| Foreword | p. vii |
| Linear Systems: Elimination Method | p. 1 |
| Examples of Linear Systems | p. 1 |
| A Review Example | p. 1 |
| Covering a Sphere with Hexagons and Pentagons | p. 2 |
| A Literal Example | p. 7 |
| Homogeneous Systems | p. 11 |
| A Chemical Reaction | p. 11 |
| Reduced Forms | p. 12 |
| Elimination Algorithm | p. 17 |
| Elementary Row Operations | p. 18 |
| Comparison of the Systems (S) and (HS) | p. 21 |
| Appendix | p. 22 |
| Potentials on a Grid | p. 22 |
| Another Illustration of the Fundamental Principle | p. 23 |
| The Euler Theorem f + v = e + 2 | p. 25 |
| Fullerenes, Radiolarians | p. 25 |
| Exercises | p. 26 |
| Vector Spaces | p. 31 |
| The Language | p. 31 |
| Axiomatic Properties | p. 31 |
| An Important Principle | p. 32 |
| Examples | p. 33 |
| Vector Subspaces | p. 35 |
| Finitely Generated Vector Spaces | p. 36 |
| Generators | p. 36 |
| Linear Independence | p. 39 |
| The Dimension | p. 41 |
| Infinite-Dimensional Vector Spaces | p. 44 |
| The Space of Polynomials | p. 45 |
| Existence of Bases: The Mathematical Credo | p. 47 |
| Infinite-Dimensional Examples | p. 49 |
| Appendix | p. 52 |
| Set Theory, Notation | p. 52 |
| Axioms for Fields of Scalars | p. 56 |
| Exercises | p. 56 |
| Matrix Multiplication | p. 60 |
| Row by Column Multiplication | p. 60 |
| Linear Fractional Transformations | p. 60 |
| Linear Changes of Variables | p. 61 |
| Definition of the Matrix Product | p. 62 |
| The Map Produced by Matrix Multiplication | p. 66 |
| Row Operations and Matrix Multiplication | p. 67 |
| Elementary Matrices | p. 68 |
| An Inversion Algorithm | p. 70 |
| LU Factorizations | p. 72 |
| Simultaneous Resolution of Linear Systems | p. 76 |
| Matrix Multiplication by Blocks | p. 76 |
| Explanation of the Method | p. 76 |
| The Field of Complex Numbers | p. 79 |
| Appendix | p. 80 |
| Affine Maps | p. 80 |
| The Field of Quaternions | p. 81 |
| The Strassen Algorithm | p. 82 |
| Exercises | p. 82 |
| Linear Maps | p. 88 |
| Linearity | p. 88 |
| Preliminary Considerations | p. 88 |
| Definition and First Properties | p. 90 |
| Examples of Linear Maps | p. 91 |
| General Results | p. 92 |
| Image and Kernel of a Linear Map | p. 92 |
| How to Construct Linear Maps | p. 94 |
| Matrix Description of Linear Maps | p. 95 |
| The Dimension Theorem for Linear Maps | p. 98 |
| The Rank-Nullity Theorem | p. 98 |
| Row-Rank versus Column-Rank | p. 99 |
| Application: Invertible Matrices | p. 101 |
| Isomorphisms | p. 102 |
| Generalities | p. 102 |
| Models of Finite-Dimensional Vector Spaces | p. 104 |
| Change of Basis: Components of Vectors | p. 105 |
| Change of Basis: Matrices of Linear Maps | p. 107 |
| The Trace of Square Matrices | p. 107 |
| Appendix | p. 108 |
| Inverting Maps Between Sets | p. 108 |
| Another Proof of Invertibility | p. 109 |
| Exercises | p. 112 |
| The Rank Theorem | p. 116 |
| More on Row- versus Column-Rank | p. 116 |
| Factorizations of a Matrix | p. 116 |
| Low Rank Examples | p. 117 |
| A Basis for the Column Space | p. 118 |
| Direct Sum of Vector Spaces | p. 119 |
| Sum of Two Subspaces | p. 119 |
| Supplementary Subspaces | p. 121 |
| Direct Sum of Two Subspaces | p. 123 |
| Independent Subspaces (General Case) | p. 125 |
| Finite Direct Sums of Vector Spaces | p. 126 |
| Projectors | p. 128 |
| An Example and General Definition | p. 128 |
| Geometrical Meaning of P[superscript 2] = P | p. 129 |
| Tricks of the Trade | p. 132 |
| Appendix | p. 133 |
| Pyramid of Ages | p. 133 |
| Color Theory | p. 134 |
| Genetics | p. 138 |
| Einstein Summation Convention | p. 139 |
| Exercises | p. 140 |
| Eigenvectors and Eigenvalues | p. 144 |
| Introduction | p. 144 |
| Definitions and Examples | p. 145 |
| Definitions | p. 145 |
| Simple 2 x 2 Examples | p. 146 |
| A 4 x 4 Example | p. 148 |
| Abstract Examples | p. 150 |
| General Results | p. 153 |
| Estimation of the Number of Eigenvalues | p. 153 |
| Localization of Eigenvalues | p. 154 |
| A Method for Finding Eigenvectors | p. 155 |
| Eigenvectors and Commutation | p. 156 |
| Applications of Eigenvectors | p. 157 |
| The Fibonacci Numbers | p. 157 |
| Diagonalization | p. 160 |
| Appendix | p. 162 |
| Eigenvectors of AB and of BA | p. 162 |
| Complements on the Fibonacci Numbers | p. 163 |
| Exercises | p. 163 |
| Inner-Product Spaces | p. 167 |
| About Multiplication and Products | p. 167 |
| The Dot Product in Plane Geometry | p. 168 |
| The Dot Product in R[superscript n] | p. 171 |
| Abstract Inner Products and Norms | p. 172 |
| Definition and Examples | p. 172 |
| The Cauchy-Schwarz-Bunyakovskii Inequality | p. 174 |
| The Pythagorean Theorem | p. 175 |
| More Identities | p. 176 |
| Orthonormal Bases | p. 179 |
| Euclidean Spaces | p. 179 |
| The Best Approximation Theorem | p. 181 |
| First Application: Periodic Functions | p. 183 |
| Second Application: Least Squares Method | p. 184 |
| Orthogonal Subspaces | p. 187 |
| Orthogonal of a Subset | p. 188 |
| The Support of a Linear Map | p. 189 |
| Least Squares Revisited | p. 192 |
| Appendix: Finite Probability Spaces | p. 194 |
| Random Variables | p. 194 |
| Algebras of Random Variables | p. 197 |
| Independence of Random Variables | p. 199 |
| Exercises | p. 200 |
| Symmetric Operators | p. 205 |
| Definition and First Properties | p. 205 |
| Intrinsic Characterization of Symmetry | p. 206 |
| General Properties of Symmetric Operators | p. 207 |
| Diagonalization | p. 208 |
| Statement of the Result | p. 208 |
| Existence of Eigenvectors | p. 209 |
| Inductive Construction | p. 211 |
| Applications | p. 212 |
| Quadratic Forms | p. 212 |
| Classification of Quadrics | p. 213 |
| Positive Definite Operators | p. 216 |
| Appendix | p. 219 |
| Principal Axes and Statistics | p. 219 |
| Functions of a Symmetric Operator | p. 220 |
| Special Configurations | p. 222 |
| Exercises | p. 225 |
| Duality | p. 227 |
| Geometric Introduction | p. 227 |
| Duality for Platonic Solids | p. 227 |
| The Pappus Theorem and its Dual | p. 229 |
| Dual of a Vector Space | p. 231 |
| Definition and First Properties | p. 231 |
| Dual Bases | p. 233 |
| Bidual of a Vector Space | p. 234 |
| Dual of a Normed Space | p. 235 |
| Dual Norm | p. 235 |
| Dual of a Euclidean Space | p. 236 |
| Dual of Important Norms in R[superscript n] | p. 238 |
| Transposition of Linear Maps | p. 240 |
| Transposition of Operators in Euclidean Spaces | p. 240 |
| Abstract Formulation of Transposition | p. 241 |
| Exercises | p. 243 |
| Determinants | p. 246 |
| From Space Geometry to Determinants | p. 247 |
| Areas in R[superscript 3] | p. 247 |
| The Cross Product in R[superscript 3] | p. 249 |
| The Scalar Triple Product | p. 251 |
| Volume Forms in Vector Spaces | p. 254 |
| Properties of Volume Forms: Uniqueness | p. 255 |
| Construction of Volume Forms in R[superscript n] | p. 258 |
| Determinant of an Operator | p. 260 |
| Volume-Amplification Factor | p. 260 |
| Determinants and Row Operations | p. 263 |
| Examples of Determinants | p. 266 |
| Geometric Examples | p. 267 |
| Arithmetic and Algebraic Examples | p. 268 |
| Examples in Calculus | p. 270 |
| Symbolic Determinants | p. 272 |
| Appendix | p. 274 |
| Permutations and Signs | p. 274 |
| More Examples | p. 275 |
| Exercises | p. 277 |
| Applications | p. 285 |
| The Characteristic Polynomial | p. 285 |
| Definition and Basic Properties | p. 285 |
| Examples | p. 287 |
| The Spectrum of an Operator | p. 288 |
| Changing the Field of Scalars | p. 288 |
| Roots of the Characteristic Polynomial | p. 289 |
| Existence of a Complex Eigenvalue | p. 293 |
| Cramer's Rule | p. 294 |
| Solution of Regular Linear Systems | p. 294 |
| Inversion of a Matrix | p. 297 |
| LU Factorizations: Necessary Condition | p. 298 |
| Construction of Orthonormal Bases | p. 299 |
| A Selection of Important Results | p. 301 |
| The Frobenius and Cayley-Hamilton Theorems | p. 301 |
| Restricting Scalars from C to R | p. 304 |
| Appendix | p. 306 |
| Back to AB and BA | p. 306 |
| Covariant Components | p. 307 |
| Series of Matrices | p. 308 |
| Exercises | p. 310 |
| Normal Operators | p. 315 |
| Orthogonal Matrices | p. 315 |
| General Properties | p. 315 |
| Geometric Properties | p. 318 |
| Spectral Properties | p. 319 |
| Transposition and Normal Operators | p. 321 |
| Skew-Symmetric Operators | p. 322 |
| Back to Orthogonal Operators | p. 323 |
| Normal Operators, Spectral Properties | p. 324 |
| Hermitian Inner Products | p. 325 |
| Hermitian Inner Product in C[superscript n] | p. 325 |
| The Adjoint of an Operator | p. 326 |
| Special Classes of Complex Operators | p. 327 |
| The Spectral Theorem for Normal Operators | p. 329 |
| Appendix | p. 330 |
| General Properties of Isometries | p. 330 |
| The Polar Decomposition | p. 331 |
| The Singular Value Decomposition | p. 333 |
| Anti-Commutation Relations | p. 337 |
| Exercises | p. 339 |
| Helpful Supplements | p. 345 |
| Some Hints for the Exercises | p. 345 |
| Answers to Some Exercises | p. 350 |
| Review Exercises | p. 354 |
| Axioms for Fields and Vector Spaces | p. 363 |
| Summary for the Cross Product in R[superscript 3] | p. 364 |
| Inner Products, Norms, and Distances | p. 366 |
| The Greek Alphabet | p. 367 |
| References | p. 368 |
| Index | p. 369 |
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