  Given two tuples of subspaces, can you tell whether the tuples are
isomorphic? We develop theory and algorithms to address this fundamental
question. We focus on isomorphisms in which the ambient vector space is acted
on by either a unitary group or general linear group. If isomorphism also
allows permutations of the subspaces, then the problem is at least as hard as
graph isomorphism. Otherwise, we provide a variety of polynomial-time
algorithms with Matlab implementations to test for isomorphism. Keywords:
subspace isomorphism, Grassmannian, Bargmann invariants, $H^\ast$-algebras,
quivers, graph isomorphism
