线性代数(第5版)(英文版)

线性代数内容包括行列式、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。

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