$\gdef\cI{\mathcal{I}}$

Let $$M_1=(E,\cI_1)$$ and $$M_2=(E,\cI_2)$$ be two matroids over the same groundset $$E$$. The matroid union of $$M_1$$ and $$M_2$$ is a new matroid $$M_1 \lor M_2$$ over the ground set $$E$$ where the independent sets are the sets which can be partitioned into an independent set of $$M_1$$ and an independent set of $$M_2$$. More formally,

$I \in \cI \iff I= I_1 \cup I_2 \text{ s.t. }I_1 \cap I_2 =\emptyset \text{, }I_1 \in \cI_1 \text{ and }I_2 \in \cI_2.$

The matroid union theorem guarantees that this is indeed a matroid and that

Theorem:

**Distance problem on the

Diameter problem on the base polytope:

 Given a matroid $$M=(E,\cI)$$ find two bases $$B_1$$, $$B_2$$ such that $$B_1 \Delta B_2$$.