Abstract
The goal of this paper is to generalize the main results of [21] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish "joint strong extremality" of arbitrary finite collection of smooth non-degenerate submanifolds of R(n). The proofs are based on generalized quantitative non-divergence estimates for translates of measures on the space of lattices.