Book description
Ward Cheney and David Kincaid have developed Linear Algebra: Theory and Applications, Second Edition, a multifaceted introductory textbook, which was motivated by their desire for a single text that meets the various requirements for differing courses within linear algebra. For theoreticallyoriented students, the text guides them as they devise proofs and deal with abstractions by focusing on a comprehensive blend between theory and applications. For applicationoriented science and engineering students, it contains numerous exercises that help them focus on understanding and learning not only vector spaces, matrices, and linear transformations, but also how software tools are used in applied linear algebra. Using a flexible design, it is an ideal textbook for instructors who wish to make their own choice regarding what material to emphasize, and to accentuate those choices with homework assignments from a large variety of exercises, both in the text and online.
Table of contents
 The Jones & Bartlett Learning Series in Mathematics
 The Jones & Bartlett Learning International Series in Mathematics
 Contents
 Preface

CHAPTER ONE Systems of Linear Equations

1.1 SOLVING SYSTEMS OF LINEAR EQUATIONS
 Linear Equations
 Systems of Linear Equations
 General Systems of Linear Equations
 Gaussian Elimination
 Elementary Replacement and Scale Operations
 RowEquivalent Pairs of Matrices
 Elementary Row Operations
 Reduced Row Echelon Form
 Row Echelon Form
 Intuitive Interpretation
 Application: Feeding Bacteria
 Mathematical Software
 Algorithm for the Reduced Row Echelon Form
 SUMMARY 1.1
 KEY CONCEPTS 1.1
 GENERAL EXERCISES 1.1
 COMPUTER EXERCISES 1.1

1.2 VECTORS AND MATRICES
 Vectors
 Linear Combinations of Vectors
 MatrixVector Products
 The Span of a Set of Vectors
 Interpreting Linear Systems
 RowEquivalent Systems
 Consistent and Inconsistent Systems
 Caution
 Application: Linear Ordinary Differential Equations
 Application: Bending of a Beam
 Mathematical Software
 SUMMARY 1.2
 KEY CONCEPTS 1.2
 GENERAL EXERCISES 1.2
 COMPUTER EXERCISES 1.2

1.3 KERNELS, RANK, HOMOGENEOUS EQUATIONS
 Kernel or Null Space of a Matrix
 Homogeneous Equations
 Uniqueness of the Reduced Row Echelon Form
 Rank of a Matrix
 General Solution of a System
 MatrixMatrix Product
 Indexed Sets of Vectors: Linear Dependence and Independence
 Using the RowReduction Process
 Determining Linear Dependence or Independence
 Application: Chemistry
 SUMMARY 1.3
 KEY CONCEPTS 1.3
 GENERAL EXERCISES 1.3
 COMPUTER EXERCISES 1.3

1.1 SOLVING SYSTEMS OF LINEAR EQUATIONS

CHAPTER TWO Vector Spaces

2.1 EUCLIDEAN VECTOR SPACES
 nTuples and Vectors
 Vector Addition and Multiplication by Scalars
 Properties of as a Vector Space
 Linear Combinations
 Span of a Set of Vectors
 Geometric Interpretation of Vectors
 Application: Elementary Mechanics
 Application: Network Problems, Traffic Flow
 Application: Electrical Circuits
 SUMMARY 2.1
 KEY CONCEPTS 2.1
 GENERAL EXERCISES 2.1
 COMPUTER EXERCISES 2.1

2.2 LINES, PLANES, AND HYPERPLANES
 Line Passing Through Origin
 Lines in
 Lines in
 Planes in
 Lines and Planes in
 General Solution of a System of Equations
 Application: The Predator–Prey Simulation
 Application: PartialFraction Decomposition
 Application: Method of Least Squares
 SUMMARY 2.2
 KEY CONCEPTS 2.2
 GENERAL EXERCISES 2.2
 COMPUTER EXERCISES 2.2

2.3 LINEAR TRANSFORMATIONS
 Functions, Mappings, and Transformations
 Domain, Codomain, and Range
 Various Examples
 Injective and Surjective Mappings
 Linear Transformations
 Using Matrices to Define Linear Maps
 Injective and Surjective Linear Transformations
 Effects of Linear Transformations
 Effects of Transformations on Geometrical Figures
 Composition of Two Linear Mappings
 Application: Data Smoothing
 SUMMARY 2.3
 KEY CONCEPTS 2.3
 GENERAL EXERCISES 2.3
 COMPUTER EXERCISES 2.3
 2.4 GENERAL VECTOR SPACES

2.1 EUCLIDEAN VECTOR SPACES

CHAPTER THREE Matrix Operations

3.1 MATRICES
 Matrix Addition and Scalar Multiplication
 Matrix–Matrix Multiplication
 Premultiplication and Postmultiplication
 Dot Product
 Special Matrices
 Matrix Transpose
 Symmetric Matrices
 Skew–Symmetric Matrices
 Noncommutativity of Matrix Multiplication
 Associativity Law for Matrix Multiplication
 Linear Transformations
 Elementary Matrices
 More on the Matrix–Matrix Product
 Vector–Matrix Product
 Application: Diet Problems
 Dangerous Pitfalls
 SUMMARY 3.1
 KEY CONCEPTS 3.1
 GENERAL EXERCISES 3.1
 COMPUTER EXERCISES 3.1

3.2 MATRIX INVERSES
 Solving Systems with a Left Inverse
 Solving Systems with a Right Inverse
 Analysis
 Square Matrices
 Invertible Matrices
 Elementary Matrices and LU Factorization
 Computing an Inverse
 More on Left and Right Inverses of Nonsquare Matrices
 Invertible Matrix Theorem
 Application: Interpolation
 Mathematical Software
 SUMMARY 3.2
 KEY CONCEPTS 3.2
 GENERAL EXERCISES 3.2
 COMPUTER EXERCISES 3.2

3.1 MATRICES

CHAPTER FOUR Determinants
 4.1 DETERMINANTS: INTRODUCTION

4.2 DETERMINANTS: PROPERTIES
 Minors and Cofactors
 Work Estimate
 Direct Methods for Computing Determinants
 Properties of Determinants
 Cramer’s Rule
 Planes in
 Computing Inverses Using Determinants
 Vandermonde Matrix
 Application: Coded Messages
 Mathematical Software
 Review of Determinant Notation and Properties
 SUMMARY 4.2
 KEY CONCEPTS 4.2
 GENERAL EXERCISES 4.2
 COMPUTER EXERCISES 4.2
 CHAPTER FIVE Vector Subspaces

CHAPTER SIX Eigensystems

6.1 EIGENVALUES AND EIGENVECTORS
 Introduction
 Eigenvectors and Eigenvalues
 Using Determinants in Finding Eigenvalues
 Linear Transformations
 Distinct Eigenvalues
 Bases of Eigenvectors
 Application: Powers of a Matrix
 Characteristic Equation and Characteristic Polynomial
 Diagonalization Involving Complex Numbers
 Application: Dynamical Systems
 Further Dynamical Systems in
 Analysis of a Dynamical System
 Application: Economic Models
 Application: Systems of Linear Differential Equations
 Epilogue: Eigensystems without Determinants
 Mathematical Software
 SUMMARY 6.1
 KEY CONCEPTS 6.1
 GENERAL EXERCISES 6.1
 COMPUTER EXERCISES 6.1

6.1 EIGENVALUES AND EIGENVECTORS

CHAPTER SEVEN Inner Product Vector Spaces

7.1 INNERPRODUCT SPACES“
 InnerProduct Spaces and Their Properties
 The Norm in an InnerProduct Space
 Distance Function
 Mutually Orthogonal Vectors
 Orthogonal Projection
 Angle between Vectors
 Orthogonal Complements
 Orthonormal Bases
 Subspaces in InnerProduct Spaces
 Application: Work and Forces
 Application: Collision
 SUMMARY 7.1
 KEY CONCEPTS 7.1
 GENERAL EXERCISES 7.1
 COMPUTER EXERCISES 7.1
 7.2 ORTHOGONALITY

7.1 INNERPRODUCT SPACES“

CHAPTER EIGHT Additional Topics

8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREM
 Introduction
 Hermitian Matrices and SelfAdjoint Mappings
 SelfAdjoint Mapping
 The Spectral Theorem
 Unitary and Orthogonal Matrices
 The Cayley–Hamilton Theorem
 Quadratic Forms
 Application: World Wide Web Searching
 Mathematical Software
 SUMMARY 8.1
 KEY CONCEPTS 8.1
 GENERAL EXERCISES 8.1
 COMPUTER EXERCISES 8.1

8.2 MATRIX FACTORIZATIONS AND BLOCK MATRICES
 Introduction
 Permutation Matrix
 LU Factorization
 LLTFactorization: Cholesky Factorization
 LDLTFactorization
 QRFactorization
 SingularValue Decomposition (SVD)
 Schur Decomposition
 Partitioned Matrices
 Solving a System Having a 2 × 2 Block Matrix
 Inverting a 2 × 2 Block Matrix
 Application: Linear LeastSquares Problem
 Mathematical Software
 SUMMARY 8.2
 KEY CONCEPTS 8.2
 GENERAL EXERCISES 8.2
 COMPUTER EXERCISES 8.2

8.3 ITERATIVE METHODS FOR LINEAR EQUATIONS
 Introduction
 Richardson Iterative Method
 Jacobi Iterative Method
 Gauss–Seidel Method
 Successive Overrelaxation (SOR) Method
 Conjugate Gradient Method
 Diagonally Dominant Matrices
 Gerschgorin’s Theorem
 Infinity Norm
 Convergence Properties
 Power Method for Computing Eigenvalues
 Application: Demographic Problems, Population Migration
 Application: Leontief Open Model
 Mathematical Software
 SUMMARY 8.3
 KEY CONCEPTS 8.3
 GENERAL EXERCISES 8.3
 COMPUTER EXERCISES 8.3

8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREM
 APPENDIX A Deductive Reasoning and Proofs
 APPENDIX B Complex Arithmetic

Answers/Hints for General Exercises
 General Exercises 1.1
 General Exercises 1.2
 General Exercises 1.3
 General Exercises 2.1
 General Exercises 2.2
 General Exercises 2.3
 General Exercises 2.4
 General Exercises 3.1
 General Exercises 3.2
 General Exercises 4.1
 General Exercises 4.2
 General Exercises 5.1
 General Exercises 5.2
 General Exercises 5.3
 General Exercises 6.1
 General Exercises 7.1
 General Exercises 7.2
 General Exercises 8.1
 General Exercises 8.2
 General Exercises 8.3
 General Exercises Appendix A
 References
 Index
Product information
 Title: Linear Algebra: Theory and Applications, 2nd Edition
 Author(s):
 Release date: December 2010
 Publisher(s): Jones & Bartlett Learning
 ISBN: 9781449613532
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