This is the second part of a two-part study of linear multimode systems. In the first part, it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant, and both first-order and polynomial representations were introduced that satisfy this requirement. Here we determine the conditions under which first-order representations are minimal. We also show how two minimal representations of the same behavior are related; this leads in particular to an appropriate state-space isomorphism theorem. The minimality conditions are given a dynamic interpretation.