Introduction to Linear Algebra, 5/e

線性代數內容包括行列式、矩陣、線性方程組與向量、矩陣的特徵值與特徵向量、二次型及Mathematica 軟件的應用等。

每章都配有習題，書後給出了習題答案。本書在編寫中力求重點突出、由淺入深、 通俗易懂，努力體現教學的適用性。

本書可作為高等院校工科專業的學生的教材，也可作為其他非數學類本科專業學生的教材或教學參考書。

作者GILBERT STRANG為Massachusetts Institute of Technology數學系教授。

從UCLA博士畢業後一直在MIT任教.教授的課程有“數據分析的矩陣方法” “線性代數” “計算機科學與工程”等，

出版的圖書有Linear Algebra and Learning from Data (NEW)、See math. mit.edu/learningfromdata、

Introduction to Linear Algebra - Fifth Edition 、Contact linearalgebrabook@gmail.com、

Complete List of Books and Articles、Differential Equations and Linear Algebra。

Table of Contents

1 Introduction to Vectors 1

1.1 VectorsandLinearCombinations...................... 2

1.2 LengthsandDotProducts............... ........... 11

1.3 Matrices ................................... 22

2 Solving Linear Equations 31

2.1 VectorsandLinearEquations........................ 31

2.2 TheIdeaofElimination................ ........... 46

2.3 EliminationUsingMatrices......................... 58

2.4 RulesforMatrixOperations ........ ................ 70

2.5 InverseMatrices............................... 83

2.6 Elimination = Factorization: A = LU .................. 97

2.7 TransposesandPermutations ........................ 108

3 Vector Spaces and Subspaces 122

3.1 SpacesofVectors ............... ............... 122

3.2 The Nullspace of A: Solving Ax = 0and Rx =0 ........... 134

3.3 The Complete Solution to Ax = b . .................... 149

3.4 Independence,BasisandDimension .................... 163

3.5 DimensionsoftheFourSubspaces .. ................... 180

4 Orthogonality 193

4.1 OrthogonalityoftheFourSubspaces . . . . . . . . . . . . . . . . . . . . 193

4.2 Projections .. ............................... 205

4.3 LeastSquaresApproximations ................ ....... 218

4.4 OrthonormalBasesandGram-Schmidt. . . . . . . . . . . . . . . . . . . 232

5 Determinants 246

5.1 ThePropertiesofDeterminants..................... .. 246

5.2 PermutationsandCofactors......................... 257

5.3 Cramer'sRule,Inverses,andVolumes . . . . . . . . . . . . . . . . . . . 272

vii

6 Eigenvalues and Eigenvectors 287

6.1 IntroductiontoEigenvalues......................... 287

6.2 DiagonalizingaMatrix ..... ...................... 303

6.3 SystemsofDifferentialEquations ..................... 318

6.4 SymmetricMatrices. ............................ 337

6.5 PositiveDe.niteMatrices.......................... 349

7 TheSingularValueDecomposition (SVD) 363

7.1 ImageProcessingbyLinearAlgebra ........... ......... 363

7.2 BasesandMatricesintheSVD ....................... 370

7.3 Principal Component Analysis (PCA by the SVD) . . . . . . . . . . . . . 381

7.4 TheGeometryoftheSVD ......................... 391

8 LinearTransformations 400

8.1 TheIdeaofaLinearTransformation ....... ............. 400

8.2 TheMatrixofaLinearTransformation. . . . . . . . . . . . . . . . . . . 410

8.3 TheSearchforaGoodBasis ............ ............ 420

9 ComplexVectorsand Matrices 429

9.1 ComplexNumbers ............................. 430

9.2 HermitianandUnitaryMatrices ................ ...... 437

9.3 TheFastFourierTransform......................... 444

10 Applications 451

10.1GraphsandNetworks .......... .................. 451

10.2MatricesinEngineering........................... 461

10.3 Markov Matrices, Population, and Economics . . . . . . . . . . . . . . . 473

10.4LinearProgramming ......................... ... 482

10.5 Fourier Series: Linear Algebra for Functions . . . . . . . . . . . . . . . . 489

10.6ComputerGraphics ................... .......... 495

10.7LinearAlgebraforCryptography...................... 501

11 NumericalLinear Algebra 507

11.1GaussianEliminationinPractice ................... ... 507

11.2NormsandConditionNumbers....................... 517

11.3 IterativeMethodsandPreconditioners . . . . . . . . . . . . . . . . . . . 523

12LinearAlgebrain Probability& Statistics 534

12.1Mean,Variance,andProbability ...................... 534

12.2 Covariance Matrices and Joint Probabilities . . . . . . . . . . . . . . . . 545

12.3 Multivariate Gaussian and Weighted Least Squares . . . . . . . . . . . . 554

MatrixFactorizations 562

Index 564

SixGreatTheorems/LinearAlgebrain aNutshell 573