This paper develops a new mathematical framework for studying the subspace-segmentation problem. We examine some important algebraic properties of subspace arrangements that are closely related to the subspace-segmentation problem. More specifically, we introduce an important class of invariants given by the Hilbert functions. We show that there exist rich relations between subspace arrangements and their corresponding Hilbert functions. We propose a new subspace-segmentation algorithm, and showcase two applications to demonstrate how the new theoretical revelation may solve subspace segmentation and model selection problems under less restrictive conditions with improved results.