Helly theorems for lattice of flats of a matroid

A basis of a matroid is adapted to a flat if it contains a basis of the flat. We prove that a family of flats in a rank r matroid has a common adapted basis whenever every subfamily of at most r+1 flats does. The ground set and the family may be infinite; only the rank must be finite. This result, in particular, contains Klyachko's vector space Helly theorem for subspaces. As an application, we obtain a splitting criterion for tropical toric vector bundles on projective spaces. As a secondary application, we derive quadratic bounds for the Helly number for distributive generation in the lattice of flats.