Linear Algebra and Matrix Analysis, 2nd Ed


Welcome yet again. This page focuses on a revised edition of my linear algebra textbook "Applied Linear Algebra and Matrix Analysis", henceforth referred to as "ALAMA", which is available in hardcopy or online at Springer-Verlag as of June, 2018. A few comments are in order:

Why this text revision? I'm still committed to a balanced blend of theory, application and computation. As I noted in the preface 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 recent 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.

Resources

A complete solutions manual to all exercises and problems in the text, along with some additional exam and project samples and solutions, is available to instructors who adopt the text. Instructors who have adopted the text and would like these materials 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.

Note: There are two new items in the list below. The first is ALAMA Calculator Files, which contains commentaries and programs written for ALAMA calculator. This calculator is an easy-to-use button calculator that I programmed in C++ off and on for the past few years. It was originally for my amusement, but I was struck by how useful this simple button calculator could be as a tool in a linear algebra course. So I designed 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. ALAMA Calculator Files contains scripts for 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.

I have decided for the time being to distribute the executables of ALAMA Calculator from my own blog site.
NB: Do not download the program from any other source and be sure to verify the download before decompressing it.
Go to the following link and click on the Linear Algebra menu item:

Tom Shores: Ends and Odds

There you will find links to versions of the calculator for Mac, Windows and some flavors of Linux along with discussions about installation of the program. Longer term, I may port it to Windows UWP or Apple iOS, but these are much more extensive projects. I have no idea how suitable the APIs of UWP are for the calculator and good old Apple, Objective C wasn't different enough from C++, so they introduced Swift, which is a problem since FLTK and Eigen are written in C++.
  • ALAMA Calculator Files ALAMA Calculator program examples.
  • ALAMA Exercises and Problems Additional exercises and problems for ALAMA, 2nd Ed.
  • ALAMA Exams and Projects Here are some sample exams and projects which I and others have used with the first edition of ALAMA; in addition to pdf files, there are Latex and LyX files, so instructors may massage them to suit their own needs. For an indication of what parts of the text were covered by the exams, consult the sample syllabi in the Documents link below. Slight adjustments will be necessary for use with the second edition of ALAMA.
  • ALAMA Documents Here are sample syllabi and class policy statements which I have used with the first edition of ALAMA. Formats are html, Latex and LyX.
  • 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.

Unrandom Notes and FAQ for ALAMA Calculator and Text

Here are some notes about topics of interest, comments and answers to questions asked by users of ALAMA text and calculator.
  1. Coming soon ...

Errata Sheet

Might there be a law of spontaneous generation of errors? My rational mind says "no", but in spite of my dedicated proofreading, they will no doubt occur. In fact, since publication, I have already found a few, and I have recorded them in the errata sheet below. If you find any unlisted errors, please report them to me and I will publish them in the errata sheet.

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


Preface

Chapter 1. LINEAR SYSTEMS OF EQUATIONS


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
          Topics: Roundoff errors, computational efficiency of Gaussian elimination,
            derivation of a time dependent form of the diffusion equation.

1.6   *Projects and Reports

Chapter 2. MATRIX ALGEBRA


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
          Topics: LU factorization, efficiency of determinants and Cramer's rule,
            digital signal processing, Isorank for graphs.

2.9   *Projects and Reports

Chapter 3. VECTOR SPACES


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
          Topics: Spaces associated with a directed graph.

3.10   *Projects and Reports

Chapter 4. GEOMETRICAL ASPECTS OF STANDARD SPACES


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
          Topics: QR factorization, practical algorithm for QR factorization,
            data compression and the Haar wavelet transform.

4.5   *Projects and Reports

Chapter 5. THE EIGENVALUE PROBLEM


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
          Topics: Jordan Canonical Form theorem, computation of eigensystems.

5.8   *Projects and Reports

Chapter 6. GEOMETRICAL ASPECTS OF ABSTRACT SPACES


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
          Topics: Introduction to Fourier analysis, digital signal processing and Fourier series.
        .
6.7 *Projects and Reports

Table of Symbols

Solutions to Selected Exercises

References

Index


[ T O P ] [ H O M E

[ U N L ] [ A & S  C O L L E G E ] [ M A T H / S T A T  D E P T ]