Primitive spaces of matrices with upper rank two over the field with two elements

Abstract

For fields with more than two elements, the classification of the vector spaces of matrices with rank at most two is already known. In this work, we complete that classification for the field . We apply the results to obtain the classification of triples of locally linearly dependent operators over , the classification of the three-dimensional subspaces of in which no matrix has a non-zero eigenvalue, and the classification of the three-dimensional affine spaces that are included in the general linear group .