Linear Matroid

Definition

Linear Matroid

A linear matroid is a matroid $M=(E,\mathcal{I})$ for which there exists a field $F$ and a matrix

$$A=[a_e{]}_{e\in E}$$

over $F$ with one column $a_e$ for each element $e\in E$, such that for every subset $X\subseteq E$,

$$X\in \mathcal{I}⟺\{a_e\mid e\in X\}\text{ is linearly independent over }F.$$

In other words, a matroid is linear if its abstract notion of independence can be represented exactly as linear independence of column vectors of a matrix over some field.

If the field $F$ is fixed, then $M$ is also called representable over $F$.