Elementary Linear Algebra with Applications

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. This text integrates applications throughout to illustrate the mathematics and to motivate the students. Numerical ideas and concepts using the computer are interspersed throughout the text, instructors can use them at their discretion. Features: * The core material of linear algebra is presented through the use of any examples and reinforced by exercises at the end of each section, chapters and cumulative reviews. * Applications are integrated throughout the text to motivate the students and illustrate the theory being discussed in each chapter. * Optional coverage of determinants in Chapter 2 (full) or Chapter 5 (brief) allows professors flexibility in how they want to present the material. * Use of many examples help to better explain the material, giving students a very concrete approach of linear algebra. New to this edition: * The Third Edition meets the requirements of the Linear Algebra Curriculum Study Group, an NSF sponsored committee designed to reform the linear algebra curriculum. * This edition features a new section, 3.1, using geometric motivation of theory, which will help students make an easier transition to theory and improve their intuition and understanding of linear algebra. * The Chapter on Determinants has been revised to make it more intuitive, and less formal, making it easier for students to learn. * MATLAB exercises have been fully integrated to the text. A new Appendix D: Introduction to MATLAB and Projects has been added to provide instructors with more projects and activities. * A more natural and intuitive development of least squares problems, making it easier for students to understand the motivation and theory.

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.

Review :
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.