{b i}s^b.}
$$
The symbol $d$ on the right means the ordinary <exterior derivative> of the coefficient functions.
Extend uniquely to $E$-valued <differential forms> by the graded <Leibniz rule>
$$
d_A(\alpha s)=d\alpha\otimes s+(-1)^r\alpha\wedge\nabla s,
\qquad \alpha\in\Omega^r(M).
$$
In the same <local trivialization>, an $E$-valued form is a column $\sigma=(\sigma^a)$ and the <covariant exterior derivative> is
$$
\boxed{(d_A\sigma)^a=d\sigma^a+\sum_b A^a{}_b\wedge\sigma^b,\qquad d_A=d+A\wedge.}
$$
Moving the one-form $A$ past the scalar $r$-form produces exactly the $(-1)^r$ in the graded rule. This explains the sign rather than replacing the rule by an ungraded product formula.
The <endomorphism bundle> has the induced <endomorphism bundle connection> characterized on sections by $\nabla^{\operatorname{End}}_X(T)s=\nabla_X(Ts)-T\nabla_Xs$. Its coefficient formula is $dT+AT-TA$. On an <endomorphism>-valued $r$-form $T$, extend as a <graded derivation> of the <endomorphism-valued exterior product>, giving
$$
\boxed{d_A^{\operatorname{End}}T=dT+A\wedge T-(-1)^rT\wedge A.}
$$
Explicitly its $(a,b)$ entry is $dT^a{}_b+\sum_cA^a{}_c\wedge T^c{}_b-(-1)^r\sum_cT^a{}_c\wedge A^c{}_b$. This version is characterized by
$$
d_A(T\wedge\sigma)=(d_A^{\operatorname{End}}T)\wedge\sigma+(-1)^rT\wedge d_A\sigma.
$$
The <endomorphism>-valued version also satisfies $d_A(T\wedge S)=d_AT\wedge S+(-1)^rT\wedge d_AS$. Products here combine composition of <endomorphisms> with wedge products of coefficient forms; <matrix> factors must retain their order.
These local formulas are independent of the <local frame>. For $e'=eg$, the <change of frame of a vector-bundle connection> gives $A'=g^{-1}Ag+g^{-1}dg$, $\sigma'=g^{-1}\sigma$ and $T'=g^{-1}Tg$. Using $d(g^{-1})=-g^{-1}(dg)g^{-1}$ gives $d_{A'}\sigma'=g^{-1}d_A\sigma$ and $d_{A'}T'=g^{-1}(d_AT)g$. Thus both extended operators are globally defined.
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