[proofplan]
The computation is a direct application of the Leibniz rule satisfied by any connection. Writing the new frame $F'$ as $\psi \cdot F$ and applying $d_A$ to the product of the matrix-valued function $\psi$ and the frame $F$, we pick up a term $d\psi \cdot F$ from differentiating $\psi$ and a term $\psi \cdot d_A F = \psi \cdot (\theta F)$ from differentiating $F$. Inserting $\psi^{-1}\psi = I$ to re-express everything in the frame $F'$ yields the formula. The converse (gluing) is a straightforward check that the compatibility condition ensures a globally well-defined connection.
[/proofplan]
[step:Express the connection in the original frame $F$]
By the defining property of the connection matrix in the frame $F$,
\begin{align*}
d_A: \Gamma(E|_U) &\to \Gamma(E|_U) \otimes \Omega^1(U), \\
\sum_i c_i e_i &\mapsto \sum_i (dc_i) e_i + \sum_i c_i\, (d_A e_i),
\end{align*}
with $d_A e_i = \sum_j \theta_{ij} e_j$. In matrix form, writing the frame as a column vector $F = (e_1, \ldots, e_k)^\top$,
\begin{align*}
d_A F = \theta \cdot F,
\end{align*}
where the product on the right is matrix-valued $1$-form $\theta \in \Omega^1(U; \mathfrak{gl}_k)$ acting on the column vector $F$ of sections.
[/step]
[step:Apply the Leibniz rule to $d_A(F') = d_A(\psi \cdot F)$]
The frame $F' = \psi \cdot F$ is a $C^\infty(U)$-linear combination of the $e_j$'s with smooth coefficients $\psi_{ij}$. By the Leibniz rule for connections (see [Leibniz Rule for Connections](/theorems/???)), for a smooth function $f: U \to \mathbb R$ and a section $s \in \Gamma(E|_U)$,
\begin{align*}
d_A(f \cdot s) = df \cdot s + f \cdot d_A s.
\end{align*}
Applied component-wise to the matrix product $\psi \cdot F$:
\begin{align*}
(d_A(\psi \cdot F))_i = d_A\!\left(\sum_j \psi_{ij} e_j\right) = \sum_j (d\psi_{ij})\, e_j + \sum_j \psi_{ij}\, (d_A e_j).
\end{align*}
In matrix form,
\begin{align*}
d_A(\psi \cdot F) = d\psi \cdot F + \psi \cdot d_A F = d\psi \cdot F + \psi \cdot (\theta \cdot F) = (d\psi + \psi \theta) \cdot F,
\end{align*}
using the associativity of matrix multiplication of matrix-valued $1$-forms acting on the frame.
[/step]
[step:Re-express the result in the new frame $F' = \psi \cdot F$]
We want the connection matrix $\theta'$ satisfying $d_A F' = \theta' \cdot F'$. We have $F = \psi^{-1} \cdot F'$ (by $\mathrm{GL}_k$-invertibility of $\psi$). Substituting into the expression for $d_A F'$ from the previous step:
\begin{align*}
d_A F' = d_A(\psi \cdot F) = (d\psi + \psi \theta) \cdot F = (d\psi + \psi\theta) \cdot (\psi^{-1} \cdot F') = \left[(d\psi) \psi^{-1} + \psi \theta \psi^{-1}\right] \cdot F'.
\end{align*}
Comparing with $d_A F' = \theta' \cdot F'$ and using that the expansion of a section in a frame is unique (the frame is a basis at each point), we conclude
\begin{align*}
\theta' = (d\psi)\psi^{-1} + \psi\, \theta\, \psi^{-1}.
\end{align*}
[guided]
Step back and think about why the extra $(d\psi)\psi^{-1}$ term appears. The connection matrix $\theta$ is *not* a tensor — it is not the coordinate expression of an intrinsic object. Connections are *affine* objects: the difference of two connections is tensorial, but a single connection is not. Under a frame change, $\theta$ transforms by an affine (inhomogeneous) rule rather than a linear one, and the inhomogeneous piece $(d\psi)\psi^{-1}$ is exactly the Maurer-Cartan form of the transition matrix $\psi$.
Let us recompute explicitly. We have $F' = \psi F$. Applying $d_A$ using the Leibniz rule:
\begin{align*}
d_A F' = d_A(\psi F) = (d\psi) F + \psi (d_A F) = (d\psi) F + \psi \theta F.
\end{align*}
Now we need to re-express the right-hand side in terms of the new frame $F' = \psi F$. Substituting $F = \psi^{-1} F'$:
\begin{align*}
d_A F' = (d\psi)\psi^{-1} F' + \psi\theta\psi^{-1} F' = [(d\psi)\psi^{-1} + \psi\theta\psi^{-1}] F'.
\end{align*}
Hence $\theta' = (d\psi)\psi^{-1} + \psi\theta\psi^{-1}$.
The first term $(d\psi)\psi^{-1}$ captures the *infinitesimal* change of frame — it is the left-invariant Maurer-Cartan form of the function $\psi: U \to \mathrm{GL}_k$. The second term $\psi\theta\psi^{-1}$ is the conjugation that would appear if $\theta$ were tensorial. The sum is the right rule for an affine object.
[/guided]
[/step]
[step:Verify that the compatibility condition is sufficient to define a global connection]
For the converse, let $\{U_\alpha\}$ be a trivialising open cover of $M$ for $E$, with local frames $F_\alpha = (e^{(\alpha)}_1, \ldots, e^{(\alpha)}_k)$ on each $U_\alpha$. On each overlap $U_\alpha \cap U_\beta$, the frames are related by a transition matrix
\begin{align*}
\psi_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{GL}_k(\mathbb R), \qquad F_\beta = \psi_{\alpha\beta} \cdot F_\alpha.
\end{align*}
Suppose we are given matrices $\theta_\alpha \in \Omega^1(U_\alpha; \mathfrak{gl}_k)$ satisfying, on each overlap,
\begin{align*}
\theta_\beta = (d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + \psi_{\alpha\beta}\, \theta_\alpha\, \psi_{\alpha\beta}^{-1}.
\end{align*}
On each $U_\alpha$, define a connection $d_A^{(\alpha)}$ by
\begin{align*}
d_A^{(\alpha)}\!\left(\sum_i c_i e^{(\alpha)}_i\right) = \sum_i (dc_i)\, e^{(\alpha)}_i + \sum_{i,j} c_i\, (\theta_\alpha)_{ij}\, e^{(\alpha)}_j.
\end{align*}
This is a genuine connection on $E|_{U_\alpha}$ (it satisfies the Leibniz rule by construction).
*Claim.* The local connections $d_A^{(\alpha)}$ agree on overlaps $U_\alpha \cap U_\beta$, i.e. for any $s \in \Gamma(E|_{U_\alpha \cap U_\beta})$,
\begin{align*}
d_A^{(\alpha)} s = d_A^{(\beta)} s.
\end{align*}
*Proof of claim.* Fix $s \in \Gamma(E|_{U_\alpha \cap U_\beta})$. In the frame $F_\alpha$, write $s = c_\alpha \cdot F_\alpha$ (a row-vector of smooth coefficients). Then $s = c_\alpha \cdot F_\alpha = c_\alpha \cdot \psi_{\alpha\beta}^{-1} \cdot F_\beta = c_\beta \cdot F_\beta$ with $c_\beta = c_\alpha \psi_{\alpha\beta}^{-1}$.
Computing $d_A^{(\alpha)} s$ in the $F_\alpha$ frame:
\begin{align*}
d_A^{(\alpha)} s = (dc_\alpha + c_\alpha \theta_\alpha) \cdot F_\alpha = (dc_\alpha + c_\alpha \theta_\alpha) \psi_{\alpha\beta}^{-1} \cdot F_\beta.
\end{align*}
Computing $d_A^{(\beta)} s$ in the $F_\beta$ frame:
\begin{align*}
d_A^{(\beta)} s = (dc_\beta + c_\beta \theta_\beta) \cdot F_\beta.
\end{align*}
We verify that the coefficients in the $F_\beta$ frame agree:
\begin{align*}
dc_\beta &= d(c_\alpha \psi_{\alpha\beta}^{-1}) = (dc_\alpha) \psi_{\alpha\beta}^{-1} + c_\alpha\, d(\psi_{\alpha\beta}^{-1}), \\
c_\beta \theta_\beta &= c_\alpha \psi_{\alpha\beta}^{-1} \big[(d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + \psi_{\alpha\beta}\theta_\alpha\psi_{\alpha\beta}^{-1}\big] = c_\alpha\big[\psi_{\alpha\beta}^{-1}(d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + \theta_\alpha \psi_{\alpha\beta}^{-1}\big].
\end{align*}
Using the identity $d(\psi_{\alpha\beta}^{-1}) = -\psi_{\alpha\beta}^{-1}(d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1}$ (obtained by differentiating $\psi_{\alpha\beta}\psi_{\alpha\beta}^{-1} = I$),
\begin{align*}
dc_\beta + c_\beta \theta_\beta = (dc_\alpha)\psi_{\alpha\beta}^{-1} - c_\alpha \psi_{\alpha\beta}^{-1}(d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + c_\alpha \psi_{\alpha\beta}^{-1}(d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + c_\alpha \theta_\alpha \psi_{\alpha\beta}^{-1},
\end{align*}
and the middle two terms cancel:
\begin{align*}
dc_\beta + c_\beta \theta_\beta = (dc_\alpha + c_\alpha \theta_\alpha)\psi_{\alpha\beta}^{-1}.
\end{align*}
Hence $d_A^{(\beta)} s = d_A^{(\alpha)} s$ on $U_\alpha \cap U_\beta$. The local connections glue to a well-defined global connection $d_A: \Gamma(E) \to \Gamma(E) \otimes \Omega^1(M)$.
[guided]
The converse direction — gluing — is a checking computation. The key algebraic fact used twice is the derivative of the inverse: for a smooth map $\psi: U \to \mathrm{GL}_k$, differentiating the identity $\psi \psi^{-1} = I_k$ gives
\begin{align*}
0 = d(\psi \psi^{-1}) = (d\psi)\psi^{-1} + \psi\, d(\psi^{-1}),
\end{align*}
hence
\begin{align*}
d(\psi^{-1}) = -\psi^{-1}(d\psi)\psi^{-1}.
\end{align*}
This is the non-commutative analogue of the scalar identity $d(1/f) = -f^{-2}df$.
With this in hand, the compatibility condition $\theta_\beta = (d\psi_{\alpha\beta})\psi_{\alpha\beta}^{-1} + \psi_{\alpha\beta}\theta_\alpha\psi_{\alpha\beta}^{-1}$ is precisely what is needed for the local formula $d_A^{(\alpha)}s = (dc_\alpha + c_\alpha\theta_\alpha)F_\alpha$ to transform covariantly under a change of frame. The middle two terms cancel because one comes from differentiating the transition matrix itself (contribution to $d c_\beta$) and the other comes from the inhomogeneous part of the transformation rule for $\theta$; they are set up to cancel exactly. This is the whole content of the compatibility condition: it makes the local recipes agree on overlaps.
A useful perspective: the set of connections on a fixed trivialising cover is an *affine space* modelled on $\Omega^1(M; \operatorname{End}(E))$. The inhomogeneous term $(d\psi)\psi^{-1}$ in the transformation rule is exactly the obstruction preventing connection matrices from being elements of a vector space. The compatibility condition is the cocycle condition identifying the affine space.
[/guided]
[/step]
[step:Conclude the bidirectional statement]
The forward direction establishes that under a change of frame $F' = \psi F$, the connection matrix transforms as $\theta' = (d\psi)\psi^{-1} + \psi\theta\psi^{-1}$. The converse direction establishes that local matrix-valued $1$-forms $\{\theta_\alpha\}$ on a trivialising cover satisfying this transformation law on overlaps glue to a globally defined connection $d_A$ on $E$. This completes the proof of both directions.
[/step]
