book

Linear Algebra: Theory and Applications, 2nd Edition

Name: Linear Algebra: Theory and Applications, 2nd Edition
Author: Cheney
ISBN: 9781449613532

by Cheney

December 2010

Beginner

624 pages

14h 57m

English

Jones & Bartlett Learning

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The Jones & Bartlett Learning Series in Mathematics
GeometryPrecalculusCalculusLinear AlgebraAdvanced Engineering MathematicsComplex AnalysisReal AnalysisTopologyDiscrete Mathematics and LogicNumerical MethodsAdvanced Mathematics
The Jones & Bartlett Learning International Series in Mathematics
Contents
Preface
Second EditionSupplementsAcknowledgmentsAuthors
CHAPTER ONE Systems of Linear Equations
1.1 SOLVING SYSTEMS OF LINEAR EQUATIONSLinear EquationsSystems of Linear EquationsGeneral Systems of Linear EquationsGaussian EliminationElementary Replacement and Scale OperationsRow-Equivalent Pairs of MatricesElementary Row OperationsReduced Row Echelon FormRow Echelon FormIntuitive InterpretationApplication: Feeding BacteriaMathematical SoftwareAlgorithm for the Reduced Row Echelon FormSUMMARY 1.1KEY CONCEPTS 1.1GENERAL EXERCISES 1.1COMPUTER EXERCISES 1.11.2 VECTORS AND MATRICESVectorsLinear Combinations of VectorsMatrix--Vector ProductsThe Span of a Set of VectorsInterpreting Linear SystemsRow-Equivalent SystemsConsistent and Inconsistent SystemsCautionApplication: Linear Ordinary Differential EquationsApplication: Bending of a BeamMathematical SoftwareSUMMARY 1.2KEY CONCEPTS 1.2GENERAL EXERCISES 1.2COMPUTER EXERCISES 1.21.3 KERNELS, RANK, HOMOGENEOUS EQUATIONSKernel or Null Space of a MatrixHomogeneous EquationsUniqueness of the Reduced Row Echelon FormRank of a MatrixGeneral Solution of a SystemMatrix--Matrix ProductIndexed Sets of Vectors: Linear Dependence and IndependenceUsing the Row-Reduction ProcessDetermining Linear Dependence or IndependenceApplication: ChemistrySUMMARY 1.3KEY CONCEPTS 1.3GENERAL EXERCISES 1.3COMPUTER EXERCISES 1.3
CHAPTER TWO Vector Spaces
2.1 EUCLIDEAN VECTOR SPACESn-Tuples and VectorsVector Addition and Multiplication by ScalarsProperties of as a Vector SpaceLinear CombinationsSpan of a Set of VectorsGeometric Interpretation of VectorsApplication: Elementary MechanicsApplication: Network Problems, Traffic FlowApplication: Electrical CircuitsSUMMARY 2.1KEY CONCEPTS 2.1GENERAL EXERCISES 2.1COMPUTER EXERCISES 2.12.2 LINES, PLANES, AND HYPERPLANESLine Passing Through OriginLines inLines inPlanes inLines and Planes inGeneral Solution of a System of EquationsApplication: The Predator–Prey SimulationApplication: Partial-Fraction DecompositionApplication: Method of Least SquaresSUMMARY 2.2KEY CONCEPTS 2.2GENERAL EXERCISES 2.2COMPUTER EXERCISES 2.22.3 LINEAR TRANSFORMATIONSFunctions, Mappings, and TransformationsDomain, Co-domain, and RangeVarious ExamplesInjective and Surjective MappingsLinear TransformationsUsing Matrices to Define Linear MapsInjective and Surjective Linear TransformationsEffects of Linear TransformationsEffects of Transformations on Geometrical FiguresComposition of Two Linear MappingsApplication: Data SmoothingSUMMARY 2.3KEY CONCEPTS 2.3GENERAL EXERCISES 2.3COMPUTER EXERCISES 2.3
2.4 GENERAL VECTOR SPACES
Vector SpacesTheorems on Vector SpacesVarious ExamplesLinearly Dependent SetsLinear MappingApplication: Models in Economic TheorySUMMARY 2.4KEY CONCEPTS 2.4GENERAL EXERCISES 2.4COMPUTER EXERCISES 2.4
CHAPTER THREE Matrix Operations
3.1 MATRICESMatrix Addition and Scalar MultiplicationMatrix–Matrix MultiplicationPre-multiplication and Post-multiplicationDot ProductSpecial MatricesMatrix TransposeSymmetric MatricesSkew–Symmetric MatricesNon-commutativity of Matrix MultiplicationAssociativity Law for Matrix MultiplicationLinear TransformationsElementary MatricesMore on the Matrix–Matrix ProductVector–Matrix ProductApplication: Diet ProblemsDangerous PitfallsSUMMARY 3.1KEY CONCEPTS 3.1GENERAL EXERCISES 3.1COMPUTER EXERCISES 3.13.2 MATRIX INVERSESSolving Systems with a Left InverseSolving Systems with a Right InverseAnalysisSquare MatricesInvertible MatricesElementary Matrices and LU FactorizationComputing an InverseMore on Left and Right Inverses of Non-square MatricesInvertible Matrix TheoremApplication: InterpolationMathematical SoftwareSUMMARY 3.2KEY CONCEPTS 3.2GENERAL EXERCISES 3.2COMPUTER EXERCISES 3.2
CHAPTER FOUR Determinants
4.1 DETERMINANTS: INTRODUCTIONProperties of DeterminantsAn Algorithm for Computing DeterminantsAlgorithm without ScalingZero DeterminantCalculating Areas and VolumesMathematical SoftwareSUMMARY 4.1KEY CONCEPTS 4.1GENERAL EXERCISES 4.1COMPUTER EXERCISES 4.14.2 DETERMINANTS: PROPERTIESMinors and CofactorsWork EstimateDirect Methods for Computing DeterminantsProperties of DeterminantsCramer’s RulePlanes inComputing Inverses Using DeterminantsVandermonde MatrixApplication: Coded MessagesMathematical SoftwareReview of Determinant Notation and PropertiesSUMMARY 4.2KEY CONCEPTS 4.2GENERAL EXERCISES 4.2COMPUTER EXERCISES 4.2
CHAPTER FIVE Vector Subspaces
5.1 COLUMN, ROW, AND NULL SPACESIntroductionLinear TransformationsRevisiting Kernels and Null SpacesThe Row Space and Column Space of a MatrixCautionSUMMARY 5.1KEY CONCEPTS 5.1GENERAL EXERCISES 5.1COMPUTER EXERCISES 5.15.2 BASES AND DIMENSIONBasis for a Vector SpaceCoordinate VectorIsomorphism and Equivalence RelationsFinite-Dimensional and Infinite-Dimensional Vector SpacesLinear Transformation of a SetDimensions of Various SubspacesCautionSUMMARY 5.2KEY CONCEPTS 5.2GENERAL EXERCISES 5.2COMPUTER EXERCISES 5.25.3 COORDINATE SYSTEMSCoordinate VectorsChanging CoordinatesLinear TransformationsMapping a Vector Space into ItselfSimilar MatricesMore on Equivalence RelationsFurther ExamplesSUMMARY 5.3KEY CONCEPTS 5.3GENERAL EXERCISES 5.3COMPUTER EXERCISES 5.3

CHAPTER SIX Eigensystems
6.1 EIGENVALUES AND EIGENVECTORSIntroductionEigenvectors and EigenvaluesUsing Determinants in Finding EigenvaluesLinear TransformationsDistinct EigenvaluesBases of EigenvectorsApplication: Powers of a MatrixCharacteristic Equation and Characteristic PolynomialDiagonalization Involving Complex NumbersApplication: Dynamical SystemsFurther Dynamical Systems inAnalysis of a Dynamical SystemApplication: Economic ModelsApplication: Systems of Linear Differential EquationsEpilogue: Eigensystems without DeterminantsMathematical SoftwareSUMMARY 6.1KEY CONCEPTS 6.1GENERAL EXERCISES 6.1COMPUTER EXERCISES 6.1
CHAPTER SEVEN Inner -Product Vector Spaces
7.1 INNER-PRODUCT SPACES“Inner-Product Spaces and Their PropertiesThe Norm in an Inner-Product SpaceDistance FunctionMutually Orthogonal VectorsOrthogonal ProjectionAngle between VectorsOrthogonal ComplementsOrthonormal BasesSubspaces in Inner-Product SpacesApplication: Work and ForcesApplication: CollisionSUMMARY 7.1KEY CONCEPTS 7.1GENERAL EXERCISES 7.1COMPUTER EXERCISES 7.17.2 ORTHOGONALITYIntroductionThe Gram–Schmidt ProcessUnnormalized Gram–Schmidt AlgorithmModified Gram–Schmidt ProcessLinear Least-Squares SolutionGram MatrixDistance from a Point to a HyperplaneMathematical SoftwareSUMMARY 7.2KEY CONCEPTS 7.2GENERAL EXERCISES 7.2COMPUTER EXERCISES 7.2
CHAPTER EIGHT Additional Topics
8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREMIntroductionHermitian Matrices and Self-Adjoint MappingsSelf-Adjoint MappingThe Spectral TheoremUnitary and Orthogonal MatricesThe Cayley–Hamilton TheoremQuadratic FormsApplication: World Wide Web SearchingMathematical SoftwareSUMMARY 8.1KEY CONCEPTS 8.1GENERAL EXERCISES 8.1COMPUTER EXERCISES 8.1
8.2 MATRIX FACTORIZATIONS AND BLOCK MATRICES
IntroductionPermutation MatrixLU -FactorizationLLT-Factorization: Cholesky FactorizationLDLT-FactorizationQR-FactorizationSingular-Value Decomposition (SVD)Schur DecompositionPartitioned MatricesSolving a System Having a 2 × 2 Block MatrixInverting a 2 × 2 Block MatrixApplication: Linear Least-Squares ProblemMathematical SoftwareSUMMARY 8.2KEY CONCEPTS 8.2GENERAL EXERCISES 8.2COMPUTER EXERCISES 8.28.3 ITERATIVE METHODS FOR LINEAR EQUATIONSIntroductionRichardson Iterative MethodJacobi Iterative MethodGauss–Seidel MethodSuccessive Overrelaxation (SOR) MethodConjugate Gradient MethodDiagonally Dominant MatricesGerschgorin’s TheoremInfinity NormConvergence PropertiesPower Method for Computing EigenvaluesApplication: Demographic Problems, Population MigrationApplication: Leontief Open ModelMathematical SoftwareSUMMARY 8.3KEY CONCEPTS 8.3GENERAL EXERCISES 8.3COMPUTER EXERCISES 8.3
APPENDIX A Deductive Reasoning and Proofs
A.1 IntroductionA.2 Deductive Reasoning and Direct VerificationA.3 ImplicationsA.4 Method of ContradictionA.5 Mathematical InductionA.6 Truth TablesA.7 Subsets and de Morgan LawsA.8 QuantifiersA.9 Denial of a Quantified AssertionA.10 Some More Questionable ‘‘Proofs’’SUMMARY APPENDIX AKEY CONCEPTS APPENDIX AGENERAL EXERCISES APPENDIX A
APPENDIX B Complex Arithmetic
B.1 Complex Numbers and ArithmeticB.2 Fundamental Theorem of AlgebraB.3 Abel–Ruffini Theorem
Answers/Hints for General Exercises
General Exercises 1.1General Exercises 1.2General Exercises 1.3General Exercises 2.1General Exercises 2.2General Exercises 2.3General Exercises 2.4General Exercises 3.1General Exercises 3.2General Exercises 4.1General Exercises 4.2General Exercises 5.1General Exercises 5.2General Exercises 5.3General Exercises 6.1General Exercises 7.1General Exercises 7.2General Exercises 8.1General Exercises 8.2General Exercises 8.3General Exercises Appendix A
References
Index

Content preview from Linear Algebra: Theory and Applications, 2nd Edition

Index

A

A = LDL^T, 497

A = LL^T, 494

A = LU, 224, 489

A = PBP⁻¹, similarity, 352

A = PDP⁻¹, diagonalizable, 372

A^k = PD^kP⁻¹, powers of a matrix, 376

A = PDQ, SVD-decomposition, 501

A = PQP⁻¹, transition, 354

A = QR, 499

A = UBU^H, unitarily similar, 468

A = UDU^H, Spectral Theorem: Matrix Version, 466

A = UTU^H, Schur decomposition, 502

Abel-Ruffini Impossibility Theorem, 584

Addition of

matrices, 187

vectors, 33

Adjacency matrix, 477, 485

Adjoint matrix, classical, 277

Adjoint operator, 482

Adjugate, 277

Algebraic multiplicity, 373

Algorithms

Cholesky factorization, 497

conjugate gradient, 530

determinant, 246

determinant, without scaling, 249

Gram-Schmidt, modified, 437

Gram-Schmidt, normalized, 435

linearly independent set of vectors, 75

reduced row echelon form, 26

Altitude, ...

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Linear Algebra: Theory and Applications, 2nd Edition

by Cheney

Index

A

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