  The Hurwitz chain gives a sequence of pairs of Farey approximations to an
irrational real number. Minkowski gave a criterion for a number to be algebraic
by using a certain generalization of the Hurwitz chain. We apply Minkowski's
generalization (the Minkowski chain) to give criteria for a real linear form to
be either badly approximable or singular. We also give a variant of Dirichlet's
approximation theorem for a real linear form that produces a whole basis of
approximating integral vectors rather than a single one. This result holds if
and only if the form is badly approximable. The proofs rely on properties of
successive minima and reduced bases of lattices.
