Given a matroid, the function DependentSets returns the dependent sets of that matroid.

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

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

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

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

$M≔\mathrm{Matroid}\left(A\right)$

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

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

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

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