Includes bibliographical references and index.

1 Vector Spaces
2 1.1 Introduction
3 1.2 Geometrical Vectors in a Plane
4 1.3 Vectors in a Cartesian (Analytic) Plane R2
5 1.4 Scalar Multiplication (The Product of a Number with a Vector)
6 1.5 The Dot Product of Two Vectors (or the Euclidean Inner Product
7 of Two Vectors in R2)
8 1.6 Applications of the Dot Product and Scalar Multiplication
9 1.7 Vectors in Three-Dimensional Space (Spatial Vectors) ............ 15
10 1.8 The Cross Product inR3 .................... ................. 18
11 1.9 The Mixed Triple Product in RR3. Applications of the Cross
12 and M ixed Products ........................... ............ 21
13 1.10 Equations of Lines in Three-Dimensional Space ................. 24
14 1.11 Equations of Planes in Three-Dimensional Space ................ 26
15 1.12 Real Vector Spaces and Subspaces ......................... 28
16 1.13 Linear Dependence and Independence. Spanning Subsets and Bases 30
17 1.14 The Three Most Important Examples of Finite-Dimensional Real
18 Vector Spaces .............................................. 33
19 1.14.1 The Vector Space Rn" (Number Columns) ................ 33
20 1.14.2 The Vector Space Rn,xn (Matrices) ...................... 35
21 1.14.3 The Vector Space P3 (Polynomials) ..................... 37
22 1.15 Some Special Topics about Matrices ......................... 39
23 1.15.1 Matrix Multiplication .............. .............. 39
24 1.15.2 Some Special Matrices ............................ 40
25 A Determinants ............................................ 45
A. I Definitions of Determinants ............... ............... 45
26 A.2 Properties of Determinants ................. ............... 49
27 2 Linear Mappings and Linear Systems ........................ 59
28 2.1 A Short Plan for the First 5 Sections of Chapter 2 ................ 59
29 2.2 Some General Statements about Mapping ...................... 60
30 2.3 The Definition of Linear Mappings (Linmaps) .................. 62
31 2.4 The Kernel and the Range of L ............................. 63
31.1.1.1.1 L'
32 2.5 The Quotient Space V,/ker L and the Isomorphism Vn/ker - ran L 65
33 2.6 Representation Theory ................ ................... 67
34 2.6.1 The Vector Space L(V,,Wm) .......................... 68
35 2.6.2 The Linear Map M:R - R'm ........................ 69
36 2.6.3 The Three Isomorphisms v, w and v - w ................. 70
37 2.6.4 How to Calculate the Representing Matrix M ............. 72
38 2.7 An Example (Representation of a Linmap Which Acts between
39 Vector Spaces of Polynomials) ............................. 75
40 2.8 Systems of Linear Equations (Linear Systems) .................. 79
41 2.9 The Four Tasks ...................................... 85
42 2.10 The Column Space and the Row Space ........................ 86
43 2.11 Two Examples of Linear Dependence of Columns
44 and Rows of a Matrix ...................................... 88
45 2.12 Elementary Row Operations (Eros) and Elementary Matrices ...... 91
46 2.12.1 Eros ........................................ 91
47 2.12.2 Elementary Matrices ............................. 93
48 2.13 The GJ Form of a Matrix ........................ ........... 95
49 2.14 An Example (Preservation of Linear Independence
50 and Dependence in GJ Form) ................................. 97
51 2.15 The Existence of the Reduced Row-Echelon (GJ)
52 Form for Every Matrix ................................. 99
53 2.16 The Standard Method for Solving A = b ..................... 101
54 2.16.1 When Does a Consistent System A-f = b Have
54.1 a Unique Solution? ........... ...................... 102
55 2.16.2 When a Consistent System A- = b Has No
55.1 Unique Solution ..................................... 108
56 2.17 The GJM Procedure - a New Approach to Solving Linear Systems
57 with Nonunique Solutions ................ ................ 109
58 2.17.1 Detailed Explanation ................................ 110
59 2.18 Summary of Methods for Solving Systems of Linear Equations .... 116
60 3 Inner-Product Vector Spaces (Euclidean and Unitary Spaces) ....... 119
61 3.1 Euclidean Spaces E, ........................................ 119
62 3.2 Unitary Spaces U, (or Complex Inner-product Vector Spaces) ..... 126
63 3.3 Orthonormal Bases and the Gram-Schmidt Procedure
64 for Orthonormalization of Bases .............................. 131
65 3.4 Direct and Orthogonal Sums of Subspaces and the Orthogonal
66 Complement of a Subspace ................... ............. 139
67 3.4.1 Direct and Orthogonal Sums of Subspaces ............... 139
68 3.4.2 The Orthogonal Complement of a Subspace .............. 141
69 4 Dual Spaces and the Change of Basis ........................ 145
70 4.1 The Dual Space U,* of a Unitary Space U, .................... 145
71 4.2 The Adjoint Operator ................ ................... . 153
72 4.3 The Change of Bases in V,(F) ................................ 157
73 4.3.1 The Change of the Matrix-Column 4 That Represents
73.1 a Vector x E V,(F) (Contravariant Vectors) ............... 158
74 4.3.2 The Change of the n x n Matrix a.d That Represents
74.1 an Operator A E L(V,,(F),V,n(F)) (Mixed Tensor
74.2 of the Second Order) ................................. 159
75 4.4 The Change of Bases in Euclidean (E,) and Unitary (U,) Vector
76 Spaces........................ . .... ...... ... . ............ 162
77 4.5 The Change of Biorthogonal Bases in V*(F)
78 (Covariant Vectors) ......................................... 164
79 4.6 The Relation between V,(F) and V(F) is Symmetric
80 (The Invariant Isomorphism between V,n(F) and V* (F)) .......... 167
81 4.7 Isodualism-The Invariant Isomorphism between the Superspaces
82 L(V,(F),V,,(F)) andL(Vn,(F), V* (F)) ......................... 168
83 5 The Eigen Problem or Diagonal Form of Representing Matrices ..... 173
84 5.1 Eigenvalues, Eigenvectors, and Eigenspaces .................... 173
85 5.2 Diagonalization of Square Matrices ......................... 180
86 5.3 Diagonalization of an Operator in U, ......................... 183
87 5.3.1 Two Examples of Normal Matrices .................. .. 188
88 5.4 The Actual Method for Diagonalization of a Normal Operator ..... 191
89 5.5 The Most Important Subsets of Normal Operators in U, .......... 194
90 5.5.1 The Unitary Operators At = A- ..................... 194
91 5.5.2 The Hermitian Operators At = A ....................... 198
92 5.5.3 The Projection Operators Pt = p = P2 .................. 200
93 5.5.4 Operations with Projection Operators ................... 203
94 5.5.5 The Spectral Form of a Normal Operator A ............... 207
95 5.6 Diagonalization of a Symmetric Operator in E3 ................. 208
96 5.6.1 The Actual Procedure for Orthogonal Diagonalization
96.1 of a Symmetric Operator in E3 ......................... 214
97 5.6.2 Diagonalization of Quadratic Forms .................... 218
98 5.6.3 Conic Sections in 1R2 .............................. 220
99 5.7 Canonical Form of Orthogonal Matrices ....................... 228
100 5.7.1 Orthogonal Matrices in 1R ......... ............... 228
101 5.7.2 Orthogonal Matrices in R2 (Rotations and Reflections) ..... 229
102 5.7.3 The Canonical Forms of Orthogonal Matrices in R3
102.1 (Rotations and Rotations with Inversions) ................ 240
103 6 Tensor Product of Unitary Spaces ........................... 243
104 6.1 Kronecker Product of Matrices ........................... . 243
105 6.2 Axioms for the Tensor Product of Unitary Spaces ................ 247
106 6.2.1 The Tensor product of Unitary Spaces Cm and Cn ......... 247
107 6.2.2 Definition of the Tensor Product of Unitary Spaces,
107.1 in Analogy with the Previous Example ............ . 249
108 6.3 Matrix Representation of the Tensor Product of Unitary Spaces .... 250
109 6.4 Multiple Tensor Products of a Unitary Space U, and of its Dual
110 Space U,* as the Principal Examples of the Notion of Unitary
111 Tensors ..................................................252
112 6.5 Unitary Space of Antilinear Operators La(U, Un) as the Main
113 Realization of Urn Un ....................... .............. 254
114 6.6 Comparative Treatment of Matrix Representations of Linear
115 Operators from L(Um, U,,) and Antimatrix Representations
116 of Antilinear Operators from La (U., Un,) = U,rn Un ............. 257
117 7 The Dirac Notation in Quantum Mechanics: Dualism
118 between Unitary Spaces (Sect. 4.1) and Isodualism
119 between Their Superspaces (Sect. 4.7) .......................... 263
120 7.1 Repeating the Statements about the Dualism D .................. 263
121 7.2 Invariant Linear and Antilinear Bijections between
122 the Superspaces L(U,,, U,) and L(U,, U,*)
123 7.2.1 Dualism between the Superspaces
124 7.2.2 Isodualism between Unitary Superspaces
125 7.3 Superspaces L(U,, Un) c L(Ul, U*) as the Tensor Product of U,
126 and Ut , i.e., U
127 7.3.1 The Tensor Product of U, and UW
128 7.3.2 Representation and the Tensor Nature of Diads
129 7.3.3 The Proof of Tensor Product Properties
130 7.3.4 Diad Representations of Operators
131 Bibliography