Elementary Linear Algebra

PREFACE ix

 



TO THE STUDENT xv

 



CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS 1



  1.1 Matrices and Vectors 1



  1.2 Linear Combinations, Matrix—Vector Products, and Special Matrices 11



  1.3 Systems of Linear Equations 25



  1.4 Gaussian Elimination 39



  1.5* Applications of Systems of Linear Equations 54



  1.6 The Span of a Set of Vectors 64



  1.7 Linear Dependence and Linear Independence 73

  Chapter 1 Review Exercises

  Chapter 1 MATLAB Exercises

 



CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS 90



  2.1 Matrix Multiplication 90



  2.2* Applications of Matrix Multiplication 101



  2.3 Invertibility and Elementary Matrices 117



  2.4 The Inverse of a Matrix 130



  2.5* Partitioned Matrices and Block Multiplication 141



  2.6* The LU Decomposition of a Matrix 147



  2.7 Linear Transformations and Matrices 162



  2.8 Composition and Invertibility of Linear Transformations 175

  Chapter 2 Review Exercises

  Chapter 2 MATLAB Exercises

 



CHAPTER 3 DETERMINANTS 192



  3.1 Cofactor Expansion 192



  3.2 Properties of Determinants 204

  Chapter 3 Review Exercises

  Chapter 3 MATLAB Exercises

 



CHAPTER 4 SUBSPACES AND THEIR PROPERTIES 218



  4.1 Subspaces 218



  4.2 Basis and Dimension 232



  4.3 The Dimension of Subspaces Associated with a Matrix 245



  4.4 Coordinate Systems 254



  4.5 Matrix Representations of Linear Operators 266

  Chapter 4 Review Exercises

  Chapter 4 MATLAB Exercises

 



CHAPTER 5 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION 282



  5.1 Eigenvalues and Eigenvectors 282



  5.2 The Characteristic Polynomial 291



  5.3 Diagonalization of Matrices 302



  5.4* Diagonalization of Linear Operators 314



  5.5* Applications of Eigenvalues 323

  Chapter 5 Review Exercises

  Chapter 5 MATLAB Exercises

 

 



CHAPTER 6 VECTOR SPACES 473



  6.1 Vector Spaces and Their Subspaces 473



  6.2 Linear Transformations 485



  6.3 Basis and Dimension 495



  6.4 Matrix Representations of Linear Operators 505



  6 .5 Inner Product Spaces 517

  Chapter 6 Review Exercises

  Chapter 6 MATLAB Exercises

 



CHAPTER 7 ORTHOGONALITY 347



  7.1 The Geometry of Vectors 347



  7.2 Orthogonal Vectors 360



  7.3 Orthogonal Projections 374



  7.4 Least-Squares Approximations and Orthogonal Projections 388



  7.5 Orthogonal Matrices and Operators 398



  7.6 Symmetric Matrices 412



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