Local linear dependence seen through duality II

TL;DR

This paper classifies small-dimensional locally linearly dependent operator spaces using duality and extends existing classification theorems, providing new insights into their structure and bounds on maximal rank.

Contribution

It introduces new classification results for LLD operator spaces, especially for larger essential ranges, extending Atkinson's work and covering more cases.

Findings

Classified all 4-dimensional LLD operator spaces over fields with more than 3 elements.

Extended Atkinson's classification to larger essential ranges.

Provided improved bounds for maximal rank in minimal LLD spaces.

Abstract

A vector space S of linear operators between finite-dimensional vector spaces U and V is called locally linearly dependent (in abbreviate form: LLD) when every vector x in U is annihilated by a non-zero operator in S. By a duality argument, one sees that studying LLD operator spaces amounts to studying vector spaces of matrices with rank less than the number of columns, or, alternatively, vector spaces of non-injective operators. In this article, this insight is used to obtain classification results for LLD spaces of small dimension or large essential range (the essential range being the sum of all the ranges of the operators in S). We show that such classification theorems can be obtained by translating into the context of LLD spaces Atkinson's classification of primitive spaces of bounded rank matrices; we also obtain a new classification theorem for such spaces that covers a range of dimensions for the essential range that is roughly twice as large as that in Atkinson's theorem. In particular, we obtain a classification of all 4-dimensional LLD operator spaces for fields with more than 3 elements (beforehand, such a classification was known only for algebraically closed fields and in the context of primitive spaces of matrices of bounded rank). These results are applied to obtain improved upper bounds for the maximal rank in a minimal LLD operator space.