  The Rado-Horn theorem provides necessary and sufficient conditions for when a
collection of vectors can be partitioned into a fixed number of linearly
independent sets. Such partitions exist if and only if every subset of the
vectors satisfies the so-called Rado-Horn inequality. Today there are at least
six proofs of the Rado-Horn theorem, but these tend to be extremely delicate or
require intimate knowledge of matroid theory. In this paper we provide an
elementary proof of the Rado-Horn theorem as well as elementary proofs for
several generalizations including results for the redundant case when the
hypotheses of the Rado-Horn theorem fail. Another problem with the existing
proofs of the Rado-Horn Theorem is that they give no information about how to
actually partition the vectors. We start by considering a specific partition of
the vectors, and the proof consists of showing that this is an optimal
partition. We further show how certain structures we construct in the proof are
at the heart of the Rado-Horn theorem by characterizing subsets of vectors
which maximize the Rado-Horn inequality. Lastly, we demonsrate how these
results may be used to select an optimal partition with respect to spanning
properties of the vectors.
