Linear Algebra and Geometry
by Kostrikin; Alexandra I.Format: Hardcover
Pub. Date: 1989-07-14
Publisher(s): CRC Press
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Summary
This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics Differing from most existing textbooks in approach, the work illustrate 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 Kahter's metric,-the theory of Hilbert polynomials, and projective and affine geometries Unusual in its extensive use of applications in physics to clarify each topic, this comprehensive volume will 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. Book jacket.
Author Biography
Aleksei I Kostrikin is currently a Corresponding Member of the USSR Academy of Sciences and holds the Chair in Algebra at Moscow State University A winner of the USSR Award in Mathematics in 1968, Professor Kostrikin's main research interests are Lie algebras and finite groups Yuri I Manin is currently Senior Research Staff Member at the Steklov Institute of the Academy of Sciences of the USSR and Professor of Algebra at Moscow State University. Professor Manin has been awarded the Lenin Prize for work in algebraic geometry and the Brouwer Gold Medal for work in number theory. His research interests also include differential equations and quantum field theory
Table of Contents
| Preface | p. vii |
| Bibliography | p. ix |
| Linear Spaces and Linear Mappings | p. 1 |
| Linear Spaces | p. 1 |
| Basis and Dimension | p. 8 |
| Linear Mappings | p. 16 |
| Matrices | p. 22 |
| Subspaces and Direct Sums | p. 34 |
| Quotient Spaces | p. 43 |
| Duality | p. 48 |
| The Structure of a Linear Mapping | p. 52 |
| The Jordan Normal Form | p. 58 |
| Normed Linear Spaces | p. 66 |
| Functions of Linear Operators | p. 72 |
| Complexification and Decomptexification | p. 75 |
| The Language of Categories | p. 81 |
| The Categorical Properties of Linear Spaces | p. 87 |
| Geometry of Spaces with an Inner Product | p. 92 |
| On Geometry | p. 92 |
| Inner Products | p. 94 |
| Classification Theorems | p. 101 |
| The Orthogonalization Algorithm and Orthogonal Polynomials | p. 109 |
| Euclidean Spaces | p. 117 |
| Unitary Spaces | p. 127 |
| Orthogonal and Unitary Operators | p. 134 |
| Self-Adjoint Operators | p. 138 |
| Self-Adjoint Operators in Quantum Mechanics | p. 148 |
| The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators | p. 156 |
| Three-Dimensional Euclidean Space | p. 164 |
| Minkowski Space | p. 173 |
| Symplectic Space | p. 182 |
| Witt's Theorem and Witt's Group | p. 187 |
| Clifford Algebras | p. 190 |
| Affine and Projective Geometry | p. 195 |
| Affine Spaces, Affine Mappings, and Affine Coordinates | p. 195 |
| Affine Groups | p. 203 |
| Affine Subspaces | p. 207 |
| Convex Polyhedra and Linear Programming | p. 215 |
| Affine Quadratic Functions and Qaadrics | p. 218 |
| Projective Spaces | p. 222 |
| Projective Duality and Projective Quadrics | p. 228 |
| Projective Groups and Projections | p. 233 |
| Desargues' and Pappus' Configurations and Classical Projective Geometry | p. 242 |
| The Kahler Metric | p. 247 |
| Algebraic Varieties and Hilbert Polynomials | p. 249 |
| Multilinear Algebra | p. 258 |
| Tensor Products of Linear Spaces | p. 258 |
| Canonical Isomorphisms and Linear Mappings of Tensor Products | p. 263 |
| The Tensor Algebra of a Linear Space | p. 269 |
| Classical Notation | p. 271 |
| Symmetric Tensors | p. 276 |
| Skew-Symmetric Tensors and the Exterior Algebra or a Linear Space | p. 279 |
| Exterior Forms | p. 290 |
| Tensor Fields | p. 293 |
| Tensor Products in Quantum Mechanics | p. 297 |
| Index | p. 303 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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