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Linear Algebra and Applications Textbook

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Welcome again. In order to enable prospective users to preview my text easily and conveniently, in the past I have put a copy of it on the web for your perusal. In the last few years I've received many helpful comments and appreciative notes for having done so. I would like to thank those of you who sent me these notes and comments. You have helped me substantially improve the text. I am now under contract with Springer-Verlag and the book has been published in their Undergraduate Texts in Mathematics series in hardbound and, more recently, soft cover editions. Therefore, I have removed the on-line copy. I will leave the table of contents below for informational purposes, along with errata sheets for the each version of the textbook. A few comments:

Why this text? 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 pure mathematician (my first research was in group and ring theory) to numerical analyst (my current speciality). I've 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 -- a natural fit for linear algebra -- and 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.

If you have any suggestions or comments, drop me a line. I appreciate any feedback.


A complete solutions manual to all exercises and problems in the text is available to instructors who have adopted the text. Instructors who would like a copy of this manual should contact Mark Spencer via email to

For the benefit of instructors and students using my text, I'm moving a number of downloadable files on my website that pertain specifically to the text into their own directories. I also have complete solution keys to the exams and projects that are found in the directories below. I will email these to instructors who are using my text upon request. The text materials come in three flavors: pdf for perusal, lyx and tex for modification and use by instructors.

  • Maple Notebooks Tutorial notebooks in Maple, some of which are the basis for linear algebra projects.
  • Mathematica Notebooks Tutorial notebooks in Mathematica, some of which are the basis for linear algebra projects (in old Mathematica .ma and new .nb formats. I'll update them at a future date, since I haven't used Mathematica in a while.)
  • Matlab Files Program files for Matlab and a Matlab-like program called Octave which I have found very useful in linear algebra.
  • Sample Documents Here are sample syllabi and class policy statements which I have used with my text. Formats are html, tex and lyx.
  • Sample Exams and Projects Here are sample exams which I have used along with my text; there are latex and lyx files, so instructors may massage them to suit their own needs.

Errata Sheets

It's frustrating to debug a text that you've written youself. I've discovered that far too often I was reading what I thought should be on the page rather than what actually on the page. I want to thank my many students and colleagues who tracked down many of the errata in the text. I've also learned that editorial help can turn up most unexpectedly. I owe a special debt of gratitude to Dr. David Taylor and Dr. Mats Desaix, who read a good deal of the text very carefully entirely at their own initiative, and brought a good number of errors to my attention. Somehow, I suspect the battle is not done, so if you find any errors not listed below, please report them to me.

The errata for the soft cover printing of the text apply also to the hardbound printing, which came out six month earlier than the soft cover text. Conversely, the errata for the hardbound printing were excised in the soft cover printing. While writing the instructors' solution manual, I rechecked all the solutions to exercises in the back of the text and found some more errors, which are recorded below in the errata Here is the table of contents of the text:

Applied Linear Algebra and Matrix Analysis
Thomas S. Shores
Copyright © 2007 Springer Science+Business Media, LLC



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


1. Matrix Addition and Scalar Multiplication

2. Matrix Multiplication

3. Applications of Matrix Arithmetic

4. Special Matrices and Transposes

5. Matrix Inverses

6. Basic Properties of Determinants

7. *Computational Notes and Projects


1. Definitions and Basic Concepts

2. Subspaces

3. Linear Combinations

4. Subspaces Associated with Matrices and Operators

5. Bases and Dimension

6. Linear Systems Revisited

7. *Computational Notes and Projects


1. Standard Norm and Inner Product

2. Applications of Norm and Inner Product

3. Orthogonal and Unitary Matrices

4. *Change of Basis and Linear Operators

5. *Computational Notes and Projects


1. Definitions and Basic Properties

2. Similarity and Diagonalization

3. Applications to Discrete Dynamical Systems

4. Orthogonal Diagonalization

5. *Schur Form and Applications

6. *The Singular Value Decomposition

7. *Computational Notes and Projects


1. Normed Spaces

2. Inner Product Spaces

3. Gram-Schmidt Algorithm

4. Linear Systems Revisited

5. *Operator Norms

6. *Computational Notes and Projects

Table of Symbols

Answers to Selected Exercises



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