인생을 바꾸는 66일: 정신적 장애물을 제거하고, 두뇌를 재프로그래밍하고, 돈을 모으는 12가지 단계

Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added. 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 simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. 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 the reviews of previous editions: Altogether, the text is a didactic masterpiece . ― zbMATH The determinant-free proofs are elegant and intuitive . ― American Mathematical Monthly The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library ― CHOICE About the Author 6 Contents 7 Preface for Students 13 Preface for Instructors 14 The author’s top ten 16 Major improvements and additions for the fourth edition 17 Acknowledgments 18 Chapter 1 Vector Spaces 19 1A Rn and Cn 20 Complex Numbers 20 Lists 23 Fn 24 Digression on Fields 28 Exercises 1A 28 1B Definition of Vector Space 30 Exercises 1B 34 1C Subspaces 36 Sums of Subspaces 37 Direct Sums 39 Exercises 1C 42 Chapter 2 Finite-Dimensional Vector Spaces 45 2A Span and Linear Independence 46 Linear Combinations and Span 46 Linear Independence 49 Exercises 2A 55 2B Bases 57 Exercises 2B 60 2C Dimension 62 Exercises 2C 66 Chapter 3 Linear Maps 69 3A Vector Space of Linear Maps 70 Definition and Examples of Linear Maps 70 Algebraic Operations on L(V, W) 73 Exercises 3A 75 3B Null Spaces and Ranges 77 Null Space and Injectivity 77 Range and Surjectivity 79 Fundamental Theorem of Linear Maps 80 Exercises 3B 84 3C Matrices 87 Representing a Linear Map by a Matrix 87 Addition and Scalar Multiplication of Matrices 89 Matrix Multiplication 90 Column–Row Factorization and Rank of a Matrix 95 Exercises 3C 97 3D Invertibility and Isomorphisms 100 Invertible Linear Maps 100 Isomorphic Vector Spaces 104 Linear Maps Thought of as Matrix Multiplication 106 Change of Basis 108 Exercises 3D 111 3E Products and Quotients of Vector Spaces 114 Products of Vector Spaces 114 Quotient Spaces 116 Exercises 3E 121 3F Duality 123 Dual Space and Dual Map 123 Null Space and Range of Dual of Linear Map 127 Matrix of Dual of Linear Map 131 Exercises 3F 133 Chapter 4 Polynomials 137 Zeros of Polynomials 140 Division Algorithm for Polynomials 141 Factorization of Polynomials over C 142 Factorization of Polynomials over R 145 Exercises 4 147 Chapter 5 Eigenvalues and Eigenvectors 150 5A Invariant Subspaces 151 Eigenvalues 151 Polynomials Applied to Operators 155 Exercises 5A 157 5B The Minimal Polynomial 161 Existence of Eigenvalues on Complex Vector Spaces 161 Eigenvalues and the Minimal Polynomial 162 Eigenvalues on Odd-Dimensional Real Vector Spaces 167 Exercises 5B 168 5C Upper-Triangular Matrices 172 Exercises 5C 178 5D Diagonalizable Operators 181 Diagonal Matrices 181 Conditions for Diagonalizability 183 Gershgorin Disk Theorem 188 Exercises 5D 190 5E Commuting Operators 193 Exercises 5E 197 Chapter 6 Inner Product Spaces 199 6A Inner Products and Norms 200 Inner Products 200 Norms 204 Exercises 6A 209 6B Orthonormal Bases 215 Orthonormal Lists and the Gram–Schmidt Procedure 215 Linear Functionals on Inner Product Spaces 222 Exercises 6B 225 6C Orthogonal Complements and Minimization Problems 229 Orthogonal Complements 229 Minimization Problems 235 Pseudoinverse 238 Exercises 6C 242 Chapter 7 Operators on Inner Product Spaces 245 7A Self-Adjoint and Normal Operators 246 Adjoints 246 Self-Adjoint Operators 251 Normal Operators 253 Exercises 7A 257 7B Spectral Theorem 261 Real Spectral Theorem 261 Complex Spectral Theorem 264 Exercises 7B 265 7C Positive Operators 269 Exercises 7C 273 7D Isometries, Unitary Operators, and Matrix Factorization 276 Isometries 276 Unitary Operators 278 QR Factorization 281 Cholesky Factorization 284 Exercises 7D 286 7E Singular Value Decomposition 288 Singular Values 288 SVD for Linear Maps and for Matrices 291 Exercises 7E 296 7F Consequences of Singular Value Decomposition 298 Norms of Linear Maps 298 Approximation by Linear Maps with Lower-Dimensional Range 301 Polar Decomposition 303 Operators Applied to Ellipsoids and Parallelepipeds 305 Volume via Singular Values 309 Properties of an Operator as Determined by Its Eigenvalues 311 Exercises 7F 312 Chapter 8 Operators on Complex Vector Spaces 315 8A Generalized Eigenvectors and Nilpotent Operators 316 Null Spaces of Powers of an Operator 316 Generalized Eigenvectors 318 Nilpotent Operators 321 Exercises 8A 324 8B Generalized Eigenspace Decomposition 326 Generalized Eigenspaces 326 Multiplicity of an Eigenvalue 328 Block Diagonal Matrices 332 Exercises 8B 334 8C Consequences of Generalized Eigenspace Decomposition 337 Square Roots of Operators 337 Jordan Form 339 Exercises 8C 342 8D Trace: A Connection Between Matrices and Operators 344 Exercises 8D 348 Chapter 9 Multilinear Algebra and Determinants 350 9A Bilinear Forms and Quadratic Forms 351 Bilinear Forms 351 Symmetric Bilinear Forms 355 Quadratic Forms 359 Exercises 9A 362 9B Alternating Multilinear Forms 364 Multilinear Forms 364 Alternating Multilinear Forms and Permutations 366 Exercises 9B 370 9C Determinants 372 Defining the Determinant 372 Properties of Determinants 375 Exercises 9C 385 9D Tensor Products 388 Tensor Product of Two Vector Spaces 388 Tensor Product of Inner Product Spaces 394 Tensor Product of Multiple Vector Spaces 396 Exercises 9D 398 Photo Credits 401 Symbol Index 402 Index 403 Colophon: Notes on Typesetting 408