A matroid basis is an independent subset of a matroid that is maximal with respect to inclusion. All bases
of a matroid have the same number of elements,
called its matroid rank.
The bases determine the matroid, since a subset is independent iff it is contained in a basis. For a matroid
represented by columns of a matrix, the bases correspond
to maximal linearly
independent collections of columns. In the graphic
matroid of a connected graph with at least
one graph vertex, the bases are its spanning
trees.
See also
Common Integer Polymatroid Basis,
Cyclic Basis Ordering,
Dual
Matroid,
Graphic Matroid,
Matroid,
Matroid Cocircuit,
Matroid
Intersection,
Matroid Rank,
Polymatroid,
Strongly Base-Orderable Matroid,
Symmetric Basis Exchange,
White's
Conjecture
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References
Oxley, J. G. Matroid
Theory. Oxford, England: Oxford University Press, 1992.
Cite this as:
Weisstein, Eric W. "Matroid Basis." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatroidBasis.html
Subject classifications
