The rank of a matroid is the cardinality of a maximal independent subset of its ground set. Equivalently, the rank of a matroid is the size of each of its bases.

The rank of a subset $S$ of the ground set $E$ of a matroid is the cardinality of a maximal independent subset of $S$.

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

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

$A≔\left[\begin{array}{cccc}1& 1& 0& 0\\ \mathrm{-1}& 0& 1& 0\\ 0& \mathrm{-1}& \mathrm{-1}& 0\end{array}\right]$

$M≔\mathrm{Matroid}\left(A\right)$

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

$r≔\mathrm{Rank}\left(M\right)$

$r≔2$

$\mathrm{r14}≔\mathrm{Rank}\left(M\,\left\{1\,4\right\}\right)$

$\mathrm{r14}≔1$

$\mathrm{r4}≔\mathrm{Rank}\left(M\,\left\{4\right\}\right)$

$\mathrm{r4}≔0$

$\mathrm{r1234}≔\mathrm{Rank}\left(M\,\mathrm{convert}\left(\mathrm{GroundSet}\left(M\right)\,\mathrm{set}\right)\right)$

$\mathrm{r1234}≔2$

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