Given a reductive group $G$, a linear $G$-representation $X$, and a choice of $G$-linearized line bundle on $X$, Geometric Invariant Theory (GIT) produces an open subset $Xss\subset X$ and a quotient $Xss/G$. The key motivation of the work in this thesis is to determine when the derived category of coherent sheaves on the GIT quotient $Xss/G$ admits a full exceptional collection consisting of vector bundles. A full exceptional collection is an important structure on a derived category with many valuable implications. For instance, such a collection produces a basis for the Grothendieck group and the Hochschild Homology of the derived category. Using ideas from local cohomology and equivariant geometry, we produce a large class of linear GIT quotients with $G$ of rank $2$ that admit a full exceptional collection. These vector bundles will come from irreducible $G$-representations whose weights lie in a particular “window” in the weight space of $G$. When $G$ has higher rank, we produce a finite list of tautological vector bundles that generate the derived category. These vector bundles do not form a full exceptional collection, but their classes still generate the Grothendieck group.