book

Linear Algebra with Applications, 10th Edition

Name: Linear Algebra with Applications, 10th Edition
ISBN: 9780136745976

by Steve Leon, Lisette de Pillis

June 2019

Beginner

553 pages

17h 41m

English

Pearson

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Linear Algebra with Applications
Dedication
Contents
Preface
What’s New in the Tenth Edition?Overview of TextSuggested Course OutlinesComputer Exercises
Chapter 1 Matrices and Systems of Equations
1.1 Systems of Linear Equations
2×2 SystemsEquivalent Systemsn×n Systems
Section 1.1 Exercises
1.2 Row Echelon Form
Overdetermined SystemsUnderdetermined SystemsReduced Row Echelon FormHomogeneous Systems
Section 1.2 Exercises
1.3 Matrix Arithmetic
Matrix NotationVectorsEqualityScalar MultiplicationMatrix AdditionMatrix Multiplication and Linear SystemsMatrix MultiplicationNotational RulesThe Transpose of a Matrix

Section 1.3 Exercises
1.4 Matrix Algebra
Algebraic RulesNotationThe Identity MatrixMatrix InversionNoteAlgebraic Rules for TransposesSymmetric Matrices and Networks
Section 1.4 Exercises
1.5 Elementary Matrices
Equivalent SystemsElementary MatricesDiagonal and Triangular MatricesTriangular Factorization
Section 1.5 Exercises
1.6 Partitioned Matrices
Block MultiplicationOuter Product Expansions
Section 1.6 Exercises
Chapter 1 Exercises
MATLAB ExercisesChapter Test A True or FalseChapter Test B
Chapter 2 Determinants
2.1 The Determinant of a Matrix
Section 2.1 Exercises
2.2 Properties of Determinants
Row Operation IRow Operation IIRow Operation IIINoteMain Results
Section 2.2 Exercises
2.3 Additional Topics and Applications
The Adjoint of a MatrixCramer’s RuleReferenceThe Cross Product
Section 2.3 Exercises
Chapter 2 Exercises
MATLAB ExercisesChapter Test A True or FalseChapter Test B
Chapter 3 Vector Spaces
3.1 Definition and Examples
Euclidean Vector SpacesThe Vector Space ℝm×nVector Space AxiomsThe Vector Space C[a,b]The Vector Space PnAdditional Properties of Vector Spaces
Section 3.1 Exercises
3.2 Subspaces
The Null Space of a MatrixThe Span of a Set of VectorsSpanning Set for a Vector SpaceLinear Systems Revisited
Section 3.2 Exercises
3.3 Linear Independence
Geometric InterpretationTheorems and ExamplesVector Spaces of FunctionsThe Vector Space PnThe Vector Space C(n−1)[a,b]
Section 3.3 Exercises
3.4 Basis and Dimension
Standard Bases
Section 3.4 Exercises
3.5 Change of Basis
Changing Coordinates in ℝ2Changing CoordinatesChange of Basis for a General Vector Space
Section 3.5 Exercises
3.6 Row Space and Column Space
Linear SystemsThe Column SpaceNote
Section 3.6 Exercises
Chapter 3 Exercises
MATLAB ExercisesChapter Test A True or FalseChapter Test B
Chapter 4 Linear Transformations
4.1 Definition and Examples
NotationLinear Operators on ℝ2Linear Transformations from ℝn to ℝmLinear Transformations from V to WThe Image and Kernel
Section 4.1 Exercises
4.2 Matrix Representations of Linear Transformations
Section 4.2 Exercises
4.3 Similarity
Section 4.3 Exercises
Chapter 4 Exercises
MATLAB ExercisesChapter Test A True or FalseChapter Test B
Chapter 5 Orthogonality
5.1 The Scalar Product in ℝn
The Scalar Product in ℝ2 and ℝ3Scalar and Vector ProjectionsNotationOrthogonality in ℝn
Section 5.1Exercises
5.2 Orthogonal Subspaces
NoteRemarksFundamental Subspaces
Section 5.2 Exercises
5.3 Least Squares Problems
Least Squares Solutions of Overdetermined Systems
Section 5.3 Exercises
5.4 Inner Product Spaces
Definition and ExamplesThe Vector Space ℝnThe Vector Space ℝm×nThe Vector Space C[a, b]The Vector Space PnBasic Properties of Inner Product SpacesObservationsNorms
Section 5.4 Exercises
5.5 Orthonormal Sets
Orthogonal MatricesPermutation MatricesOrthonormal Sets and Least SquaresApproximation of FunctionsApproximation by Trigonometric Polynomials
Section 5.5 Exercises
5.6 The Gram–Schmidt Orthogonalization Process
The Modified Gram–Schmidt Process
Section 5.6 Exercises
5.7 Orthogonal Polynomials
Orthogonal SequencesClassical Orthogonal PolynomialsLegendre PolynomialsChebyshev PolynomialsJacobi PolynomialsHermite PolynomialsLaguerre Polynomials
Section 5.7 Exercises
Chapter 5 Exercises
MATLAB ExercisesChapter Test A True or FalseChapter Test B
Chapter 6 Eigenvalues
6.1 Eigenvalues and Eigenvectors
Geometric Visualization of Eigenvalues and EigenvectorsFinding Eigenvalues and EigenvectorsComplex EigenvaluesThe Product and Sum of the EigenvaluesSimilar Matrices
Section 6.1 Exercises
6.2 Systems of Linear Differential Equations
Complex EigenvaluesHigher-Order Systems
Section 6.2 Exercises
6.3 Diagonalization
RemarksThe Exponential of a Matrix
Section 6.3 Exercises
6.4 Hermitian Matrices
Complex Inner ProductsHermitian MatricesThe Real Schur DecompositionNormal Matrices
Section 6.4 Exercises
6.5 The Singular Value Decomposition
ObservationsVisualizing the SVDNumerical Rank and Lower Rank Approximations
Section 6.5 Exercises
6.6 Quadratic Forms
Conic SectionsOptimization: An Application to the Calculus
Section 6.6 Exercises
6.7 Positive Definite Matrices
Section 6.7 Exercises
6.8 Nonnegative Matrices
Section 6.8 Exercises
Chapter 6 Exercises
MATLAB ExercisesCritical Loads for a BeamDiagonalizable and Defective MatricesApplication: Sex-Linked GenesSimilarityHermitian MatricesOptimizationPositive Definite MatricesChapter Test A True or FalseChapter Test B
Chapter 7 Numerical Linear Algebra
7.1 Floating-Point Numbers
The IEEE Standard 754 Floating-Point RepresentationLoss of Accuracy and Instability
Section 7.1 Exercises
7.2 Gaussian Elimination
Gaussian Elimination without InterchangesUsing the Triangular Factorization to Solve Ax=bOperation Count
Section 7.2 Exercises
7.3 Pivoting Strategies
Gaussian Elimination with InterchangesRemarksPartial Pivoting
Section 7.3 Exercises
7.4 Matrix Norms and Condition Numbers
Matrix NormsSubordinate Matrix NormsCondition Numbers
Section 7.4 Exercises
7.5 Orthogonal Transformations
Elementary Orthogonal TransformationsHouseholder TransformationsRemarksOperation CountRotations and ReflectionsOperation CountThe QR Factorization for Solving General Linear Systems
Section 7.5 Exercises
7.6 The Eigenvalue Problem
The Power MethodDeflationReduction to Hessenberg FormQR AlgorithmRemarks
Section 7.6 Exercises
7.7 Least Squares Problems
Normal EquationsModified Gram–Schmidt Method for Solving Least Squares ProblemsThe Householder QR FactorizationThe PseudoinverseBidiagonalizationThe Golub–Reinsch Algorithm
Section 7.7 Exercises
7.8 Iterative Methods
Matrix SplittingsJacobi IterationGauss–Seidel Iteration
Section 7.8 Exercises
Chapter 7 Exercises
MATLAB ExercisesSensitivity of Linear SystemsSensitivity of EigenvaluesHouseholder TransformationsRotations and ReflectionsSingular Value DecompositionGerschgorin CirclesDistribution of Condition Numbers and Eigenvalues of Random MatricesChapter Test A True or FalseChapter Test B
Chapter 8 Canonical Forms
8.1 Nilpotent Operators
Section 8.1 Exercises
8.2 The Jordan Canonical Form
Section 8.2 Exercises
Appendix: Matlab
The MATLAB Desktop Display
Basic Data Elements
Submatrices
Generating Matrices
Matrix Arithmetic
Addition and Multiplication of MatricesBackslash or Matrix Left DivisionExponentiation
MATLAB Functions
Programming Features
M-files
Script FilesFunction FilesThe MATLAB Path
Relational and Logical Operators
Columnwise Array Operators
Graphics
Symbolic Toolbox
Help Facility
Conclusions
Bibliography
A Linear Algebra and Matrix Theory
B Applied and Numerical Linear Algebra
C Books of Related Interest
Answers to Selected Exercises
Chapter 1
1.1
1.2
1.3
1.4
1.5
1.6
Chapter Test A
Chapter Test B
Chapter 2
2.1
2.2
2.3
Chapter Test A
Chapter Test B
Chapter 3
3.1
3.2
3.3
3.4
3.5
3.6
Chapter Test A
Chapter Test B
Chapter 4
4.1
4.2
4.3
Chapter Test A
Chapter Test B
Chapter 5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
MATLAB Exercises
Chapter Test A
Chapter Test B
Chapter 8
8.1
8.2
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z

Content preview from Linear Algebra with Applications, 10th Edition

7.4 Matrix Norms and Condition Numbers

In this section, we are concerned with the accuracy of computed solutions of linear systems. How accurate can we expect the computed solutions to be, and how can we test their accuracy? The answer to these questions depends largely on how sensitive the coefficient matrix of the system is to small changes. The sensitivity of the matrix can be measured in terms of its condition number. The condition number of a nonsingular matrix is defined in terms of its norm and the norm of its inverse. Before discussing condition numbers, it is necessary to establish some important results regarding the standard types of matrix norms.

Matrix Norms

Just as vector norms are used to measure the size of vectors, matrix norms ...

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Linear Algebra with Applications, 10th Edition

by Steve Leon, Lisette de Pillis

7.4 Matrix Norms and Condition Numbers

Matrix Norms

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