$GL(n)$-dependence of matrices

Abstract

We introduce the notion of $GL(n)$-dependence of matrices, which is a generalization of linear dependence taking into account the matrix structure. Then we prove a theorem, which generalizes, on the one hand, the fact that $n+1$ vectors in an $n$-dimensional vector space are linearly dependent and, on the other hand, the fact that the natural action of the group $GL(n,\mathcal{K})$ on ${\mathcal{K}}^n\setminus \{0\}$ is transitive.