Linear Algebra and Applications Textbook

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Welcome yet again. I have written a revised edition of my linear algebra textbook "Applied Linear Algebra and Matrix Analysis", henceforth referred to as "ALAMA". This new edition of ALAMA will appear some time in 2018. A few comments:

Why this text revision? I'm still committed to a balanced blend of theory, application and computation. As I noted in the page for the first edition of ALAMA: "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."

But times change, don't they? The notion of mathematical experimentation as an important part of a linear algebra course has become commonplace, and I continue to support this perspective. I took this revision as an opportunity to incorporate the many helpful corrections and suggestions for improvement of the text that I received from instructors who used the text; for that I thank all of them! And a contemporary perspective on linear algebra has to incorporate some of the extraordinary technological developments that depend heavily on concepts from linear algebra, and indeed these developments reinforce the centrality of this discipline. A prime example is the PageRank technology that was developed by Google. This technology found a place in every chapter but one of the revised text. Likewise, digital signal processing (DSP) finds a place in three chapters of the revised text. A fairly substantial introduction to linear programming is found in Chapter 3. A proof of the Jordan Canonical Form theorem can be found in Chapter 5. Even a brief introduction to Fourier analysis and its connections to linear algebra (needed for an understanding of some of the underlying principles of DSP) can be found in Chapter 6.

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 will be available to instructors who adopt the text. Once the text has become available, instructors who have adopted the text and would like a copy of this manual should contact me via email.

For the benefit of instructors and students using my text, (some of) the resources available for the first edition are still relevant and listed below. I also have complete solution keys to the exams and projects that are found in the directories below. I will email these to adopting instructors upon request. The text materials come in three flavors: pdf for perusal, LyX and Latex for modification and use by instructors.

Note: There are two new items in the list below: The first is ALAMA Calculator Files. This calculator is an easy to use button calculator that I programmed off and on for the past few years (a busy mathematician is a happy mathematician). It was originally for my amusement, but I wrote it with an eye to the ALAMA textbook, both first and revised editions. It has enough capabilities that I could even carry out a few projects with it and used it to create some of the graphs that appear in the revised ALAMA, along with pretty much all the routine matrix calculations in ALAMA -- nice tool if you want to check answers. I'm not quite finished with it (revising the Users Guide to conform more closely to revised ALAMA). I wrote it in C++, so that I could use the excellent packages FLTK and Eigen. When finished, it will be available on machines with operating systems OS X, Windows 10 and a few flavors of linux. Longer term, I may port it to iOS, but this is a much more extensive project (good old Apple, Objective C wasn't different enough, so they introduced Swift!). Packages for the various executables will be available soon. ALAMA Calculator Files contains worked from nearly all the examples from ALAMA that require technology tools, as well as a few programs that I wrote (yes, it's a fully programmable calculator, but it takes a bit of careful thinking to do it).

The second item is a directory which will contain exercises and problems (and even projects) that are nice supplements to the text. They will be submitted by me or instructors using the textbook (with attribution, of course). So if you have any exercises, problems or projects that you found useful and would like to share, by all means send them in to me.

  • ALAMA Calculator Files ALAMA Calculator executables and program examples.
  • ALAMA Exercises and Problems Additional exercises, problems and projects for ALAMA, 2nd Ed.
  • 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 may 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.
  • ALAMA Documents Here are sample syllabi and class policy statements which I have used with ALAMA. Formats are html, tex and lyx.
  • ALAMA Exams Here are sample exams which I (and others) have used along with ALAMA; there are latex and lyx files, so instructors may massage them to suit their own needs.

Errata Sheet

None yet, but if you find any unposted errors, please report them to me and I will publish them in the errata sheet.

Here is the table of contents of the revised text:

Applied Linear Algebra and Matrix Analysis, 2nd Ed.
Thomas S. Shores
Copyright © 2018 Springer Science+Business Media, LLC



1.1   Some Examples

1.2   Notations and a Review of Numbers

1.3   Gaussian Elimination: Basic Ideas

1.4   Gaussian Elimination: General Procedure

1.5   *Applications and Computational Notes

1.6   *Projects and Reports


2.1   Matrix Addition and Scalar Multiplication

2.2   Matrix Multiplication

2.3   Applications of Matrix Arithmetic

2.4   Special Matrices and Transposes

2.5   Matrix Inverses

2.6   Determinants

2.7   *Tensor products

2.8   *Applications and Computational Notes

2.9   *Projects and Reports


3.1   Definitions and Basic Concepts

3.2   Subspaces

3.3   Linear Combinations

3.4   Subspaces Associated with Matrices and Operators
3.5   Bases and Dimension

3.6   Linear Systems Revisited

3.7   *Change of Basis and Linear Operators

3.8   *Introduction to linear programming

3.9   *Applications and Computational Notes

3.10   *Projects and Reports


4.1   Standard Norm and Inner Product

4.2   Applications of Norm and Inner Product

4.3   Orthogonal and Unitary Matrices

4.4   *Applications and Computational Notes

4.5   *Projects and Reports


5.1   Definitions and Basic Properties

5.2   Similarity and Diagonalization

5.3   Applications to Discrete Dynamical Systems

5.4   Orthogonal Diagonalization

5.5   *Schur Form and Applications

5.6   *The Singular Value Decomposition

5.7   *Applications and Computational Notes

5.8   *Projects and Reports


6.1 Normed Spaces

6.2 Inner Product Spaces

6.3 Gram-Schmidt Algorithm

6.4 Orthogonal Vectors and Projection

6.5 *Operator Norms

6.6 *Applications and Computational Notes

6.7 *Projects and Reports

Table of Symbols

Solutions to Selected Exercises



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