内容简介
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本书内容包括行列式、矩阵、线性方程组与向量、矩阵的特征值与特征向量、二次型及Mathematica软件的应用等。每章都配有习题，书后给出了习题答案。本书在编写中力求重点突出、由浅入深、通俗易懂，努力体现教学的适用性。本书可作为高等院校工科专业的学生的教材，也可作为其 他非数学类本科专业学生的教材或教学参考书。
作者简介
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Gilbert Strang，MIT 数学系教授。从 UCLA 博士毕业后一直在 MIT 任教，教授的课程有“数据分析的矩阵方法”、“线性代数”、“计算机科学与工程”等，出版的图书有 Linear Algebra and Learning from Data、Introduction to Linear Algebra、Differential Equations and Linear Algebra。
目录
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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
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