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The hyperplanes of a matroid are those flats whose rank is one less than the rank of the matroid. Given a matroid, the Hyperplanes command returns the hyperplanes of that matroid as a list of sets.

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

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

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

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

$M≔\mathrm{Matroid}\left(A\right)$

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

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

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

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