In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent set.

As an example, consider the matroid over the ground-set R² (the vectors in the two-dimensional Euclidean plane), with the following independent sets:

It has two bases, which are the sets {(0,1),(2,0)} , {(0,3),(2,0)}. These are the only independent sets that are maximal under inclusion.

The basis has a specialized name in several specialized kinds of matroids:^[1]

• In a graphic matroid, where the independent sets are the forests, the bases are called the spanning forests of the graph.
• In a transversal matroid, where the independent sets are endpoints of matchings in a given bipartite graph, the bases are called transversals.
• In a linear matroid, where the independent sets are the linearly-independent sets of vectors in a given vector-space, the bases are just called bases of the vector space. Hence, the concept of basis of a matroid generalizes the concept of basis from linear algebra.
• In a uniform matroid, where the independent sets are all sets with cardinality at most k (for some integer k), the bases are all sets with cardinality exactly k.
• In a partition matroid, where elements are partitioned into categories and the independent sets are all sets containing at most k_c elements from each category c, the bases are all sets which contain exactly k_c elements from category c.
• In a free matroid, where all subsets of the ground-set E are independent, the unique basis is E.

All matroids satisfy the following properties, for any two distinct bases ${\displaystyle A}$ and ${\displaystyle B}$:^[2][3]

• Basis-exchange property: if ${\displaystyle a\in A\setminus B}$, then there exists an element ${\displaystyle b\in B\setminus A}$ such that ${\displaystyle (A\setminus \{a\})\cup \{b\))$ is a basis.
• Symmetric basis-exchange property: if ${\displaystyle a\in A\setminus B}$, then there exists an element ${\displaystyle b\in B\setminus A}$ such that both ${\displaystyle (A\setminus \{a\})\cup \{b\))$ and ${\displaystyle (B\setminus \{b\})\cup \{a\))$ are bases. Brualdi^[4] showed that it is in fact equivalent to the basis-exchange property.
• Multiple symmetric basis-exchange property: if ${\displaystyle X\subseteq A\setminus B}$, then there exists a subset ${\displaystyle Y\subseteq B\setminus A}$ such that both ${\displaystyle (A\setminus X)\cup Y}$ and ${\displaystyle (B\setminus Y)\cup X}$ are bases. Brylawski, Greene, and Woodall, showed (independently) that it is in fact equivalent to the basis-exchange property.
• Bijective basis-exchange property: There is a bijection ${\displaystyle f}$ from ${\displaystyle A}$ to ${\displaystyle B}$, such that for every ${\displaystyle a\in A\setminus B}$, ${\displaystyle (A\setminus \{a\})\cup \{f(a)\))$ is a basis. Brualdi^[4] showed that it is equivalent to the basis-exchange property.
• Partition basis-exchange property: For each partition ${\displaystyle (A_{1},A_{2},\ldots ,A_{m})}$ of ${\displaystyle A}$ into m parts, there exists a partition ${\displaystyle (B_{1},B_{2},\ldots ,B_{m})}$ of ${\displaystyle B}$ into m parts, such that for every ${\displaystyle i\in [m]}$, ${\displaystyle (A\setminus A_{i})\cup B_{i))$ is a basis.^[5]

However, a basis-exchange property that is both symmetric and bijective is not satisfied by all matroids: it is satisfied only by base-orderable matroids.

In general, in the symmetric basis-exchange property, the element ${\displaystyle b\in B\setminus A}$ need not be unique. Regular matroids have the unique exchange property, meaning that for some ${\displaystyle a\in A\setminus B}$, the corresponding b is unique.^[6]

It follows from the basis exchange property that no member of ${\displaystyle {\mathcal {B))}$ can be a proper subset of another.

Moreover, all bases of a given matroid have the same cardinality. In a linear matroid, the cardinality of all bases is called the dimension of the vector space.

It is conjectured that all matroids satisfy the following property:^[2] For every integer t ≥ 1, If B and B' are two t-tuples of bases with the same multi-set union, then there is a sequence of symmetric exchanges that transforms B to B'.

The bases of a matroid characterize the matroid completely: a set is independent if and only if it is a subset of a basis. Moreover, one may define a matroid ${\displaystyle M}$ to be a pair ${\displaystyle (E,{\mathcal {B)))}$, where ${\displaystyle E}$ is the ground-set and ${\displaystyle {\mathcal {B))}$ is a collection of subsets of ${\displaystyle E}$, called "bases", with the following properties:^[7][8]

(B1) There is at least one base -- ${\displaystyle {\mathcal {B))}$ is nonempty;

(B2) If ${\displaystyle A}$ and ${\displaystyle B}$ are distinct bases, and ${\displaystyle a\in A\setminus B}$, then there exists an element ${\displaystyle b\in B\setminus A}$ such that ${\displaystyle (A\setminus \{a\})\cup \{b\))$ is a basis (this is the basis-exchange property).

(B2) implies that, given any two bases A and B, we can transform A into B by a sequence of exchanges of a single element. In particular, this implies that all bases must have the same cardinality.

If ${\displaystyle (E,{\mathcal {B)))}$ is a finite matroid, we can define the orthogonal or dual matroid ${\displaystyle (E,{\mathcal {B))^{*})}$ by calling a set a basis in ${\displaystyle {\mathcal {B))^{*))$ if and only if its complement is in ${\displaystyle {\mathcal {B))}$. It can be verified that ${\displaystyle (E,{\mathcal {B))^{*})}$ is indeed a matroid. The definition immediately implies that the dual of ${\displaystyle (E,{\mathcal {B))^{*})}$ is ${\displaystyle (E,{\mathcal {B)))}$.^{[9]: 32 [10]}

Using duality, one can prove that the property (B2) can be replaced by the following:

(B2*) If ${\displaystyle A}$ and ${\displaystyle B}$ are distinct bases, and ${\displaystyle b\in B\setminus A}$, then there exists an element ${\displaystyle a\in A\setminus B}$ such that ${\displaystyle (A\setminus \{a\})\cup \{b\))$ is a basis.

A dual notion to a basis is a circuit. A circuit in a matroid is a minimal dependent set—that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs.

One may define a matroid ${\displaystyle M}$ to be a pair ${\displaystyle (E,{\mathcal {C)))}$, where ${\displaystyle E}$ is the ground-set and ${\displaystyle {\mathcal {C))}$ is a collection of subsets of ${\displaystyle E}$, called "circuits", with the following properties:^[8]

(C1) The empty set is not a circuit;

(C2) A proper subset of a circuit is not a circuit;

(C3) If C₁ and C₂ are distinct circuits, and x is an element in their intersection, then ${\displaystyle C_{1}\cup C_{2}\setminus \{x\))$ contains a circuit.

Another property of circuits is that, if a set ${\displaystyle J}$ is independent, and the set ${\displaystyle J\cup \{x\))$ is dependent (i.e., adding the element ${\displaystyle x}$ makes it dependent), then ${\displaystyle J\cup \{x\))$ contains a unique circuit ${\displaystyle C(x,J)}$, and it contains ${\displaystyle x}$. This circuit is called the fundamental circuit of ${\displaystyle x}$ w.r.t. ${\displaystyle J}$. It is analogous to the linear algebra fact, that if adding a vector ${\displaystyle x}$ to an independent vector set ${\displaystyle J}$ makes it dependent, then there is a unique linear combination of elements of ${\displaystyle J}$ that equals ${\displaystyle x}$.^[10]

• Matroid polytope - a polytope in Rⁿ (where n is the number of elements in the matroid), whose vertices are indicator vectors of the bases of the matroid.