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Elementary Linear Algebra with Applications

Elementary Linear Algebra with Applications - 3rd edition

ISBN13: 978-0030103476

Cover of Elementary Linear Algebra with Applications 3RD 96 (ISBN 978-0030103476)
ISBN13: 978-0030103476
ISBN10: 0030103479
Cover type:
Edition/Copyright: 3RD 96
Publisher: Saunders College Division
Published: 1996
International: No
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Elementary Linear Algebra with Applications - 3RD 96 edition

ISBN13: 978-0030103476

Richard O. Hill

ISBN13: 978-0030103476
ISBN10: 0030103479
Cover type:
Edition/Copyright: 3RD 96
Publisher: Saunders College Division

Published: 1996
International: No
Summary

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.

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