Linear Algebra and Geometry de P. K. Suetin

Linear Algebra and Geometry: Algebra, Logic and Applications

P. K. Suetin

Alexandra I. Kostrikin

Yu I Manin

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.

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Cuprins

Preface CHAPTER 1 Linear Spaces and Linear Mappings 1 Linear Spaces 2 Basis and Dimension 3 Linear Mappings 4 Matrices 5 Subspaces and Direct Sums 6 Quotient Spaces 7 Duality 8 The Structure of a Linear Mapping 9 The Jordan Normal Form 10 Normed Linear Spaces 11 Functions of Linear Operators 12 Complexification and Decomplexification 13 The Language of Categories 14 The Categorical Properties of Linear Spaces CHAPTER 2 Geometry of Spaces with an Inner Product 1 On Geometry 2 Inner Products 3 Classification Theorems 4 The Orthogonalization Algorithm and Orthogonal Polynomials 5 Euclidean Spaces 6 Unitary Spaces 7 Orthogonal and Unitary Operators 8 Self-Adjoint Operators 9 Self-Adjoint Operators in Quantum Mechanics 10 The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators 111 Three-Dimensional Euclidean Space 12 Minkowski Space 13 Symplectic Space 14 Witt's Theorem and Witt's Group 15 Clifford Algebras CHAPTER 3 Affine and Projective Geometry 1 Affine Spaces, Affine Mappings, and Affine Coordinates 2 Affine Groups 3 Affine Subspaces 4 Convex Polyhedra and Linear Programming 5 Affine Quadratic Functions and Quadrics 6 Projective Spaces 7 Projective Duality and Projective Quadrics 8 Projective Groups and Projections 9 Desargues' and Pappus' Configurations and Classical Projective Geometry 10 The Kahler Metric 11 Algebraic Varieties and Hilbert Polynomials CHAPTER 4 Multilinear Algebra 1 Tensor Products of Linear Spaces 2 Canonical Isomorphisms and Linear Mappings of Tensor Products 3 The Tensor Algebra of a Linear Space 4 Classical Notation 5 Symmetric Tensors 6 Skew-Symmetric Tensors and the Exterior Algebra of a Linear Space 7 Exterior Forms 8 Tensor Fields 9 Tensor Products in Quantum Mechanics