  The goal of this paper is to generalize the main results of [KM] 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 nondegenerate
submanifolds of ${\bold R}^n$. The proofs are based on generalized quantitative
nondivergence estimates for translates of measures on the space of lattices.
