Matroid representation

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.

A (finite) matroid ${\displaystyle (E,{\mathcal {I}})}$ is defined by a finite set ${\displaystyle E}$ (the elements of the matroid) and a family ${\displaystyle {\mathcal {I}}}$ of the subsets of ${\displaystyle E}$, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one independent set ${\displaystyle A}$ is larger than a second independent set ${\displaystyle B}$ then there exists an element ${\displaystyle x\in A\setminus B}$ that can be added to ${\displaystyle B}$ to form a larger independent set. One of the key motivating examples in the formulation of matroids was the notion of linear independence of vectors in a vector space: if ${\displaystyle E}$ is a finite set or multiset of vectors, and ${\displaystyle {\mathcal {I}}}$ is the family of linearly independent subsets of ${\displaystyle E}$, then ${\displaystyle (E,{\mathcal {I}})}$ is a matroid.

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