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This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some detail include normed linear spaces, functions of linear operators, the basic structures of quantum mechanics and an introduction to linear programming. Also discussed are Kahler's metic, the theory of Hilbert polynomials, and projective and affine geometries. Unusual in its extensive use of applications in physics to clarify each topic, this comprehensice volume should be of particular interest to advanced undergraduates and graduates in mathematics and physics, and to lecturers in linear and multilinear algebra, linear programming and quantum mechanics.
Part 1 Linear spaces and linear mappings: linear spaces; basis and dimension; linear mappings; matrices; subspaces and direct sums; quotient spaces; duality; the structure of a linear mapping; the Jordan normal form; normed linear spaces; functions of linear operators; complexification and decomplexification; the language of categories; the categorical properties of linear spaces. Part 2 Geometry of spaces with an inner product: on geometry; inner products; classification theorems; the orthogonalization algorithm and orthogonal polynomials; Euclidian spaces; unitary spaces; orthogonal and unitary operators; self-adjoint operators; self-adjoint operators in quantum mechanics; the geometry of quadratic forms and the Eigenvalues of self-adjoint operators; three-dimensional Euclidean space; Minkowski space; symplectic space; Witt's theorem and Witt's group; Clifford algebras. Part 3 Affine and projective geometry: affine spaces, affine mappings and affine coordinates; affine groups; affine subspaces; convex polyhedra and linear programming; affine quadratic functions and quadrics; projective duality and projective quadrics; projective groups and projections; Desargues' and Pappus' configurations and classical projective geometry; the Kahler metric; algebraic varieties and Hilbert polynomials. Part 4 Multilinear algebra: tensor products of linear spaces; canonical isomorphisms and linear mappings of tensor products; the tensor algebra of a linear space; classical notation; symmetric tensors; skew-symmetric tensors and the exterior algebra of a linear space; exterior forms; tensor fields; tensor products in quantum mechanics.
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- ID: 9782881246838
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