An elementary, illustrative proof of the Rado-Horn Theorem

TL;DR

This paper presents an elementary, accessible proof of the Rado-Horn theorem, including generalizations and methods for constructing optimal partitions of vectors based on the theorem's conditions.

Contribution

It offers a simplified proof of the Rado-Horn theorem and its extensions, along with techniques for explicitly constructing optimal partitions of vectors.

Findings

Elementary proof of the Rado-Horn theorem

Methods for constructing optimal partitions

Generalizations including redundant cases

Abstract

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.