Dual matroid

The dual $M^*$ of a matroid $M$ on $E$ is the matroid whose bases are the complements $E \setminus B$ of the bases $B$ of $M$. It has rank $n - r$. The circuits of $M^*$ are the cocircuits of $M$ (complements of hyperplanes), and the loops of $M^*$ are the coloops of $M$. If $M$ is realized by $[\,I_r \mid A\,]$, then $M^*$ is realized by $[\,-A^{T} \mid I_{n-r}\,]$.

See also self-dual.