It turns out that what I'll be mentioning only in brief now -- the topic of duality -- is dreadfully important in matroids. However, coming up with examples of duals of matroids is not an easy thing, especially since I'm trying not to lose you in this. So I'll do my best to keep this very short.

Take a matroid M, and let B be its collection of bases (remember, a basis is a maximal independent subset of the ground set E). Let B* = {E\B | B in B}. Then the set B* is a set of bases of a matroid as well, using the same ground set as M. This matroid is called the dual matroid of M, and is denoted M*.