I am thinking about the following problem: Suppose $(E, \mathcal F)$ is a matroid and $(E, \mathcal F^*)$ is the dual matroid of $(E,\mathcal F)$. Let $C$ be a circuit of $(E, \mathcal F)$ and $C^*$ be a circuit of $(E, \mathcal F^*)$. Show that then $|C\cap C^*|\neq 1$.
If I just consider matroids which come from planar graphs, I can show this. But how do I proceed in the general case?
I have an answer:
Suppose there exists just on element $x$ in the intersection. Then, as $C$ is a circuit, $C\backslash \{x\}$ is independent. Hence, we can expand it to a basis $B$ of $E\backslash C^*$. But as $C^*$ is a circuit too, we see that $B \cup \{x\}$ is still independent. But by the second Matroid axiom, $C$ is then independent. This gives us a contradiction.
EDIT: Also I just saw that this question is already posted here, so it should be closed.