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Linear Algebra: Theory and Applications, 2nd Edition

Book Description

Ward Cheney and David Kincaid have developed Linear Algebra: Theory and Applications, Second Edition, a multi-faceted introductory textbook, which was motivated by their desire for a single text that meets the various requirements for differing courses within linear algebra. For theoretically-oriented students, the text guides them as they devise proofs and deal with abstractions by focusing on a comprehensive blend between theory and applications. For application-oriented 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

  1. The Jones & Bartlett Learning Series in Mathematics
    1. Geometry
    2. Precalculus
    3. Calculus
    4. Linear Algebra
    5. Advanced Engineering Mathematics
    6. Complex Analysis
    7. Real Analysis
    8. Topology
    9. Discrete Mathematics and Logic
    10. Numerical Methods
    11. Advanced Mathematics
  2. The Jones & Bartlett Learning International Series in Mathematics
  3. Contents
  4. Preface
    1. Second Edition
    2. Supplements
    3. Acknowledgments
    4. Authors
  5. CHAPTER ONE Systems of Linear Equations
    1. 1.1 SOLVING SYSTEMS OF LINEAR EQUATIONS
      1. Linear Equations
      2. Systems of Linear Equations
      3. General Systems of Linear Equations
      4. Gaussian Elimination
      5. Elementary Replacement and Scale Operations
      6. Row-Equivalent Pairs of Matrices
      7. Elementary Row Operations
      8. Reduced Row Echelon Form
      9. Row Echelon Form
      10. Intuitive Interpretation
      11. Application: Feeding Bacteria
      12. Mathematical Software
      13. Algorithm for the Reduced Row Echelon Form
      14. SUMMARY 1.1
      15. KEY CONCEPTS 1.1
      16. GENERAL EXERCISES 1.1
      17. COMPUTER EXERCISES 1.1
    2. 1.2 VECTORS AND MATRICES
      1. Vectors
      2. Linear Combinations of Vectors
      3. Matrix--Vector Products
      4. The Span of a Set of Vectors
      5. Interpreting Linear Systems
      6. Row-Equivalent Systems
      7. Consistent and Inconsistent Systems
      8. Caution
      9. Application: Linear Ordinary Differential Equations
      10. Application: Bending of a Beam
      11. Mathematical Software
      12. SUMMARY 1.2
      13. KEY CONCEPTS 1.2
      14. GENERAL EXERCISES 1.2
      15. COMPUTER EXERCISES 1.2
    3. 1.3 KERNELS, RANK, HOMOGENEOUS EQUATIONS
      1. Kernel or Null Space of a Matrix
      2. Homogeneous Equations
      3. Uniqueness of the Reduced Row Echelon Form
      4. Rank of a Matrix
      5. General Solution of a System
      6. Matrix--Matrix Product
      7. Indexed Sets of Vectors: Linear Dependence and Independence
      8. Using the Row-Reduction Process
      9. Determining Linear Dependence or Independence
      10. Application: Chemistry
      11. SUMMARY 1.3
      12. KEY CONCEPTS 1.3
      13. GENERAL EXERCISES 1.3
      14. COMPUTER EXERCISES 1.3
  6. CHAPTER TWO Vector Spaces
    1. 2.1 EUCLIDEAN VECTOR SPACES
      1. n-Tuples and Vectors
      2. Vector Addition and Multiplication by Scalars
      3. Properties of as a Vector Space
      4. Linear Combinations
      5. Span of a Set of Vectors
      6. Geometric Interpretation of Vectors
      7. Application: Elementary Mechanics
      8. Application: Network Problems, Traffic Flow
      9. Application: Electrical Circuits
      10. SUMMARY 2.1
      11. KEY CONCEPTS 2.1
      12. GENERAL EXERCISES 2.1
      13. COMPUTER EXERCISES 2.1
    2. 2.2 LINES, PLANES, AND HYPERPLANES
      1. Line Passing Through Origin
      2. Lines in
      3. Lines in
      4. Planes in
      5. Lines and Planes in
      6. General Solution of a System of Equations
      7. Application: The Predator–Prey Simulation
      8. Application: Partial-Fraction Decomposition
      9. Application: Method of Least Squares
      10. SUMMARY 2.2
      11. KEY CONCEPTS 2.2
      12. GENERAL EXERCISES 2.2
      13. COMPUTER EXERCISES 2.2
    3. 2.3 LINEAR TRANSFORMATIONS
      1. Functions, Mappings, and Transformations
      2. Domain, Co-domain, and Range
      3. Various Examples
      4. Injective and Surjective Mappings
      5. Linear Transformations
      6. Using Matrices to Define Linear Maps
      7. Injective and Surjective Linear Transformations
      8. Effects of Linear Transformations
      9. Effects of Transformations on Geometrical Figures
      10. Composition of Two Linear Mappings
      11. Application: Data Smoothing
      12. SUMMARY 2.3
      13. KEY CONCEPTS 2.3
      14. GENERAL EXERCISES 2.3
      15. COMPUTER EXERCISES 2.3
    4. 2.4 GENERAL VECTOR SPACES
      1. Vector Spaces
      2. Theorems on Vector Spaces
      3. Various Examples
      4. Linearly Dependent Sets
      5. Linear Mapping
      6. Application: Models in Economic Theory
      7. SUMMARY 2.4
      8. KEY CONCEPTS 2.4
      9. GENERAL EXERCISES 2.4
      10. COMPUTER EXERCISES 2.4
  7. CHAPTER THREE Matrix Operations
    1. 3.1 MATRICES
      1. Matrix Addition and Scalar Multiplication
      2. Matrix–Matrix Multiplication
      3. Pre-multiplication and Post-multiplication
      4. Dot Product
      5. Special Matrices
      6. Matrix Transpose
      7. Symmetric Matrices
      8. Skew–Symmetric Matrices
      9. Non-commutativity of Matrix Multiplication
      10. Associativity Law for Matrix Multiplication
      11. Linear Transformations
      12. Elementary Matrices
      13. More on the Matrix–Matrix Product
      14. Vector–Matrix Product
      15. Application: Diet Problems
      16. Dangerous Pitfalls
      17. SUMMARY 3.1
      18. KEY CONCEPTS 3.1
      19. GENERAL EXERCISES 3.1
    2. COMPUTER EXERCISES 3.1
    3. 3.2 MATRIX INVERSES
      1. Solving Systems with a Left Inverse
      2. Solving Systems with a Right Inverse
      3. Analysis
      4. Square Matrices
      5. Invertible Matrices
      6. Elementary Matrices and LU Factorization
      7. Computing an Inverse
      8. More on Left and Right Inverses of Non-square Matrices
      9. Invertible Matrix Theorem
      10. Application: Interpolation
      11. Mathematical Software
    4. SUMMARY 3.2
    5. KEY CONCEPTS 3.2
    6. GENERAL EXERCISES 3.2
    7. COMPUTER EXERCISES 3.2
  8. CHAPTER FOUR Determinants
    1. 4.1 DETERMINANTS: INTRODUCTION
      1. Properties of Determinants
      2. An Algorithm for Computing Determinants
      3. Algorithm without Scaling
      4. Zero Determinant
      5. Calculating Areas and Volumes
      6. Mathematical Software
      7. SUMMARY 4.1
      8. KEY CONCEPTS 4.1
      9. GENERAL EXERCISES 4.1
      10. COMPUTER EXERCISES 4.1
    2. 4.2 DETERMINANTS: PROPERTIES
      1. Minors and Cofactors
      2. Work Estimate
      3. Direct Methods for Computing Determinants
      4. Properties of Determinants
      5. Cramer’s Rule
      6. Planes in
      7. Computing Inverses Using Determinants
      8. Vandermonde Matrix
      9. Application: Coded Messages
      10. Mathematical Software
      11. Review of Determinant Notation and Properties
      12. SUMMARY 4.2
      13. KEY CONCEPTS 4.2
      14. GENERAL EXERCISES 4.2
      15. COMPUTER EXERCISES 4.2
  9. CHAPTER FIVE Vector Subspaces
    1. 5.1 COLUMN, ROW, AND NULL SPACES
      1. Introduction
      2. Linear Transformations
      3. Revisiting Kernels and Null Spaces
      4. The Row Space and Column Space of a Matrix
      5. Caution
      6. SUMMARY 5.1
      7. KEY CONCEPTS 5.1
      8. GENERAL EXERCISES 5.1
      9. COMPUTER EXERCISES 5.1
    2. 5.2 BASES AND DIMENSION
      1. Basis for a Vector Space
      2. Coordinate Vector
      3. Isomorphism and Equivalence Relations
      4. Finite-Dimensional and Infinite-Dimensional Vector Spaces
      5. Linear Transformation of a Set
      6. Dimensions of Various Subspaces
      7. Caution
      8. SUMMARY 5.2
      9. KEY CONCEPTS 5.2
      10. GENERAL EXERCISES 5.2
      11. COMPUTER EXERCISES 5.2
    3. 5.3 COORDINATE SYSTEMS
      1. Coordinate Vectors
      2. Changing Coordinates
      3. Linear Transformations
      4. Mapping a Vector Space into Itself
      5. Similar Matrices
      6. More on Equivalence Relations
      7. Further Examples
      8. SUMMARY 5.3
      9. KEY CONCEPTS 5.3
      10. GENERAL EXERCISES 5.3
      11. COMPUTER EXERCISES 5.3
  10. CHAPTER SIX Eigensystems
    1. 6.1 EIGENVALUES AND EIGENVECTORS
      1. Introduction
      2. Eigenvectors and Eigenvalues
      3. Using Determinants in Finding Eigenvalues
      4. Linear Transformations
      5. Distinct Eigenvalues
      6. Bases of Eigenvectors
      7. Application: Powers of a Matrix
      8. Characteristic Equation and Characteristic Polynomial
      9. Diagonalization Involving Complex Numbers
      10. Application: Dynamical Systems
      11. Further Dynamical Systems in
      12. Analysis of a Dynamical System
      13. Application: Economic Models
      14. Application: Systems of Linear Differential Equations
      15. Epilogue: Eigensystems without Determinants
      16. Mathematical Software
      17. SUMMARY 6.1
      18. KEY CONCEPTS 6.1
      19. GENERAL EXERCISES 6.1
      20. COMPUTER EXERCISES 6.1
  11. CHAPTER SEVEN Inner -Product Vector Spaces
    1. 7.1 INNER-PRODUCT SPACES“
      1. Inner-Product Spaces and Their Properties
      2. The Norm in an Inner-Product Space
      3. Distance Function
      4. Mutually Orthogonal Vectors
      5. Orthogonal Projection
      6. Angle between Vectors
      7. Orthogonal Complements
      8. Orthonormal Bases
      9. Subspaces in Inner-Product Spaces
      10. Application: Work and Forces
      11. Application: Collision
      12. SUMMARY 7.1
      13. KEY CONCEPTS 7.1
      14. GENERAL EXERCISES 7.1
      15. COMPUTER EXERCISES 7.1
    2. 7.2 ORTHOGONALITY
      1. Introduction
      2. The Gram–Schmidt Process
      3. Unnormalized Gram–Schmidt Algorithm
      4. Modified Gram–Schmidt Process
      5. Linear Least-Squares Solution
      6. Gram Matrix
      7. Distance from a Point to a Hyperplane
      8. Mathematical Software
      9. SUMMARY 7.2
      10. KEY CONCEPTS 7.2
      11. GENERAL EXERCISES 7.2
      12. COMPUTER EXERCISES 7.2
  12. CHAPTER EIGHT Additional Topics
    1. 8.1 HERMITIAN MATRICES AND THE SPECTRAL THEOREM
      1. Introduction
      2. Hermitian Matrices and Self-Adjoint Mappings
      3. Self-Adjoint Mapping
      4. The Spectral Theorem
      5. Unitary and Orthogonal Matrices
      6. The Cayley–Hamilton Theorem
      7. Quadratic Forms
      8. Application: World Wide Web Searching
      9. Mathematical Software
      10. SUMMARY 8.1
      11. KEY CONCEPTS 8.1
      12. GENERAL EXERCISES 8.1
      13. COMPUTER EXERCISES 8.1
    2. 8.2 MATRIX FACTORIZATIONS AND BLOCK MATRICES
      1. Introduction
      2. Permutation Matrix
      3. LU -Factorization
      4. LLT-Factorization: Cholesky Factorization
      5. LDLT-Factorization
      6. QR-Factorization
      7. Singular-Value Decomposition (SVD)
      8. Schur Decomposition
      9. Partitioned Matrices
      10. Solving a System Having a 2 × 2 Block Matrix
      11. Inverting a 2 × 2 Block Matrix
      12. Application: Linear Least-Squares Problem
      13. Mathematical Software
      14. SUMMARY 8.2
      15. KEY CONCEPTS 8.2
      16. GENERAL EXERCISES 8.2
      17. COMPUTER EXERCISES 8.2
    3. 8.3 ITERATIVE METHODS FOR LINEAR EQUATIONS
      1. Introduction
      2. Richardson Iterative Method
      3. Jacobi Iterative Method
      4. Gauss–Seidel Method
      5. Successive Overrelaxation (SOR) Method
      6. Conjugate Gradient Method
      7. Diagonally Dominant Matrices
      8. Gerschgorin’s Theorem
      9. Infinity Norm
      10. Convergence Properties
      11. Power Method for Computing Eigenvalues
      12. Application: Demographic Problems, Population Migration
      13. Application: Leontief Open Model
      14. Mathematical Software
      15. SUMMARY 8.3
      16. KEY CONCEPTS 8.3
      17. GENERAL EXERCISES 8.3
      18. COMPUTER EXERCISES 8.3
  13. APPENDIX A Deductive Reasoning and Proofs
    1. A.1 Introduction
    2. A.2 Deductive Reasoning and Direct Verification
    3. A.3 Implications
    4. A.4 Method of Contradiction
    5. A.5 Mathematical Induction
    6. A.6 Truth Tables
    7. A.7 Subsets and de Morgan Laws
    8. A.8 Quantifiers
    9. A.9 Denial of a Quantified Assertion
    10. A.10 Some More Questionable ‘‘Proofs’’
      1. SUMMARY APPENDIX A
      2. KEY CONCEPTS APPENDIX A
      3. GENERAL EXERCISES APPENDIX A
  14. APPENDIX B Complex Arithmetic
    1. B.1 Complex Numbers and Arithmetic
    2. B.2 Fundamental Theorem of Algebra
    3. B.3 Abel–Ruffini Theorem
  15. Answers/Hints for General Exercises
    1. General Exercises 1.1
    2. General Exercises 1.2
    3. General Exercises 1.3
    4. General Exercises 2.1
    5. General Exercises 2.2
    6. General Exercises 2.3
    7. General Exercises 2.4
    8. General Exercises 3.1
    9. General Exercises 3.2
    10. General Exercises 4.1
    11. General Exercises 4.2
    12. General Exercises 5.1
    13. General Exercises 5.2
    14. General Exercises 5.3
    15. General Exercises 6.1
    16. General Exercises 7.1
    17. General Exercises 7.2
    18. General Exercises 8.1
    19. General Exercises 8.2
    20. General Exercises 8.3
    21. General Exercises Appendix A
  16. References
  17. Index