| Vector Spaces | p. 1 |
| Introduction | p. 1 |
| Geometrical Vectors in a Plane | p. 2 |
| Vectors in a Cartesian (Analytic) Plane R[superscript 2] | p. 5 |
| Scalar Multiplication (The Product of a Number with a Vector) | p. 7 |
| The Dot Product of Two Vectors (or the Euclidean Inner Product of Two Vectors in R[superscript 2]) | p. 8 |
| Applications of the Dot Product and Scalar Multiplication | p. 10 |
| Vectors in Three-Dimensional Space (Spatial Vectors) | p. 15 |
| The Cross Product in R[superscript 3] | p. 18 |
| The Mixed Triple Product in R[superscript 3]. Applications of the Cross and Mixed Products | p. 21 |
| Equations of Lines in Three-Dimensional Space | p. 24 |
| Equations of Planes in Three-Dimensional Space | p. 26 |
| Real Vector Spaces and Subspaces | p. 28 |
| Linear Dependence and Independence. Spanning Subsets and Bases | p. 30 |
| The Three Most Important Examples of Finite-Dimensional Real Vector Spaces | p. 33 |
| The Vector Space R[superscript n] (Number Columns) | p. 33 |
| The Vector Space R[subscript n x n] (Matrices) | p. 35 |
| The Vector Spaces P[subscript 3] (Polynomials) | p. 37 |
| Some Special Topics about Matrices | p. 39 |
| Matrix Multiplication | p. 39 |
| Some Special Matrices | p. 40 |
| Determinants | p. 45 |
| Definitions of Determinants | p. 45 |
| Properties of Determinants | p. 49 |
| Linear Mappings and Linear Systems | p. 59 |
| A Short Plan for the First 5 Sections of Chapter 2 | p. 59 |
| Some General Statements about Mapping | p. 60 |
| The Definition of Linear Mappings (Linmaps) | p. 62 |
| The Kernel and the Range of L | p. 63 |
| The Quotient Space V[subscript n]/ker L and the Isomorphism V[subscript n]/ker [characters not reproducible] ran L | p. 65 |
| Representation Theory | p. 67 |
| The Vector Space L(V[subscript n], W[subscript m]) | p. 68 |
| The Linear Map M : R[superscript n] to R[superscript m] | p. 69 |
| The Three Isomorphisms v, w and v - w | p. 70 |
| How to Calculate the Representing Matrix M | p. 72 |
| An Example (Representation of a Linmap Which Acts between Vector Spaces of Polynomials) | p. 75 |
| Systems of Linear Equations (Linear Systems) | p. 79 |
| The Four Tasks | p. 85 |
| The Column Space and the Row Space | p. 86 |
| Two Examples of Linear Dependence of Columns and Rows of a Matrix | p. 88 |
| Elementary Row Operations (Eros) and Elementary Matrices | p. 91 |
| Eros | p. 91 |
| Elementary Matrices | p. 93 |
| The GJ Form of a Matrix | p. 95 |
| An Example (Preservation of Linear Independence and Dependence in GJ Form) | p. 97 |
| The Existence of the Reduced Row-Echelon (GJ) Form for Every Matrix | p. 99 |
| The Standard Method for Solving A X = b | p. 101 |
| When Does a Consistent System A X = b Have a Unique Solution? | p. 102 |
| When a Consistent System A X = b Has No Unique Solution | p. 108 |
| The GJM Procedure - a New Approach to Solving Linear Systems with Nonunique Solutions | p. 109 |
| Detailed Explanation | p. 110 |
| Summary of Methods for Solving Systems of Linear Equations | p. 116 |
| Inner-Product Vector Spaces (Euclidean and Unitary Spaces) | p. 119 |
| Euclidean Spaces E[subscript n] | p. 119 |
| Unitary Spaces U[subscript n] (or Complex Inner-product Vector Spaces) | p. 126 |
| Orthonormal Bases and the Gram-Schmidt Procedure for Orthonormalization of Bases | p. 131 |
| Direct and Orthogonal Sums of Subspaces and the Orthogonal Complement of a Subspace | p. 139 |
| Direct and Orthogonal Sums of Subspaces | p. 139 |
| The Orthogonal Complement of a Subspace | p. 141 |
| Dual Spaces and the Change of Basis | p. 145 |
| The Dual Space U*[subscript n] of a Unitary Space U[subscript n] | p. 145 |
| The Adjoint Operator | p. 153 |
| The Change of Bases in V[subscript n](F) | p. 157 |
| The Change of the Matrix-Column [xi] that Represents a Vector x [set membership] V[subscript n](F) (Contravariant Vectors) | p. 158 |
| The Change of the n x n Matrix A That Represents an Operator A [set membership] L(V[subscript n](F), V[subscript n](F)) (Mixed Tensor of the Second Order) | p. 159 |
| The Change of Bases in Euclidean (E[subscript n]) and Unitary (U[subscript n]) Vector Spaces | p. 162 |
| The Change of Biorthogonal Bases in V*[subscript n](F) (Covariant Vectors) | p. 164 |
| The Relation between V[subscript n](F) and V*[subscript n](F) is Symmetric (The Invariant Isomorphism between V[subscript n](F) and V**[subscript n] (F)) | p. 167 |
| Isodualism-The Invariant Isomorphism between the Superspaces L(V[subscript n](F), V[subscript n](F)) and L(V*[subscript n](F),V*[subscript n](F)) | p. 168 |
| The Eigen Problem or Diagonal Form of Representing Matrices | p. 173 |
| Eigenvalues, Eigenvectors, and Eigenspaces | p. 173 |
| Diagonalization of Square Matrices | p. 180 |
| Diagonalization of an Operator in U[subscript n] | p. 183 |
| Two Examples of Normal Matrices | p. 188 |
| The Actual Method for Diagonalization of a Normal Operator | p. 191 |
| The Most Important Subsets of Normal Operators in U[subscript n] | p. 194 |
| The Unitary Operators A[superscript dagger] = A[superscript -1] | p. 194 |
| The Hermitian Operators A[superscript dagger] = A | p. 198 |
| The Projection Operators P[superscript dagger] = P = P[superscript 2] | p. 200 |
| Operations with Projection Operators | p. 203 |
| The Spectral Form of a Normal Operator A | p. 207 |
| Diagonalization of a Symmetric Operator in E[subscript 3] | p. 208 |
| The Actual Procedure for Orthogonal Diagonalization of a Symmetric Operator in E[subscript 3] | p. 214 |
| Diagonalization of Quadratic Forms | p. 218 |
| Conic Sections in R[superscript 2] | p. 220 |
| Canonical Form of Orthogonal Matrices | p. 228 |
| Orthogonal Matrices in R[superscript n] | p. 228 |
| Orthogonal Matrices in R[superscript 2] (Rotations and Reflections) | p. 229 |
| The Canonical Forms of Orthogonal Matrices in R[superscript 3] (Rotations and Rotations with Inversions) | p. 240 |
| Tensor Product of Unitary Spaces | p. 243 |
| Kronecker Product of Matrices | p. 243 |
| Axioms for the Tensor Product of Unitary Spaces | p. 247 |
| The Tensor product of Unitary Spaces C[superscript m] and C[superscript n] | p. 247 |
| Definition of the Tensor Product of Unitary Spaces, in Analogy with the Previous Example | p. 249 |
| Matrix Representation of the Tensor Product of Unitary Spaces | p. 250 |
| Multiple Tensor Products of a Unitary Space U[subscript n] and of its Dual Space U*[subscript n] as the Principal Examples of the Notion of Unitary Tensors | p. 252 |
| Unitary Space of Antilinear Operators L[subscript a] (U[subscript m], U[subscript n]) as the Main Realization of U[subscript m] [plus sign in circle] U[subscript n] | p. 254 |
| Comparative Treatment of Matrix Representations of Linear Operators from L(U[subscript m], U[subscript n]) and Antimatrix Representations of Antilinear Operators from L[subscript a] (U[subscript m], U[subscript n]) = U[subscript m] [plus sign in circle] U[subscript n] | p. 257 |
| The Dirac Notation in Quantum Mechanics: Dualism between Unitary Spaces (Sect. 4.1) and Isodualism between Their Superspaces (Sect. 4.7) | p. 263 |
| Repeating the Statements about the Dualism D | p. 263 |
| Invariant Linear and Antilinear Bijections between the Superspaces L(U[subscript n], U[subscript n]) and L(U*[subscript n], U*[subscript n]) | p. 266 |
| Dualism between the Superspaces | p. 266 |
| Isodualism between Unitary Superspaces | p. 267 |
| Superspaces L(U[subscript n], U[subscript n]) [characters not reproducible] L(U*[subscript n], U*[subscript n]) as the Tensor Product of U[subscript n] and U*[subscript n], i.e., U[subscript n] [plus sign in circle] U*[subscript n] | p. 270 |
| The Tensor Product of U[subscript n] and U*[subscript n] | p. 270 |
| Representation and the Tensor Nature of Diads | p. 271 |
| The Proof of Tensor Product Properties | p. 272 |
| Diad Representations of Operators | p. 274 |
| Bibliography | p. 279 |
| Index | p. 281 |
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