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- Why this text? You might read the preface to see my philosophy on linear
algebra courses. In particular, I'm committed to a balanced blend of theory,
application and computation. Mathematicians are beginning to see their discipline
as more of an experimental science, with computer software as the "laboratory"
for mathematical experimentation. I believe that the teaching of linear
algebra should incorporate this new perspective. My own experience ranges
from that of a pure mathematician (my first research was in group and ring
theory) to numerical analysis (my current speciality). I have seen linear
algebra from many viewpoints and I think they all have something to offer.
My computational experience makes me like the use of technology in the course
-- it's a natural fit for linear algebra -- and I think that computer exercises
and group projects also fit very well into the context of linear algebra.
My applied math background colors my choice and emphasis of applications
and topics. At the same time, I have a traditionalist streak that expects
a text to be rigorous, correct and complete. After all, linear algebra also
serves as a bridge course between lower and higher level mathematics.
- Many thanks to John Bakula, our McGraw-Hill text representative. John
prodded and goaded me into moving this project into the final stages and
pointed me in the direction of McGraw-Hill custom publishing, which will
do the first run of this text. They make a very nice soft copy text, and
the price to students is very attractive: about $22.
- Thanks also to my colleagues Jamie Radcliffe, Lynn Erbe, Brian Harbourne,
Roger Wiegand, Kristie Pfabe, Barton Willis and a number of other colleagues
who have used the text and made many suggestions and corrections.
- While I'm at it, special thanks also go to Jackie Kohles for her excellent
work on solutions to the exercises. And to my linear algebra students, who
are relentlessly tracking down typos and errors.
- About the process: I am writing the text in Latex. The pages you will
see have been converted to gif files for universal viewing with most browsers.
The downside of this conversion is that the pages appear at a fairly crude
resolution. I hope that they are still readable to all. Hardcopy of the
text is much prettier, to say the least. The converter that I use
has the annoying habit of cropping all images right at the edge of text,
and at the top and bottom as well. So don't judge the print qualtiy of the
text by these images.
- I'm not completely finished with the text. A a first and second edition
have been printed, and most recently a revision of the second edition that
has erlier typos and errors excised, but I'm adding projects, exercises
and a few more topics. If you have any suggestions, drop me a line. I really
appreciate any feedback. Right now, the revised second edition has incorporated
nearly all of the errors of which I am aware.
- Errata: Since many are still using the earlier version, so I'm keeping
an active Errata Sheet for the unrevised
second edition of this text. If you are using this copy of the text, be
sure to have a look at this information.
Table of Contents
Preface
Chapter 1. LINEAR SYSTEMS OF EQUATIONS
1.
Some Examples
2.
Notations and a Review of Numbers
3.
Gaussian Elimination: Basic Ideas
4.
Gaussian Elimination: General Procedure
5.
*Computational Notes and Projects
6.
Review Exercises
Chapter 2. MATRIX ALGEBRA
1.
Matrix Addition and Scalar Multiplication
2.
Matrix Multiplication
4.
Applications of Matrix Multiplication
3.
Special Matrices and Transposes
5.
Matrix Inverses
6.
Basic Properties of Determinants
7.
*Applications and Proofs for Determinants
8.
*Computational Notes and Projects
9.
Review Exercises
Chapter 3. VECTOR SPACES
1.
Definitions and Basic Concepts
2.
Subspaces
3.
Subspaces Associated with Matrices and Operators
4.
Standard Norm and Inner Product
5.
Applications of Norm and Inner Product
6.
Bases and Dimension
7.
Linear Systems Revisited
8.
*Computational Notes and Projects
9.
Review Exercises
Chapter 4. THE EIGENVALUE PROBLEM
1.
Definitions and Basic Properties
2.
Similarity and Diagonalization
3.
Applications to Discrete Dynamical Systems
4.
*Computational Notes and Projects
5.
Review Exercises
Chapter 5. GEOMETRICAL ASPECTS OF VECTORS
1.
Normed Linear Spaces
2.
Inner Product Spaces
3.
Gram-Schmidt Algorithm
4.
Unitary and Orthogonal Matrices
5.
Linear Systems Revisited
6.
*Computational Notes and Projects
7.
Review Exercises
Chapter 6. ADDITIONAL TOPICS
1.
*Tensor Products
2.
*Change of Basis and Linear Operators
2.
*Operator Norms
3.
*Schur Form and Applications
2.
*The Singular Value Decomposition
Appendix A.
Table
of Symbols
Solutions
to Selected Exercises
Bibliography
Index
Errata Sheet for the revised second edition