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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