The Revolt of the Masses

Cover......Page 1 Title page......Page 4 Copyright page......Page 5 Contents......Page 6 Preface for the Instructor......Page 12 Preface for the Student......Page 16 Acknowledgments......Page 17 Vector Spaces......Page 18 1.2 Example......Page 19 1.4 Example......Page 20 1.6 Notation......Page 21 1.8 Definition......Page 22 1.11 Example......Page 23 1.14 Definition......Page 24 1.15 Example......Page 25 1.16 Definition......Page 26 Digression on Fields......Page 27 EXERCISES 1.A......Page 28 1.19 Definition......Page 29 1.22 Example......Page 30 1.24 Example......Page 31 1.27 Notation......Page 32 1.30 A number times the vector 0......Page 33 EXERCISES 1.B......Page 34 1.34 Conditions for a subspace......Page 35 1.35 Example......Page 36 1.39 Sum of subspaces is the smallest containing subspace......Page 37 1.41 Example......Page 38 1.43 Example......Page 39 1.45 Direct sum of two subspaces......Page 40 EXERCISES 1.C......Page 41 2.1 Notation......Page 44 2.4 Example......Page 45 2.7 Span is the smallest containing subspace......Page 46 2.11 Definition......Page 47 2.15 Definition......Page 48 2.17 Definition......Page 49 2.20 Example......Page 50 2.22......Page 51 2.23 Length of linearly independent list ≤ length of spanning list......Page 52 2.26 Finite-dimensional subspaces......Page 53 EXERCISES 2.A......Page 54 2.30......Page 56 2.31 Spanning list contains a basis......Page 57 2.33 Linearly independent list extends to a basis......Page 58 2.34 Every subspace of V is part of a direct sum equal to V......Page 59 EXERCISES 2.B......Page 60 2.37 Example......Page 61 2.40 Example......Page 62 2.42 Spanning list of the right length is a basis......Page 63 2.43 Dimension of a sum......Page 64 EXERCISES 2.C......Page 65 3.1 Notation......Page 67 3.4 Example......Page 68 3.5 Linear maps and basis of domain......Page 70 3.8 Definition......Page 71 3.10 Example......Page 72 EXERCISES 3.A......Page 73 3.13 Example......Page 75 3.15 Definition......Page 76 3.18 Example......Page 77 3.21 Example......Page 78 3.22 Fundamental Theorem of Linear Maps......Page 79 3.24 A map to a larger dimensional space is not surjective......Page 80 3.26 Homogeneous system of linear equations......Page 81 3.29 Inhomogeneous system of linear equations......Page 82 EXERCISES 3.B......Page 83 3.32 Definition......Page 86 3.33 Example......Page 87 3.35 Definition......Page 88 3.39 Notation......Page 89 Matrix Multiplication......Page 90 3.43 The matrix of the product of linear maps......Page 91 3.48 Example......Page 92 3.52 Linear combination of columns......Page 93 EXERCISES 3.C......Page 94 3.56 Invertibility is equivalent to injectivity and surjectivity......Page 96 3.57 Example......Page 97 3.59 Dimension shows whether vector spaces are isomorphic......Page 98 3.61 dimL(V;W) = (dimV)(dimW)......Page 99 3.63 Example......Page 100 3.66......Page 101 3.68 Example......Page 102 3.70 Example......Page 103 EXERCISES 3.D......Page 104 3.73 Product of vector spaces is a vector space......Page 107 3.76 Dimension of a product is the sum of dimensions......Page 108 3.78 A sum is a direct sum if and only if dimensions add up......Page 109 3.82 Example......Page 110 3.85 Two affine subsets parallel to U are equal or disjoint......Page 111 3.87 Quotient space is a vector space......Page 112 3.90 Definition......Page 113 EXERCISES 3.E......Page 114 3.95 dim V' = dim V......Page 117 3.98 Dual basis is a basis of the dual space......Page 118 3.100 Example......Page 119 3.103 Example......Page 120 3.105 The annihilator is a subspace......Page 121 3.107 The null space of T'......Page 122 3.109 The range of T'......Page 123 3.110 T' injective is equivalent to T' surjective......Page 124 3.113 The transpose of the product of matrices......Page 125 3.114 The matrix of T' is the transpose of the matrix of T'......Page 126 3.116 Example......Page 127 3.119 Definition......Page 128 EXERCISES 3.F......Page 129 4.1 Notation......Page 133 4.4 Example......Page 134 4.5 Properties of complex numbers......Page 135 4.7 If a polynomial is the zero function, then all coefficients are 0......Page 136 4.8 Division Algorithm for Polynomials......Page 137 4.11 Each zero of a polynomial corresponds to a degree-1 factor......Page 138 Factorization of polynomials over C......Page 139 4.13 Fundamental Theorem of Algebra......Page 140 4.14 Factorization of a polynomial over C......Page 141 4.16 Factorization of a quadratic polynomial......Page 142 4.17 Factorization of a polynomial over R......Page 144 EXERCISES 4......Page 145 5.1 Notation......Page 147 5.3 Example......Page 148 Eigenvalues and Eigenvectors......Page 149 5.7 Definition......Page 150 5.9......Page 151 5.13 Number of eigenvalues......Page 152 5.14 Definition......Page 153 EXERCISES 5.A......Page 154 5.17 Definition......Page 159 5.20 Multiplicative properties......Page 160 5.21 Operators on complex vector spaces have an eigenvalue......Page 161 5.23 Example......Page 162 5.25 Definition......Page 163 5.26 Conditions for upper-triangular matrix......Page 164 5.29......Page 165 5.31......Page 166 5.33 Example......Page 168 EXERCISES 5.B......Page 169 5.37 Example......Page 171 5.40 Example......Page 172 5.41 Conditions equivalent to diagonalizability......Page 173 5.43 Example......Page 174 5.45 Example......Page 175 EXERCISES 5.C......Page 176 6.1 Notation......Page 178 6.2 Definition......Page 179 6.4 Example......Page 181 6.7 Basic properties of an inner product......Page 182 6.9 Example......Page 183 6.11 Definition......Page 184 6.13 Pythagorean Theorem......Page 185 6.14 An orthogonal decomposition......Page 186 6.17 Example......Page 187 6.21......Page 188 6.22 Parallelogram Equality......Page 189 EXERCISES 6.A......Page 190 6.26 An orthonormal list is linearly independent......Page 195 6.29 Example......Page 196 6.30 Writing a vector as linear combination of orthonormal basis......Page 197 6.32......Page 198 6.33 Example......Page 199 6.35 Orthonormal list extends to orthonormal basis......Page 200 6.38 Schur’s Theorem......Page 201 6.41 Example......Page 202 6.44 Example......Page 203 EXERCISES 6.B......Page 204 6.46 Basic properties of orthogonal complement......Page 208 6.49......Page 209 6.53 Definition......Page 210 6.55 Properties of the orthogonal projection PU......Page 211 6.57......Page 213 6.60......Page 214 6.61......Page 215 EXERCISES 6.C......Page 216 7.1 Notation......Page 218 7.3 Example......Page 219 7.5 The adjoint is a linear map......Page 220 7.6 Properties of the adjoint......Page 221 7.9 Example......Page 222 7.10 The matrix of T*......Page 223 7.12 Example......Page 224 7.14 Over C, T v is orthogonal to v for all v only for the 0 operator......Page 225 7.17......Page 226 7.19 Example......Page 227 7.21 For T normal, T and T* have the same eigenvectors......Page 228 EXERCISES 7.A......Page 229 7.23 Example......Page 232 7.25......Page 233 7.26 Invertible quadratic expressions......Page 234 7.27 Self-adjoint operators have eigenvalues......Page 235 7.29 Real Spectral Theorem......Page 236 7.30 Example......Page 237 EXERCISES 7.B......Page 238 7.34 Example......Page 240 7.35 Characterization of positive operators......Page 241 7.36 Each positive operator has only one positive square root......Page 242 7.41......Page 243 7.42 Characterization of isometries......Page 244 EXERCISES 7.C......Page 246 7.45 Polar Decomposition......Page 248 7.47......Page 249 7.48......Page 250 7.50 Example......Page 251 7.51 Singular Value Decomposition......Page 252 EXERCISES 7.D......Page 253 8.1 Notation......Page 256 8.3 Equality in the sequence of null spaces......Page 257 8.6......Page 258 8.8......Page 259 8.10 Definition......Page 260 8.12 Example......Page 261 8.15......Page 262 8.18 Nilpotent operator raised to dimension of domain is 0......Page 263 EXERCISES 8.A......Page 264 8.21 Description of operators on complex vector spaces......Page 267 8.22......Page 268 8.25 Example......Page 269 8.27 Definition......Page 270 8.29 Block diagonal matrix with upper-triangular blocks......Page 271 8.30 Example......Page 272 8.32......Page 273 EXERCISES 8.B......Page 274 8.54 Example......Page 285 8.57......Page 286 8.58......Page 287 8.60 Jordan Form......Page 288 EXERCISES 8.D......Page 289 8.37 Cayley–Hamilton Theorem......Page 276 8.40 Minimal polynomial......Page 277 8.43 Definition......Page 278 8.46 q(T) = 0 implies q is a multiple of the minimal polynomial......Page 279 8.49 Eigenvalues are the zeros of the minimal polynomial......Page 280 8.50 Example......Page 281 EXERCISES 8.C......Page 282 9.1 Notation......Page 290 9.3 VC is a complex vector space.......Page 291 9.5 Definition......Page 292 9.7 Matrix of TC equals matrix of T......Page 293 9.10 Minimal polynomial of TC equals minimal polynomial of T......Page 294 9.12 TC –λI and TC – λI......Page 295 9.16 Nonreal eigenvalues of TC come in pairs......Page 296 9.19 Operator on odd-dimensional vector space has eigenvalue......Page 297 9.22 Example......Page 298 9.25 Example......Page 299 EXERCISES 9.A......Page 300 9.28......Page 302 9.30 Normal operators and invariant subspaces......Page 303 9.32......Page 304 9.33......Page 305 9.34 Characterization of normal operators when F = R......Page 306 9.37......Page 307 EXERCISES 9.B......Page 309 10.1 Notation......Page 310 10.3 Definition......Page 311 10.5 Matrix of the identity with respect to two bases......Page 312 10.8......Page 313 10.10 Example......Page 314 10.13 Definition......Page 315 10.14 Trace of AB equals trace of BA......Page 316 10.16 Trace of an operator equals trace of its matrix......Page 317 10.18 Trace is additive......Page 318 EXERCISES 10.A......Page 319 10.21 Example......Page 322 10.24 Invertible is equivalent to nonzero determinant......Page 323 Determinant of a Matrix......Page 324 10.26 Example......Page 325 10.28 Example......Page 326 10.30 Definition......Page 327 10.32 Interchanging two entries in a permutation......Page 328 10.35 Example......Page 329 10.37 Matrices with two equal columns......Page 330 10.38 Permuting the columns of a matrix......Page 331 10.40 Determinant is multiplicative......Page 332 10.41 Determinant of matrix of operator does not depend on basis......Page 333 10.43 Example......Page 334 The Sign of the Determinant......Page 335 10.45 Isometries have determinant with absolute value 1......Page 336 10.46 Example......Page 337 10.48 Definition......Page 338 10.51 Notation......Page 339 10.53 An isometry does not change volume......Page 340 10.54 T changes volume by factor of |det T|......Page 341 10.56 Definition......Page 342 10.58 Change of variables in an integral......Page 343 10.59 Example......Page 344 EXERCISES 10.B......Page 345 Photo Credits......Page 347 Symbol Index......Page 348 Index......Page 349 This Best-selling Textbook For A Second Course In Linear Algebra Is Aimed At Undergrad Math Majors And Graduate Students. The Novel Approach Taken Here Banishes Determinants To The End Of The Book. The Text Focuses On The Central Goal Of Linear Algebra: Understanding The Structure Of Linear Operators On Vector Spaces. The Author Has Taken Unusual Care To Motivate Concepts And To Simplify Proofs. A Variety Of Interesting Exercises In Each Chapter Helps Students Understand And Manipulate The Objects Of Linear Algebra. -- Back Cover. Machine Generated Contents Note: 1.vector Spaces -- 1.a.rn And Cn -- Complex Numbers -- Lists -- Fn -- Digression On Fields -- Exercises 1.a -- 1.b.definition Of Vector Space -- Exercises 1.b -- 1.c.subspaces -- Sums Of Subspaces -- Direct Sums -- Exercises 1.c -- 2.finite-dimensional Vector Spaces -- 2.a.span And Linear Independence -- Linear Combinations And Span -- Linear Independence -- Exercises 2.a -- 2.b.bases -- Exercises 2.b -- 2.c.dimension -- Exercises 2.c -- 3.linear Maps -- 3.a.the Vector Space Of Linear Maps -- Definition And Examples Of Linear Maps -- Algebraic Operations On L(v, W) -- Exercises 3.a -- 3.b.null Spaces And Ranges -- Null Space And Injectivity -- Range And Surjectivity -- Fundamental Theorem Of Linear Maps -- Exercises 3.b -- 3.c.matrices -- Representing A Linear Map By A Matrix -- Addition And Scalar Multiplication Of Matrices -- Matrix Multiplication -- Exercises 3.c -- 3.d.invertibility And Isomorphic Vector Spaces -- Invertible Linear Maps Note Continued: Isomorphic Vector Spaces -- Linear Maps Thought Of As Matrix Multiplication -- Operators -- Exercises 3.d -- 3.e.products And Quotients Of Vector Spaces -- Products Of Vector Spaces -- Products And Direct Sums -- Quotients Of Vector Spaces -- Exercises 3.e -- 3.f.duality -- The Dual Space And The Dual Map -- The Null Space And Range Of The Dual Of A Linear Map -- The Matrix Of The Dual Of A Linear Map -- The Rank Of A Matrix -- Exercises 3.f -- 4.polynomials -- Complex Conjugate And Absolute Value -- Uniqueness Of Coefficients For Polynomials -- The Division Algorithm For Polynomials -- Zeros Of Polynomials -- Factorization Of Polynomials Over C -- Factorization Of Polynomials Over R -- Exercises 4 -- 5.eigenvalues, Eigenvectors, And Invariant Subspaces -- 5.a.invariant Subspaces -- Eigenvalues And Eigenvectors -- Restriction And Quotient Operators -- Exercises 5.a -- 5.b.eigenvectors And Upper-triangular Matrices -- Polynomials Applied To Operators Note Continued: Existence Of Eigenvalues -- Upper-triangular Matrices -- Exercises 5.b -- 5.c.eigenspaces And Diagonal Matrices -- Exercises 5.c -- 6.inner Product Spaces -- 6.a.inner Products And Norms -- Inner Products -- Norms -- Exercises 6.a -- 6.b.orthonormal Bases -- Linear Functionals On Inner Product Spaces -- Exercises 6.b -- 6.c.orthogonal Complements And Minimization Problems -- Orthogonal Complements -- Minimization Problems -- Exercises 6.c -- 7.operators On Inner Product Spaces -- 7.a.self-adjoint And Normal Operators -- Adjoints -- Self-adjoint Operators -- Normal Operators -- Exercises 7.a -- 7.b.the Spectral Theorem -- The Complex Spectral Theorem -- The Real Spectral Theorem -- Exercises 7.b -- 7.c.positive Operators And Isometries -- Positive Operators -- Isometries -- Exercises 7.c -- 7.d.polar Decomposition And Singular Value Decomposition -- Polar Decomposition -- Singular Value Decomposition -- Exercises 7.d Note Continued: 8.operators On Complex Vector Spaces -- 8.a.generalized Eigenvectors And Nilpotent Operators -- Null Spaces Of Powers Of An Operator -- Generalized Eigenvectors -- Nilpotent Operators -- Exercises 8.a -- 8.b.decomposition Of An Operator -- Description Of Operators On Complex Vector Spaces -- Multiplicity Of An Eigenvalue -- Block Diagonal Matrices -- Square Roots -- Exercises 8.b -- 8.c.characteristic And Minimal Polynomials -- The Cayley -- Hamilton Theorem -- The Minimal Polynomial -- Exercises 8.c -- 8.d.jordan Form -- Exercises 8.d -- 9.operators On Real Vector Spaces -- 9.a.complexification -- Complexification Of A Vector Space -- Complexification Of An Operator -- The Minimal Polynomial Of The Complexification -- Eigenvalues Of The Complexification -- Characteristic Polynomial Of The Complexification -- Exercises 9.a -- 9.b.operators On Real Inner Product Spaces -- Normal Operators On Real Inner Product Spaces Note Continued: Isometries On Real Inner Product Spaces -- Exercises 9.b -- 10.trace And Determinant -- 10.a.trace -- Change Of Basis -- Trace: A Connection Between Operators And Matrices -- Exercises 10.a -- 10.b.determinant -- Determinant Of An Operator -- Determinant Of A Matrix -- The Sign Of The Determinant -- Volume -- Exercises 10.b. Sheldon Axler. Includes Indexes. This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator. From reviews of previous editions: “... a didactic masterpiece” —Zentralblatt MATH “... a tour de force in the service of simplicity and clarity ... The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library.” —CHOICE “The determinant-free proofs are elegant and intuitive.” —American Mathematical Monthly “Clarity through examples is emphasized ... the text is ideal for class exercises ... I congratulate the author and the publisher for a well-produced textbook on linear algebra.” —Mathematical Reviews This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra : understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator (4e de couverture) New edition extensively revised and updated Covers new topics such as product spaces, quotient spaces, and dual spaces Features new visually appealing format for both print and electronic versions Includes almost three times the number of exercises as the previous edition