Let $M=(E,\mathcal{I})$ be a matroid with rank function $r$. A hyperplane is a maximal set with rank $r(E)-1$.
What can we say about the rank of the intersection of $k$ hyperplanes? Is it true it has rank at least $r(E)-k$ or $0$? This is true for an arrangement of hyperplanes in $\mathbb{R}^n$.
This seems true for linear matroids.
In the rank $4$ projective geometry $\text{PG}(3,2)$, every hyperplane is isomorphic to the Fano matroid. Now delete two points on the same line. The two hyperplanes that share that line intersect at a single point, so the rank of their intersection is $1$.
And of course, any minor of a projective geometry is a linear matroid.