The dual $M^{\*}$ of a matroid $M$ is itself a matroid on the same ground set. The bases of $M^{\*}$ are the complements of bases of $M$.

The operations of deletion and contraction are dual to each other. That is, ${\left(M\setminus \left\{e\right\}\right)}^{\*}$ is isomorphic to $M^{\*}/\left\{e\right\}$.

The Tutte polynomial of the dual of a matroid simply swaps the roles of the variables.

The dual of a representable matroid is the matroid on its orthogonal complement. More precisely, if $M$ is the matroid representable by the columns of a matrix $A$, and $B$ is a matrix whose rows form the orthogonal complement to the rows of $A$, then $M^{\*}$ is the matroid representable by the columns of $B$.

Given a planar graph $G$, the dual of the matroid underlying $G$ is the graph dual of $G$.

The dual of the dual of a matroid $M$ is itself.

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

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

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

$M≔\mathrm{Matroid}\left(A\right)$

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

$B≔\mathrm{Bases}\left(M\right)$

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

$\mathrm{M2}≔\mathrm{Dual}\left(M\right)$

$\mathrm{M2}≔⟨\mathrm{a\; matroi\ⅆ\; on\; 6\; \ⅇl\ⅇm\ⅇnts\; with\; 4\; bas\ⅇs\; of\; siz\ⅇ\; 1}⟩$

$\mathrm{B2}≔\mathrm{Bases}\left(\mathrm{M2}\right)$

$\mathrm{B2}≔\left[\left\{1\right\}\,\left\{2\right\}\,\left\{4\right\}\,\left\{5\right\}\right]$

$T≔\mathrm{TuttePolynomial}\left(M\,x\,y\right)$

$T≔x^5+x^4+x^3+x^{2}y$

$\mathrm{T2}≔\mathrm{TuttePolynomial}\left(\mathrm{M2}\,y\,x\right)$

$\mathrm{T2}≔x^5+x^4+x^3+x^{2}y$

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