Let $A$ be a Noetherian Commutative $k$-algebra. A locally free sheaf $E$ of rank $r$ on $Spec(A)\times_k X$ is defined by an open covering $\{U_i\}_{i\in I}$ of $X$ and transition fucntions $\theta_{ij}:U_i\cap U_j\rightarrow H^0(Spec(A),GL(r,\mathcal{O}_{Spec(A)}))=GL(r,A)$.
We can consider $GL(r,A)$ as the set of global sections of the sheaf of groups $Spec(A)\times_k GL(r)$.
I am just totally lost in understanding this statement.
I know that by the definition of locally free sheaf there exists isomorphisms $\phi_i:E|_{Spec(A)\times U_i}\rightarrow \oplus^r \mathcal{O}_{Spec(A)\times U_i} $ for all $i\in I $. Therefore we have an isomorphism $ \phi_j\circ\phi_i^{-1} :\oplus^r \mathcal{O}_{Spec(A)\times (U_i\cap U_j)} \rightarrow \oplus^r \mathcal{O}_{Spec(A)\times (U_i\cap U_j)}$. I don't understand how from this how do we get those transition functions. I am quite aware how do we get transition functions in case of Riemann Surfaces and Manifolds etc. I think it's just too many notations which is bothering me to fully reveal the meaning of transition functions in this context. Any kind of help will be appreciated.
Given the last map you gave:$$ \phi_j\circ\phi_i^{-1} :\oplus^r \mathcal{O}_{Spec(A)\times (U_i\cap U_j)} \rightarrow \oplus^r \mathcal{O}_{Spec(A)\times (U_i\cap U_j)},$$if you precompose it with the injection of the $k$th summand and post-compose it with the projection to the $l$th summand, you get a section in the structure sheaf of $\text{Spec}(A) \times (U_i \cap U_j)$, which we may think of as a function on $U_i \cap U_j$ with values in $A$. Do you agree so far?
Doing this for all $k$ and all $l$ and collecting the various functions into a matrix, we get a function on $U_I \cap U_j$ with values in the ring of matrices over $A$. That these matrix valued functions are invertible follows from the cocycle conditions. Is this helpful?