I would say the theory of quivers and in particular Gabriel's theorem on finite representation type and its extensions to tame type. Representations of quivers are essentially linear algebra problems in a different language. For instance Jordan canonical form is the description of indecomposable reps of a quiver with one vertex and a loop. In general things like the classification of two endomorphisms of vector spaces, matrix pencils and the n-subspace problem are all problems in the rep theory of quivers. The intro to the book of Gabriel-Roiter says more.
Added. A quiver is a directed multigraph, often assumed finite in this context. A representation of a quiver Q is an assignment of a vector space to each vertex and a linear transformation to each edge from the vector space at its source to the vector space if its target. Isomorphisms are isomorphisms of vertex spaces making commuting squares with the edge linear transformations. There is a fairly straightforward notion of direct sum and hence indecomposable rep. Finite rep type means finitely many isoclasses of indecomposables, tame type essentially means indecomposables come in 1-parameter families (plus finitely many exceptions) if you fix the dimensions of the vertex spaces. Wild means its representation theory contains that of all finite dinensional (and hence all finitely generated) algebras. In particular the first order theory is undecidable. Only finite, tame and wild occur.