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.
Author(s): Alexei I. Kostrikin, Yu. I. Manin
Series: Algebra, Logic and Applications, Volume 1
Edition: 1
Publisher: CRC Press
Year: 1989
Language: English
Pages: 322
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface to the Paperback Edition......Page 8
Bibliography......Page 0
1 Linear Spaces......Page 12
2 Basis and Dimension......Page 19
3 Linear Mappings......Page 27
4 Matrices......Page 33
5 Subspaces and Direct Sums......Page 45
6 Quotient Spaces......Page 54
7 Duality......Page 59
8 The Structure of a Linear Mapping......Page 63
9 The Jordan Normal Form......Page 69
10 Normed Linear Spaces......Page 77
11 Functions of Linear Operators......Page 83
12 Complexification and Decomplexification......Page 86
13 The Language of Categories......Page 92
14 The Categorical Properties of Linear Spaces......Page 98
1 On Geometry......Page 103
2 Inner Products......Page 105
3 Classification Theorems......Page 112
4 The Orthogonalization Algorithm and Orthogonal Polynomials......Page 120
5 Euclidean Spaces......Page 128
6 Unitary Spaces......Page 138
7 Orthogonal and Unitary Operators......Page 145
8 Self-Adjoint Operators......Page 149
9 Self-Adjoint Operators in Quantum Mechanics......Page 159
10 The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators......Page 167
11 Three-Dimensional Euclidean Space......Page 175
12 Minkowski Space......Page 184
13 Symplectic Space......Page 193
14 Witt's Theorem and Witt's Group......Page 198
15 Clifford Algebras......Page 201
1 Affine Spaces, Affine Mappings, and Affine Coordinates......Page 206
2 Affine Groups......Page 214
3 Affine Subspaces......Page 218
4 Convex Polyhedra and Linear Programming......Page 226
5 Affine Quadratic Functions and Quadrics......Page 229
6 Projective Spaces......Page 233
7 Projective Duality and Projective Quadrics......Page 239
8 Projective Groups and Projections......Page 244
9 Desargues' and Pappus' Configurations and Classical Projective Geometry......Page 253
10 The Kahier Metric......Page 258
11 Algebraic Varieties and Hilbert Polynomials......Page 260
1 Tensor Products of Linear Spaces......Page 269
2 Canonical Isomorphisms and Linear Mappings of Tensor Products......Page 274
3 The Tensor Algebra of a Linear Space......Page 280
4 Classical Notation......Page 282
5 Symmetric Tensors......Page 287
6 Skew-Symmetric Tensors and the Exterior Algebra of a Linear Space......Page 290
7 Exterior Forms......Page 301
8 Tensor Fields......Page 304
9 Tensor Products in Quantum Mechanics......Page 308
Index......Page 314
Back Cover......Page 322
