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For a sophomore-level course in Linear Algebra.
Based on the recommendations of the Linear Algebra Curriculum Study Group, this introduction to linear algebra offers a matrix-oriented approach with more emphasis on problem solving and applications. Throughout the text, use of technology is encouraged. The focus is on matrix arithmetic, systems of linear equations, properties of Euclidean n-space, eigenvalues and eigenvectors, and orthogonality. Although matrix-oriented, the text provides a solid coverage of vector spaces
List of contents
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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Summary
For a sophomore-level course in Linear Algebra.
Based on the recommendations of the Linear Algebra Curriculum Study Group, this introduction to linear algebra offers a matrix-oriented approach with more emphasis on problem solving and applications. Throughout the text, use of technology is encouraged. The focus is on matrix arithmetic, systems of linear equations, properties of Euclidean n-space, eigenvalues and eigenvectors, and orthogonality. Although matrix-oriented, the text provides a solid coverage of vector spaces
