According to Nitsure's Construction of Hilbert and Quot Schemes (p. 16), if $S$ is Noetherian, a \textit{linear scheme over $S$} is a scheme of the form $V=\operatorname{Spec}\operatorname{Sym}_{\mathsf{O}_{S}}\mathsf{Q}$ for some coherent sheaf $\mathsf{Q}$ over $S$.
The author makes the claim that this is naturally a group scheme. Why?
Also, how does this generalize vector bundles?
Best Answer
The answer to your first question is because $V$ represents the group-valued functor taking an $S$-scheme $f: T \to S$ to $(f^* Q^\vee)(T)$, where $Q^\vee$ is the dual of $Q$.
If $Q$ is a locally free $\mathcal{O}_S$-module of rank $n$, then $V$ is locally (over an open cover $U_i$ of $S$) of the form $U_i \times \mathbb{A}^n$, hence a vector bundle.
For reference see EGA II sections 1.3 and 1.7, Stacks, or section 17.1 of Vakil. There's also an exercise about this in Hartshorne.