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A vector space S of linear operators between finite-dimensional vector spacesU and V is called locally linearly dependent (in abbreviate form: LLD) whenevery vector x in U is annihilated by a non-zero operator in S. By a dualityargument, one sees that studying LLD operator spaces amounts to studying vectorspaces 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 forLLD spaces of small dimension or large essential range (the essential rangebeing the sum of all the ranges of the operators in S). We show that suchclassification theorems can be obtained by translating into the context of LLDspaces Atkinson's classification of primitive spaces of bounded rank matrices;we also obtain a new classification theorem for such spaces that covers a rangeof dimensions for the essential range that is roughly twice as large as that inAtkinson's theorem. In particular, we obtain a classification of all4-dimensional LLD operator spaces for fields with more than 3 elements(beforehand, such a classification was known only for algebraically closedfields and in the context of primitive spaces of matrices of bounded rank).  These results are applied to obtain improved upper bounds for the maximalrank in a minimal LLD operator space.

Local linear dependence seen through duality II