I am reading pilloni's lecture note "THE STACK OF VECTOR BUNDLES ON A CURVE". On page 35 he states that
Let $X$ be a scheme and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$. We denote $\mathbb{V}(\mathcal{E}):=\operatorname{Spec}\left(\operatorname{Sym}\left(\mathcal{E}^{\vee}\right)\right)$.
Define $$ T(\mathcal{E})=\underline{\operatorname{Isom}}_{{}_X}\left(\mathscr{O}_X^r, \mathcal{E}\right), $$ with action of $G=\mathrm{GL}_r=\underline{\operatorname{Isom}}_{{}_X}\left(\mathscr{O}_X^r, \mathscr{O}_X^r\right)$ by $(g, \phi) \mapsto \phi \circ g^{-1}$.
Then $T(\mathcal{E})$ is a $\mathrm{GL}_{\mathrm{r}}$-torsor.
Conversely, let $T$ be a $\mathrm{GL}_{\mathrm{r}}$-torsor. Then $T$ is representable by a scheme. Indeed, let $X=\cup U_i$ be an open cover such that $T\left(U_i\right) \neq \emptyset$ for all $i$. Then using $T_{\mid U_i} \stackrel{\sim}{\rightarrow} G_{\mid U_i}$, we can glue them to get a scheme. We can define a geometric vector bundle $\mathcal{V}(T)$ on $X$ via the following rule : $$ \mathcal{V}(T)=\left(T \times_X \mathbb{V}\left(\mathscr{O}_X^r\right)\right) / \mathrm{GL}_r $$ for the diagonal action $(\phi, v) g=\left(\phi \circ g, g^{-1} v\right)$.
Lemma 7.6. The space $\mathcal{V}(T)$ is a geometric vector bundle.
Proof. By working locally on $X$, we may assume that $T=\mathrm{GL}_r$. In this case, we have a morphism : $T \times \mathbb{V}\left(\mathscr{O}_X^r\right) \rightarrow T \times \mathbb{V}\left(\mathscr{O}_X^r\right),(t, v) \mapsto(t, t v)$. This isomorphism intertwins the diagonal action of $\mathrm{GL}_r$ and the action of $\mathrm{GL}_r$ on $T$. We deduce that $\mathcal{V}(T) \simeq T / \mathrm{GL}_r \times$ V $\left(\mathscr{O}_X^r\right)$.
Proposition 7.2. The above rule defines an equivalence of categories between : $$ \begin{aligned} \text { \{vector bundles of rank } r, \text { maps are isomorphisms }\} & \rightarrow\left\{\mathrm{GL}_r \text {-torsors }\right\} \\ \mathcal{E} & \mapsto T(\mathcal{E}) \\ \mathcal{V}(T) & \leftrightarrow T \end{aligned} $$
My question is
(1)How to associate T with a scheme. Does it mean that on every affine open set $U_i=Spec(A)$ of $X$ such that $T_{\mid U_i} \stackrel{\sim}{\rightarrow} G_{\mid U_i}$ we get $Spec(A[T_{11}\cdots T_{rr},f]/ （detT\cdot f-1）)$ and glue on a affine open set $V=B$ of $U_i \cap U_j$ by an isomorphism $G_{\mid V} \rightarrow G_{\mid V}$ which is compitible with the group action, hence is a multiplication sending g to gx for some x. This G-set isomorphism induces an isomorphism of scheme by sending $T_{ij}$ to the sum of $x_{ik}T_{kj}$ Moreover if we get such a scheme L over X. Then will T be the sheaf of section from X to L ?
(2)What is the mean of this space. $$ \mathcal{V}(T)=\left(T \times_X \mathbb{V}\left(\mathscr{O}_X^r\right)\right) / \mathrm{GL}_r $$ Does the fibre product take on the scheme associated with T? And the action of $GL_r$ on this shceme is described above? And what is the meaning for a quotient scheme?