To lie or not to lie in a subspace

We give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data from a union of subspaces, it is because such data really lies in a subspace. Furthermore, we give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data, such subspace is unique. We do this by characterizing when and only when a set of incomplete vectors behaves as a single but complete one.