Linear Algebra and Geometry (Algebra, Logic and Applications)

✍ Scribed by P. K. Suetin, Alexandra I. Kostrikin, Yu I Manin

• Publisher: CRC Press
• Year: 1989
• Tongue: English
• Leaves: 321
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
ALGEBRA, LOGIC AND APPLICATIONS
ISBN 9056990497
Contents
Preface to the Paperback Edition
Bibliography
CHAPTER 1 Linear Spaces and Linear Mappings
1 Linear Spaces
2 Basis and Dimension
3 Linear Mappings
4 Matrices
5 Subspaces and Direct S urns
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
11 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 Kahier 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
Index