In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra.
A linear matroid is a matroid that has a representation, and an F-linear matroid (for a field F) is a matroid that has a representation using a vector space over F. Matroid representation theory studies the existence of representations and the properties of linear matroids.