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The orthogonal group O(1, n ) acts by norm-preserving transformations on Minkowski space R 1, n , and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic n -space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O( n ) and O(1), where O( n ) acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.