Linear Algebra: Theory and Applications, Second Edition
by Cheney
Linear Algebra: Theory and Applications, 2nd Edition
- 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
- Row-Equivalent 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
- Matrix--Vector Products
- The Span of a Set of Vectors
- Interpreting Linear Systems
- Row-Equivalent 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
- Matrix--Matrix Product
- Indexed Sets of Vectors: Linear Dependence and Independence
- Using the Row-Reduction 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
- n-Tuples 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: Partial-Fraction 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, Co-domain, 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
- Pre-multiplication and Post-multiplication
- Dot Product
- Special Matrices
- Matrix Transpose
- Symmetric Matrices
- Skew–Symmetric Matrices
- Non-commutativity 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 Non-square 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 INNER-PRODUCT SPACES“
- Inner-Product Spaces and Their Properties
- The Norm in an Inner-Product Space
- Distance Function
- Mutually Orthogonal Vectors
- Orthogonal Projection
- Angle between Vectors
- Orthogonal Complements
- Orthonormal Bases
- Subspaces in Inner-Product 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 INNER-PRODUCT SPACES“
- CHAPTER EIGHT Additional Topics
- 8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREM
- Introduction
- Hermitian Matrices and Self-Adjoint Mappings
- Self-Adjoint 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
- LLT-Factorization: Cholesky Factorization
- LDLT-Factorization
- QR-Factorization
- Singular-Value Decomposition (SVD)
- Schur Decomposition
- Partitioned Matrices
- Solving a System Having a 2 × 2 Block Matrix
- Inverting a 2 × 2 Block Matrix
- Application: Linear Least-Squares 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
-
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