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Based on material taught to final-year students of theoretical physics, this is an introduction to the essentials of vector spaces and linear operators. It builds bridges between the concepts and mathematics of classical physics, and the mathematical framework employed in quantum mechanics. The axioms of nonrelativistic quantum theory are introduced, and shown to lead to a variety of new conceptual problems. Subjects discussed include state-vector reduction, the problem of measurement, quantum entanglement, the Kochen-Specker theorem, and the Bell inequalities. The book includes 25 problems with worked solutions.