Further definition and properties
A subset of a matroid M is called dependent if it is not independent. A dependent set all of whose subsets are independent is called a circuit (with the terminology coming from the graph example above). An independent set all of whose supersets are dependent is called a basis (with the terminology coming from the vector space example above). An important fact is that all bases of a matroid have the same number of elements, called the rank of M. Removing any element from a circuit yields a basis, so all circuits have the same number of elements too, one more than the rank.
In the first example matroid above, a basis is a basis in the sense of linear algebra of the subspace spanned by M. In the second example, a basis is the same as a spanning tree of the graph G. In the third example, a basis is any subsets of M with k elements.
If N is a subset of the matroid M, then N becomes a matroid by considering a subset of N independent if and only if it is independent in M. This allows to talk about the rank of any subset of M.
The rank function r assigns a natural number to every subset of M and has the following properties:
In fact, these three properties can be used as an alternative definition of matroids: the independent sets are then defined as those subsets N of M with r(N) = N.
If M is a matroid, we can define the dual matroid M* by taking the same underlying set and calling a set independent in M* if and only if it is contained in the complement of some basis of M.
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