# Introduction to Linear Algebra, 5/e

### Gilbert Strang

• 出版商:
• 出版日期: 2019-08-01
• 定價: \$648
• 售價: 8.5\$551
• 語言: 英文
• 頁數: 573
• 裝訂: 平裝
• ISBN: 7302535566
• ISBN-13: 9787302535560
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## 作者簡介

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

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

## 目錄大綱

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

## 目錄大綱(中文翻譯)

1 向量介紹 1
1.1 向量和線性組合...................... 2
1.2 長度和點積............... ........... 11
1.3 矩陣 ................................... 22

2 解線性方程組 31
2.1 向量和線性方程組........................ 31
2.2 消元法的概念................ ........... 46
2.3 利用矩陣進行消元......................... 58
2.4 矩陣運算規則 ........ ................ 70
2.5 逆矩陣............................... 83
2.6 消元=因式分解: A = LU .................. 97
2.7 轉置和排列 ........................ 108

3 向量空間和子空間 122
3.1 向量空間 ............... ............... 122
3.2 A的零空間: 解Ax = 0和Rx =0 ........... 134
3.3 Ax = b的完整解 . .................... 149
3.4 獨立性、基底和維度 .................... 163
3.5 四個子空間的維度 .. ................... 180

4 正交性 193
4.1 四個子空間的正交性 . . . . . . . . . . . . . . . . . . . . 193
4.2 投影 .. ............................... 205
4.3 最小二乘逼近 ................ ....... 218
4.4 正交基和格拉姆-施密特. . . . . . . . . . . . . . . . . . . 232

5 行列式 246
5.1 行列式的性質..................... .. 246
5.2 排列和餘子式......................... 257
5.3 克拉默法則、逆矩陣和體積 . . . . . . . . . . . . . . . . . . . 272

6 特徵值和特徵向量 287
6.1 特徵值介紹......................... 287
6.2 矩陣對角化 ..... ...................... 303
6.3 微分方程組 ..................... 318
6.4 對稱矩陣. ............................ 337
6.5 正定矩陣.......................... 349

7 奇異值分解 (SVD) 363
7.1 線性代數在圖像處理中的應用 ........... ......... 363
7.2 SVD中的基底和矩陣 ....................... 370
7.3 主成分分析 (PCA by the SVD) . . . . . . . . . . . . . . 381
7.4 SVD的幾何性質 ......................... 391

8 線性變換 400
8.1 線性變換的概念 ....... ............. 400
8.2 線性變換的矩陣. . . . . . . . . . . . . . . . . . . 410
8.3 尋找一個好的基底 ............ ............ 420

9 複數向量和矩陣 429
9.1 複數數字 ............................. 430
9.2 共軛和酉矩陣 ................ ...... 437
9.3 快速傅立葉變換......................... 444

10 應用 451
10.1 圖形和網絡 .......... .................. 451
10.2 工程中的矩陣........................... 461
10.3 馬可夫矩陣、人口和經濟 . . . . . . . . . . . . . . . . . . . 473
10.4 線性規劃 ......................... ... 482
10.5 傅立葉級數: 函數的線性代數 . . . . . . . . . . . . . . . . . . . 489
10.6 電腦圖形 ................... .......... 495
10.7 密碼學的線性代數...................... 501

11 數值線性代數 507
11.1 實踐中的高斯消元 ................... ... 507
11.2 范數和條件數....................... 517
11.3 迭代方法和預條件子 . . . . . . . . . . . . . . . . . . . 523

12 概率和統計中的線性代數 534
12.1 平均值、變異數和概率 ...................... 534
12.2 協方差矩陣