Abstract: Roughly speaking, this talk is about questions of the following general type on collections of matrices or linear operators: if we know something about the collection to be "small" in some sense, when can we say the collection is itself small? A classical example of the kind of results we are interested in is the old theorem that if a group G of complex matrices is irreducible (i.e., the members of G have no nontrivial invariant subspace in common), and if the traces of members of G form a finite set, then G is itself finite. We'll consider (a) measures of smallness other than finiteness, e.g., boundedness; and (b) linear functionals other than just trace.