A central problem in built-in self test (BIST) is how to efficiently generate a small set of test vectors that detect all targeted faults. We propose a novel solution that uses linear algebraic concepts to partition the vector space of tests into subspaces (clusters). A subspace is defined by a compact set of basis vectors. We give an algorithm to compute sets of basis vectors defining the clusters. We also describe a low-cost logic circuit based on Gray codes that reproduces the subspaces from these basis vectors. Experimental results are presented which show that this approach reduces on-chip hardware overhead and test application time, while also guaranteeing full fault coverage.