Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in $L^{\infty }\left(\Omega \right)$; they were first introduced by Bownik and Ross (2014). In this paper we work with MI spaces that are finitely generated. We prove that almost every set of functions constructed by taking linear combinations of the generators of a finitely generated MI space is a new set of generators for the same space, and we give necessary and sufficient conditions on the linear...