TITLE: A Family of Bases for a Vector Space
QUESTION [9 upvotes]: Let $V$ be a multiset of $kn$ nonzero vectors in $\mathbf{R}^n$.  Suppose that for $1 \leq d \leq n$, each $d$-dimensional subspace of $\mathbf{R}^n$ contains at most $kd$ members of $V$.  Then $V$ can be partitioned into $k$ bases of $\mathbf{R}^n$.
A proof or a reference would be appreciated.

REPLY [5 votes]: Apparently the question was answered in the comments:  Edmonds proves a more general theorem for a family of matroids.  If the matroids are all the same and if that matroid is a vector matroid, then it is exactly the question stated.