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This book is intended for the first course in linear algebra, taken by mathematics, science, engineering and economics majors. The new edition presents a stronger geometric intuition for the ensuing concepts of span and linear independence. Applications are integrated throughout to illustrate the mathematics and to motivate the student.
Preface.
List of Applications.
1. Introduction to Linear Equations and Matrices.
Introduction to Linear Systems and Matrices.
Gaussian Elimination.
The Algebra of Matrices: Four Descriptions of the Product.
Inverses and Elementary Matrices.
Gaussian Elimination as a Matrix Factorization.
Transposes, Symmetry, and Band Matrices: An Application. Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems.
Review Exercises.
2. Determinants.
The Determinant Function.
Properties of Determinants.
Finding det A Using Signed Elementary Products.
Cofactor Expansion: Cramer's Rule.
Applications.
Review Exercises.
3. Vector Spaces.
Vectors in 2- and 3-Spaces.
Euclidean n-Space.
General Vector Spaces.
Subspaces, Span, Null Spaces.
Linear Independence.
Basis and Dimension.
The Fundamental Subspaces of a Matrix; Rank.
Coordinates and Change of Basis.
An Application: Error-Correcting Codes.
Review Exercises.
Cumulative Review Exercises.
4. Linear Transformations, Orthogonal Projections and Least Squares.
Matrices as Linear Transformation.
Relationships Involving Inner Products.
Least Squares and Orthogonal Projections.
Orthogonal Bases and the Gram-Schmidt Process.
Orthogonal Matrices, QR Decompositions, and Least Squares (Revisited).
Encoding the QR Decompositions: A Geometric Approach.
General Matrices of Linear of Linear Transformations; Similarity.
Review Exercises.
Cumulative Review Exercises.
5. Eigenvectors and Eigenvalues.
A Brief Introduction to Determinants.
Eigenvalues and Eigenvectors.
Diagonalization.
Symmetric Matrices.
An Application - Difference Equations: Fibonacci Sequences and Markov Processes.
An Application -Differential Equations.
An Application -- Quadratic Forms.
Solving the Eigenvalue Problem Numerically.
Review Exercises.
Cumulative Review Exercises.
6. Further Directions. Function Spaces.
The Singular Value Decomposition -- Generalized Inverses, the General Least-Squares Problem, and an Approach to Ill-Conditioned Systems.
Iterative Method. Matrix Norms.
General Vector Spaces and Linear Transformations Over an Arbitrary Field.
Review Exercises.
Appendix A: More on LU Decompositions.
Appendix B: Counting Operations and Gauss-Jordan Elimination.
Appendix C: Another Application.
Appendix D: Introduction to MATLAB and Projects.
Bibliography and Further Readings.
Index.
This book is intended for the first course in linear algebra, taken by mathematics, science, engineering and economics majors. The new edition presents a stronger geometric intuition for the ensuing concepts of span and linear independence. Applications are integrated throughout to illustrate the mathematics and to motivate the student.
Table of Contents
Preface.
List of Applications.
1. Introduction to Linear Equations and Matrices.
Introduction to Linear Systems and Matrices.
Gaussian Elimination.
The Algebra of Matrices: Four Descriptions of the Product.
Inverses and Elementary Matrices.
Gaussian Elimination as a Matrix Factorization.
Transposes, Symmetry, and Band Matrices: An Application. Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems.
Review Exercises.
2. Determinants.
The Determinant Function.
Properties of Determinants.
Finding det A Using Signed Elementary Products.
Cofactor Expansion: Cramer's Rule.
Applications.
Review Exercises.
3. Vector Spaces.
Vectors in 2- and 3-Spaces.
Euclidean n-Space.
General Vector Spaces.
Subspaces, Span, Null Spaces.
Linear Independence.
Basis and Dimension.
The Fundamental Subspaces of a Matrix; Rank.
Coordinates and Change of Basis.
An Application: Error-Correcting Codes.
Review Exercises.
Cumulative Review Exercises.
4. Linear Transformations, Orthogonal Projections and Least Squares.
Matrices as Linear Transformation.
Relationships Involving Inner Products.
Least Squares and Orthogonal Projections.
Orthogonal Bases and the Gram-Schmidt Process.
Orthogonal Matrices, QR Decompositions, and Least Squares (Revisited).
Encoding the QR Decompositions: A Geometric Approach.
General Matrices of Linear of Linear Transformations; Similarity.
Review Exercises.
Cumulative Review Exercises.
5. Eigenvectors and Eigenvalues.
A Brief Introduction to Determinants.
Eigenvalues and Eigenvectors.
Diagonalization.
Symmetric Matrices.
An Application - Difference Equations: Fibonacci Sequences and Markov Processes.
An Application -Differential Equations.
An Application -- Quadratic Forms.
Solving the Eigenvalue Problem Numerically.
Review Exercises.
Cumulative Review Exercises.
6. Further Directions. Function Spaces.
The Singular Value Decomposition -- Generalized Inverses, the General Least-Squares Problem, and an Approach to Ill-Conditioned Systems.
Iterative Method. Matrix Norms.
General Vector Spaces and Linear Transformations Over an Arbitrary Field.
Review Exercises.
Appendix A: More on LU Decompositions.
Appendix B: Counting Operations and Gauss-Jordan Elimination.
Appendix C: Another Application.
Appendix D: Introduction to MATLAB and Projects.
Bibliography and Further Readings.
Index.