  We answer a number of open problems in frame theory concerning the
decomposition of frames into linearly independent and/or spanning sets. We
prove that in finite dimensional Hilbert spaces, Parseval frames with norms
bounded away from 1 can be decomposed into a number of sets whose complements
are spanning, where the number of these sets only depends on the norm bound. We
also prove, assuming the Kadison-Singer conjecture is true, that this holds for
infinite dimensional Hilbert spaces. Further, we prove a stronger result for
Parseval frames whose norms are uniformly small, which shows that in addition
to the spanning property, the sets can be chosen to be independent, and the
complement of each set to contain a number of disjoint, spanning sets.
