Linear Algebra and Geometry

Differing from most existing textbooks in approach, this work illustrates the various applications and connections of linear algebra with functional analysis, quantum mechanics, and algebraic and differential geometry.

Preface to the Paperback Edition

vii

Bibliography

ix

Linear Spaces and Linear Mappings

1

(91)

Linear Spaces

1

(7)

Basis and Dimension

8

(8)

Linear Mappings

16

(6)

Matrices

22

(12)

Subspaces and Direct Sums

34

(9)

Quotient Spaces

43

(5)

Duality

48

(4)

The Structure of a Linear Mapping

52

(6)

The Jordan Normal Form

58

(8)

Normed Linear Spaces

66

(6)

Functions of Linear Operators

72

(3)

Complexification and Decomplexification

75

(6)

The Language of Categories

81

(6)

The Categorical Properties of Linear Spaces

87

(5)

Geometry of Spaces with an Inner Product

92

(103)

On Geometry

92

(2)

Inner Products

94

(7)

Classification Theorems

101

(8)

The Orthogonalization Algorithm and Orthogonal Polynomials

109

(8)

Euclidean Spaces

117

(10)

Unitary Spaces

127

(7)

Orthogonal and Unitary Operators

134

(4)

Self-Adjoint Operators

138

(10)

Self-Adjoint Operators in Quantum Mechanics

148

(8)

The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators

156

(8)

Three-Dimensional Euclidean Space

164

(9)

Minkowski Space

173

(9)

Symplectic Space

182

(5)

Witt's Theorem and Witt's Group

187

(3)

Clifford Algebras

190

(5)

Affine and Projective Geometry

195

(63)

Affine Spaces, Affine Mappings, and Affine Coordinates

195

(8)

Affine Groups

203

(4)

Affine Subspaces

207

(8)

Convex Polyhedra and Linear Programming

215

(3)

Affine Quadratic Functions and Quadrics

218

(4)

Projective Spaces

222

(6)

Projective Duality and Projective Quadrics

228

(5)

Projective Groups and Projections

233

(9)

Desargues' and Pappus' Configurations and Classical Projective Geometry

242

(5)

The Kahler Metric

247

(2)

Algebraic Varieties and Hilbert Polynomials

249

(9)

Multilinear Algebra

258

(45)

Tensor Products of Linear Spaces

258

(5)

Canonical Isomorphisms and Linear Mappings of Tensor Products

263

(6)

The Tensor Algebra of a Linear Space

269

(2)

Classical Notation

271

(5)

Symmetric Tensors

276

(3)

Skew-Symmetric Tensors and the Exterior Algebra of a Linear Space

279

(11)

Exterior Forms

290

(3)

Tensor Fields

293

(4)

Tensor Products in Quantum Mechanics

297

(6)

Index

303

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Linear Algebra and Geometry

Summary

Table of Contents

Supplemental Materials

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Linear Algebra and Geometry

Summary

Table of Contents

Supplemental Materials

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