The bases of a matroid are its maximal independent sets. Given a matroid, the function Bases returns the bases of that matroid as a list of sets.

If the bases of this matroid haven't been computed before, they are computed by this command and stored for any future computations that use the bases.

$\mathrm{with}\left(\mathrm{Matroids}\right)\:$

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,1\,1\,0\,0\,0\right]\,\left[0\,0\,0\,1\,1\,1\right]\,\left[1\,0\,0\,1\,0\,0\right]\,\left[0\,1\,0\,0\,1\,0\right]\,\left[0\,0\,1\,0\,0\,2\right]\right]\right)$

$A≔\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]$

$M≔\mathrm{Matroid}\left(A\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]\end{array}⟩$

$\mathrm{Bases}\left(M\right)$

$\left[\left\{1\,2\,3\,4\,6\right\}\,\left\{1\,2\,3\,5\,6\right\}\,\left\{1\,3\,4\,5\,6\right\}\,\left\{2\,3\,4\,5\,6\right\}\right]$

$G≔\mathrm{Graph}\left(\left\{\left\{a\,x\right\}\,\left\{a\,y\right\}\,\left\{b\,x\right\}\,\left\{b\,y\right\}\,\left\{c\,x\right\}\,\left\{c\,y\right\}\right\}\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$M≔\mathrm{Matroid}\left(G\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; graphic\; matroi\ⅆ\; on\; th\ⅇ\; graph:}\\ \mathrm{PLOT}\left(\mathrm{...}\right)\end{array}⟩$

$\mathrm{Bases}\left(M\right)$

$\left[\left\{''a\_x''\,''a\_y''\,''b\_x''\,''c\_x''\right\}\,\left\{''a\_x''\,''a\_y''\,''b\_y''\,''c\_x''\right\}\,\left\{''a\_x''\,''b\_x''\,''b\_y''\,''c\_x''\right\}\,\left\{''a\_y''\,''b\_x''\,''b\_y''\,''c\_x''\right\}\,\left\{''a\_x''\,''a\_y''\,''b\_x''\,''c\_y''\right\}\,\left\{''a\_x''\,''a\_y''\,''b\_y''\,''c\_y''\right\}\,\left\{''a\_x''\,''b\_x''\,''b\_y''\,''c\_y''\right\}\,\left\{''a\_y''\,''b\_x''\,''b\_y''\,''c\_y''\right\}\,\left\{''a\_x''\,''b\_x''\,''c\_x''\,''c\_y''\right\}\,\left\{''a\_y''\,''b\_x''\,''c\_x''\,''c\_y''\right\}\,\left\{''a\_x''\,''b\_y''\,''c\_x''\,''c\_y''\right\}\,\left\{''a\_y''\,''b\_y''\,''c\_x''\,''c\_y''\right\}\right]$

James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.