Can LP for matroid polytopes be solved using the greedy algorithm?
You can associate with each matroid two polytopes in $\mathbb{R}^n$, where $n$ is the size of the universe over which the matroid is defined:


The matroid polytope is the convex hull of the characteristic vectors of all independent sets.
The base polytope is the convex hull of the characteristic vectors of all bases (maximal independent sets).


The maximum of a linear function over a polytope is always attained (not necessarily uniquely) at a vertex. Therefore maximizing a linear function over a base polytope is like maximizing it over the collection of bases, for which the greedy algorithm can be used. The same goes for the matroid polytope, with the necessary changes.