Elementary Linear Algebra

1st Edition - January 1, 1986
Author: Richard O. Hill
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 5 1 7 - 9

Elementary Linear Algebra reviews the elementary foundations of linear algebra in a student-oriented, highly readable way. The many examples and large number and variety of… Read more

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Elementary Linear Algebra reviews the elementary foundations of linear algebra in a student-oriented, highly readable way. The many examples and large number and variety of exercises in each section help the student learn and understand the material. The instructor is also given flexibility by allowing the presentation of a traditional introductory linear algebra course with varying emphasis on applications or numerical considerations. In addition, the instructor can tailor coverage of several topics. Comprised of six chapters, this book first discusses Gaussian elimination and the algebra of matrices. Applications are interspersed throughout, and the problem of solving AX = B, where A is square and invertible, is tackled. The reader is then introduced to vector spaces and subspaces, linear independences, and dimension, along with rank, determinants, and the concept of inner product spaces. The final chapter deals with various topics that highlight the interaction between linear algebra and all the other branches of mathematics, including function theory, analysis, and the singular value decomposition and generalized inverses. This monograph will be a useful resource for practitioners, instructors, and students taking elementary linear algebra.

Preface

1 Introduction to Linear Equations and Matrices

1.1 Introduction to Linear Systems and Matrices

1.2 Gaussian Elimination

1.3 The Algebra of Matrices

1.4 Inverses and Elementary Matrices

1.5 Gaussian Elimination as a Matrix Factorization

1.6 Transposes, Symmetry, and Band Matrices; An Application

1.7 Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems

Review Exercises

2 Vector Spaces

2.1 Vectors in 2- and 3- Spaces

2.2 Euclidean n-Space

2.3 General Vector Spaces

2.4 Subspaces, Span, Null Spaces

2.5 Linear Independence

2.6 Basis and Dimension

2.7 The Fundamental Subspaces of a Matrix; Rank

2.8 An Application: Error-Correcting Codes

Review Exercises

3 Linear Transformations, Orthogonal Projections, and Least Squares

3.1 Matrices as Linear Transformations

3.2 Relationships Involving Inner Products

3.3 Least Squares and Orthogonal Projections

3.4 Orthogonal Bases and the Gram-Schmidt Process

3.5 Orthogonal Matrices, QR Decompositions, and Least Squares (Revisited)

3.6 Encoding the QR Decomposition—A Geometric Approach

Review Exercises

4 Eigenvectors and Eigenvalues

4.1 A Brief Introduction to Determinants

4.2 Eigenvalues and Eigenvectors

4.3 Diagonalization

4.4 Symmetric Matrices

4.5 An Application—Difference Equations: Fibonacci Sequences and Markov Processes

4.6 An Application—Differential Equations

4.7 An Application—Quadratic Forms

4.8 Solving the Eigenvalue Problem Numerically

Review Exercises

5 Determinantst

5.1 The Determinant Function

5.2 Evaluating Determinants

5.3 Properties of Determinants

5.4 Cofactor Expansion; Cramer's Rule

Review Exercises

6 Further Directions

6.1 Function Spaces

6.2 Singular Value Decomposition and Generalized Inverses

6.3 General Vector Spaces and Linear Transformations Over an Arbitrary Field

Review Exercises

Appendices

Appendix A. More on LU-Decompositions

Appendix B. Counting Operations and Gauss-Jordan Elimination

Appendix C. Another Application

Appendix D. Software and Codes for Linear Algebra

Bibliography and Further Readings

Answers to Odd-Numbered Exercises

Index