A redundant version of the Rado–Horn Theorem

The Rado–Horn Theorem gives a characterization of those sets of vectors which can be written as the union of a fixed number of linearly independent sets. In this paper, we study the redundant case. We show that then the span of the vectors can be written as the direct sum of a subspace which directly fails the Rado–Horn criteria and a subspace for which the Rado–Horn criteria hold. As a corollary, we characterize those sets of vectors, which, after the deletion of a fixed number of vectors, can be written as the finite union of linearly independent sets.