1 Introduction to Vectors 1
1.1 Vectors and Linear Combinations...................... 2
1.2 Lengths and Dot Products.......................... 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
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
Six Great Theorems / Linear Algebra in a Nutshell 573
· · · · · · (收起)
1.1 Vectors and Linear Combinations...................... 2
1.2 Lengths and Dot Products.......................... 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
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
Six Great Theorems / Linear Algebra in a Nutshell 573
· · · · · · (收起)
具體描述
本書內容包括行列式、矩陣、綫性方程組與嚮量、矩陣的特徵值與特徵嚮量、二次型及Mathematica軟件的應用等。每章都配有習題，書後給齣瞭習題答案。本書在編寫中力求重點突齣、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。
用戶評價
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##明白透徹
評分##也可能是自己的問題，自己會覺得MIT的課有其特點，但是也沒有大傢說的那麼好 相比之下，清華學堂在綫馬輝老師的課或許更加符閤我們的學習思維（雖然他們也有藉鑒MIT的課） 更新: 《高等代數簡明教程》yyds，大傢快去看！
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評分##越來越能體會到Linear Algebra的elegant和great ，書排版很好的，口語化一些
評分##多年後用此書重溫綫性代數，此書基本都是從二維、三維實例來進行講解，使得讀者容易把握問題的內在，而不是僅僅記住一個n維的公式。
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評分##（隻看瞭大概一半…）很妙。????????
評分##比同濟版不知高到哪裏去瞭 @2021-06-23 19:43:41 @2021-07-05 00:40:58 @2021-06-23 19:43:41
評分##越來越能體會到Linear Algebra的elegant和great ，書排版很好的，口語化一些
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