[proofplan] The transition functions of $E$ satisfy the identity and cocycle relations because they come from composing local trivializations. Since $\rho$ is a [group homomorphism](/page/Group%20Homomorphism), applying $\rho$ preserves these relations and gives a smooth $GL(V)$-valued cocycle. We then construct the associated bundle by gluing the products $U_i\times V$ using this cocycle, verify that the quotient has compatible smooth local trivializations, and finally check that a change of frame produces an isomorphic glued bundle. [/proofplan]

[step:Apply the representation to the transition cocycle]For each pair $i,j\in I$, define the map $h_{ij}:U_i\cap U_j\to GL(V)$ by $h_{ij}(x)=\rho(g_{ij}(x))$ for $x\in U_i\cap U_j$. Since $g_{ij}$ is smooth and $\rho$ is smooth, the composition $h_{ij}=\rho\circ g_{ij}$ is smooth. Let $I_k$ denote the identity element of $GL(k,\mathbb{R})$. The transition functions of $E$ satisfy $g_{ii}(x)=I_k$ for every $x\in U_i$. They also satisfy $g_{ij}(x)g_{jk}(x)=g_{ik}(x)$ for every $x\in U_i\cap U_j\cap U_k$. Since $\rho$ is a group homomorphism, for $x\in U_i$ we have \begin{align*} h_{ii}(x)=\rho(g_{ii}(x))=\rho(I_k)=\operatorname{id}_V. \end{align*} For $x\in U_i\cap U_j\cap U_k$, we similarly obtain \begin{align*} h_{ij}(x)h_{jk}(x)=\rho(g_{ij}(x))\rho(g_{jk}(x))=\rho(g_{ij}(x)g_{jk}(x))=\rho(g_{ik}(x))=h_{ik}(x). \end{align*} Thus $(h_{ij})$ is a smooth $GL(V)$-valued cocycle on the open cover $\{U_i\}_{i\in I}$.[/step]

[guided]The point of this step is that the representation $\rho$ converts changes of frame in $\mathbb{R}^k$ into changes of frame in $V$, and it does so compatibly with multiplication. For each pair $i,j\in I$, define the map $h_{ij}:U_i\cap U_j\to GL(V)$ by $h_{ij}(x)=\rho(g_{ij}(x))$ for $x\in U_i\cap U_j$. This is a legitimate transition map candidate because $g_{ij}(x)\in GL(k,\mathbb{R})$ and $\rho$ maps $GL(k,\mathbb{R})$ into $GL(V)$. It is smooth because it is the composition of the smooth map $g_{ij}:U_i\cap U_j\to GL(k,\mathbb{R})$ with the smooth map $\rho:GL(k,\mathbb{R})\to GL(V)$. Now we check the two cocycle identities. Since the maps $g_{ij}$ are transition functions for the vector bundle $E$, they satisfy $g_{ii}(x)=I_k$ for every $x\in U_i$, and they satisfy $g_{ij}(x)g_{jk}(x)=g_{ik}(x)$ for every $x\in U_i\cap U_j\cap U_k$. The representation $\rho$ is a group homomorphism, so it preserves identity elements and products. Hence, for $x\in U_i$, \begin{align*} h_{ii}(x)=\rho(g_{ii}(x))=\rho(I_k)=\operatorname{id}_V. \end{align*} For a triple overlap point $x\in U_i\cap U_j\cap U_k$, we compute \begin{align*} h_{ij}(x)h_{jk}(x)=\rho(g_{ij}(x))\rho(g_{jk}(x))=\rho(g_{ij}(x)g_{jk}(x))=\rho(g_{ik}(x))=h_{ik}(x). \end{align*} This proves that the maps $h_{ij}$ satisfy precisely the transition-function identities required to glue the local products $U_i\times V$ into a vector bundle with fibre $V$.[/guided]

[step:Glue the local products using the represented cocycle]Let \begin{align*} X:=\bigsqcup_{i\in I}(U_i\times V\times\{i\}) \end{align*} be the disjoint union of the local model spaces. Define a relation $\sim$ on $X$ by declaring \begin{align*} (x,v,i)\sim (y,w,j) \end{align*} if and only if $x=y\in U_i\cap U_j$ and \begin{align*} v=h_{ij}(x)w. \end{align*} The identity $h_{ii}=\operatorname{id}_V$ gives reflexivity. For symmetry, if $x\in U_i\cap U_j$, then the cocycle identity with indices $i,j,i$ gives \begin{align*} h_{ij}(x)h_{ji}(x)=h_{ii}(x)=\operatorname{id}_V, \end{align*} and the cocycle identity with indices $j,i,j$ gives \begin{align*} h_{ji}(x)h_{ij}(x)=h_{jj}(x)=\operatorname{id}_V. \end{align*} Thus $h_{ji}(x)=h_{ij}(x)^{-1}$, so $(x,v,i)\sim (x,w,j)$ implies $(x,w,j)\sim (x,v,i)$. For transitivity, if $(x,v,i)\sim (x,w,j)$ and $(x,w,j)\sim (x,z,k)$, then $v=h_{ij}(x)w$ and $w=h_{jk}(x)z$, so \begin{align*} v=h_{ij}(x)h_{jk}(x)z=h_{ik}(x)z. \end{align*} Hence $(x,v,i)\sim (x,z,k)$, and $\sim$ is an [equivalence relation](/page/Equivalence%20Relation). Define \begin{align*} E_\rho:=X/\sim \end{align*} and let $q:X\to E_\rho$ be the quotient map. Define $\pi_\rho:E_\rho\to M$ by \begin{align*} \pi_\rho(q(x,v,i))=x. \end{align*} This is well-defined because equivalent triples have the same base point. For each $i\in I$, define $\Psi_i:\pi_\rho^{-1}(U_i)\to U_i\times V$ as follows. If a class in $\pi_\rho^{-1}(U_i)$ is represented by $q(x,w,j)$, then $x\in U_i\cap U_j$, and the defining relation gives \begin{align*} (x,h_{ij}(x)w,i)\sim (x,w,j). \end{align*} Thus every class over $U_i$ has a representative with index $i$. Define \begin{align*} \Psi_i(q(x,v,i))=(x,v). \end{align*} This definition is independent of the representative with index $i$: if $q(x,v,i)=q(y,w,i)$, then $x=y$ and $v=h_{ii}(x)w=w$, because $h_{ii}=\operatorname{id}_V$. The inverse map $\Theta_i:U_i\times V\to \pi_\rho^{-1}(U_i)$ is defined by \begin{align*} \Theta_i(x,v)=q(x,v,i), \end{align*} and the preceding existence and uniqueness show that $\Psi_i\circ\Theta_i=\operatorname{id}_{U_i\times V}$ and $\Theta_i\circ\Psi_i=\operatorname{id}_{\pi_\rho^{-1}(U_i)}$. Hence $\Psi_i$ is a bijection. Give $E_\rho$ the topology for which a subset $O\subset E_\rho$ is open exactly when $\Psi_i(O\cap \pi_\rho^{-1}(U_i))$ is open in $U_i\times V$ for every $i\in I$. With this topology each set $\pi_\rho^{-1}(U_i)$ is open and each $\Psi_i$ is a homeomorphism onto the open product $U_i\times V$. Since this is the standard Vector Bundle gluing construction from a smooth $GL(V)$-valued transition cocycle over a smooth manifold, the compatible product charts give the total space the required Hausdorff and second-countable smooth-manifold topology. If the original trivializing cover is not countable, one first restricts the cocycle to a countable smooth refinement of the cover; the same overlap formulas give a canonically isomorphic glued bundle. On an overlap $U_i\cap U_j$, the transition map $\Psi_i\circ \Psi_j^{-1}:(U_i\cap U_j)\times V\to (U_i\cap U_j)\times V$ is given by \begin{align*} (\Psi_i\circ \Psi_j^{-1})(x,w)=(x,h_{ij}(x)w). \end{align*} This map is smooth because $h_{ij}$ is smooth and the action $GL(V)\times V\to V$ is smooth. Its inverse is the smooth map $(x,v)\mapsto (x,h_{ji}(x)v)$, since $h_{ji}(x)=h_{ij}(x)^{-1}$. Hence the charts $(\pi_\rho^{-1}(U_i),\Psi_i)$ are smoothly compatible and define the smooth structure on $E_\rho$ by declaring a map into $E_\rho$, or out of $E_\rho$, to be smooth precisely when its coordinate representatives through the maps $\Psi_i$ are smooth. For $x\in M$, the fibre $\pi_\rho^{-1}(x)$ receives its vector-space operations by transporting the vector-space operations of $V$ through any chart $\Psi_i$ with $x\in U_i$. This is independent of $i$ because the overlap map on fibres is $v\mapsto h_{ij}(x)v$, and $h_{ij}(x)\in GL(V)$ is linear. Therefore the charts $\Psi_i$ give $E_\rho$ the structure of a smooth vector bundle over $M$ with fibre $V$ and transition functions $h_{ij}$.[/step]

[guided]The quotient construction has two separate jobs: first it must identify the local products consistently, and then it must leave behind the local vector-space structure on each fibre. The equivalence relation was defined by \begin{align*} (x,v,i)\sim (y,w,j) \end{align*} exactly when $x=y\in U_i\cap U_j$ and $v=h_{ij}(x)w$. Reflexivity follows from $h_{ii}(x)=\operatorname{id}_V$. Symmetry follows because the cocycle identities give $h_{ij}(x)h_{ji}(x)=\operatorname{id}_V$ and $h_{ji}(x)h_{ij}(x)=\operatorname{id}_V$, so $h_{ji}(x)=h_{ij}(x)^{-1}$. Transitivity follows by multiplying the two relations: if $v=h_{ij}(x)w$ and $w=h_{jk}(x)z$, then \begin{align*} v=h_{ij}(x)h_{jk}(x)z=h_{ik}(x)z. \end{align*} Thus the quotient $E_\rho=X/\sim$ is well-defined. For each $i\in I$, the map $\Psi_i:\pi_\rho^{-1}(U_i)\to U_i\times V$ sends the class $q(x,v,i)$ to $(x,v)$. This is well-defined because every class over $U_i$ has a unique representative with index $i$: existence is obtained from $(x,h_{ij}(x)w,i)\sim (x,w,j)$, and uniqueness follows from $h_{ii}(x)=\operatorname{id}_V$. Its inverse is $\Theta_i:U_i\times V\to\pi_\rho^{-1}(U_i)$, defined by $\Theta_i(x,v)=q(x,v,i)$. We now put the topology and smooth structure on the quotient in local coordinates. Declare a subset $O\subset E_\rho$ to be open exactly when $\Psi_i(O\cap \pi_\rho^{-1}(U_i))$ is open in $U_i\times V$ for every $i\in I$. Then each $\pi_\rho^{-1}(U_i)$ is open, and each $\Psi_i$ is a homeomorphism onto the [open set](/page/Open%20Set) $U_i\times V$. This is the local-triviality part of the construction: around every base point $x\in M$, choose an index $i$ with $x\in U_i$, and the map $\Psi_i$ identifies the part of the quotient over $U_i$ with the product $U_i\times V$. The standard Vector Bundle gluing construction from a smooth $GL(V)$-valued transition cocycle also supplies the required manifold-topology properties of the total space: the product charts are compatible, the resulting total space is Hausdorff and second countable, and a non-countable original cover may be replaced by a countable smooth refinement without changing the glued bundle up to the canonical isomorphism induced by the same transition formulas. On $U_i\cap U_j$, the transition map between these two local descriptions is \begin{align*} (\Psi_i\circ \Psi_j^{-1})(x,w)=(x,h_{ij}(x)w). \end{align*} This map is smooth because $h_{ij}:U_i\cap U_j\to GL(V)$ is smooth and the evaluation action $GL(V)\times V\to V$ is smooth. It is a diffeomorphism, with inverse $(x,v)\mapsto (x,h_{ji}(x)v)$, because $h_{ji}(x)=h_{ij}(x)^{-1}$. Therefore the charts $(\pi_\rho^{-1}(U_i),\Psi_i)$ are smoothly compatible. We define the smooth structure on $E_\rho$ by this compatible atlas, so each $\Psi_i$ is a diffeomorphism by construction. The transition map is fibrewise linear because $h_{ij}(x)$ is a linear automorphism of $V$ for each $x$. Therefore the vector-space operations transported from $V$ through any chart $\Psi_i$ agree on overlaps, so each fibre $\pi_\rho^{-1}(x)$ is canonically a [vector space](/page/Vector%20Space) in these local coordinates. The quotient therefore carries the topology, smooth structure, projection, and fibrewise linear structure of a smooth vector bundle with fibre $V$ and transition functions $h_{ij}$.[/guided]

[step:Compare the construction after passing to a common refinement] Suppose first that $\{\widetilde{\Phi}_i\}_{i\in I}$ is another family of local trivializations over the same cover. If a second choice of trivializations is given over a different cover, replace the two covers by their common refinement and restrict all transition functions to the refined overlaps; the following same-cover computation then applies on that refinement. Let $a_i:U_i\to GL(k,\mathbb{R})$ be the smooth change-of-frame maps defined by \begin{align*} \widetilde{\Phi}_i\circ \Phi_i^{-1}(x,b)=(x,a_i(x)b). \end{align*} The corresponding transition functions $\widetilde{g}_{ij}:U_i\cap U_j\to GL(k,\mathbb{R})$ satisfy \begin{align*} \widetilde{g}_{ij}(x)=a_i(x)g_{ij}(x)a_j(x)^{-1}. \end{align*} Define $\widetilde{h}_{ij}:U_i\cap U_j\to GL(V)$ and $A_i:U_i\to GL(V)$ by \begin{align*} \widetilde{h}_{ij}(x)=\rho(\widetilde{g}_{ij}(x)), \qquad A_i(x)=\rho(a_i(x)). \end{align*} Using that $\rho$ is a homomorphism, \begin{align*} \widetilde{h}_{ij}(x)=\rho(a_i(x)g_{ij}(x)a_j(x)^{-1})=\rho(a_i(x))\rho(g_{ij}(x))\rho(a_j(x))^{-1}=A_i(x)h_{ij}(x)A_j(x)^{-1}. \end{align*} Thus the two represented cocycles differ by the smooth coboundary $(A_i)$. Let $E_\rho$ and $\widetilde{E}_\rho$ be the glued bundles obtained from $(h_{ij})$ and $(\widetilde{h}_{ij})$. Define local maps $F_i:U_i\times V\to U_i\times V$ by \begin{align*} F_i(x,v)=(x,A_i(x)v). \end{align*} On $U_i\cap U_j$, the identity $\widetilde{h}_{ij}=A_i h_{ij}A_j^{-1}$ says exactly that the maps $F_i$ respect the gluing relations. Hence they descend to a smooth vector-bundle isomorphism $F:E_\rho\to \widetilde{E}_\rho$. Therefore the associated bundle is independent, up to vector-bundle isomorphism, of the chosen local trivializations. If $E$ and $E'$ are isomorphic vector bundles, let $\varphi:E\to E'$ be a vector-bundle isomorphism. Choose local trivializations $\Phi_\alpha:E|_{W_\alpha}\to W_\alpha\times\mathbb{R}^k$ and $\Phi'_\alpha:E'|_{W_\alpha}\to W_\alpha\times\mathbb{R}^k$ over a common refinement $\{W_\alpha\}$ of their covers. Define the smooth maps $b_\alpha:W_\alpha\to GL(k,\mathbb{R})$ by \begin{align*} \Phi'_\alpha\circ\varphi\circ\Phi_\alpha^{-1}(x,b)=(x,b_\alpha(x)b). \end{align*} If $g_{\alpha\beta}$ and $g'_{\alpha\beta}$ are the corresponding transition functions for $E$ and $E'$, then comparing the two ways of passing from the $\beta$-trivialization of $E$ to the $\alpha$-trivialization of $E'$ gives \begin{align*} g'_{\alpha\beta}(x)=b_\alpha(x)g_{\alpha\beta}(x)b_\beta(x)^{-1} \end{align*} for $x\in W_\alpha\cap W_\beta$. Applying $\rho$ and setting $B_\alpha(x)=\rho(b_\alpha(x))$ gives \begin{align*} \rho(g'_{\alpha\beta}(x))=B_\alpha(x)\rho(g_{\alpha\beta}(x))B_\beta(x)^{-1}. \end{align*} Thus the represented cocycles are related by the same coboundary formula as above, so the glued associated bundles are isomorphic. Hence the isomorphism class of $E_\rho$ depends only on the isomorphism class of $E$ and the representation $\rho$. [/step]

[step:Conclude the represented cocycle defines the associated bundle] The maps $h_{ij}=\rho\circ g_{ij}$ are smooth, satisfy the identity and cocycle relations, and therefore glue the products $U_i\times V$ into a smooth vector bundle $\pi_\rho:E_\rho\to M$ with fibre $V$. The change-of-frame computation shows that this construction is intrinsic up to vector-bundle isomorphism. This proves the theorem. [/step]