Preface

SUGGESTED COURSE OUTLINES

This text includes the basis for a two-semester course in linear algebra.

• Chapters [chap:1]–[chap:4] provide a standard one-semester course of 35 lectures, including linear equations, matrix algebra, determinants, diagonalization, and geometric vectors, with applications as time permits. At Calgary, we cover Sections [sec:1_1]–[sec:1_3], [sec:2_1]–[sec:2_6], [sec:3_1]–[sec:3_3], and [sec:4_1]–[sec:4_4] and the course is taken by all science and engineering students in their first semester. Prerequisites include a working knowledge of high school algebra (algebraic manipulations and some familiarity with polynomials); calculus is not required.

• Chapters [chap:5]–[chap:9] contain a second semester course including \(\mathbb{R}^n\), abstract vector spaces, linear transformations (and their matrices), orthogonality, complex matrices (up to the spectral theorem) and applications. There is more material here than can be covered in one semester, and at Calgary we cover Sections [sec:5_1]–[sec:5_5], [sec:6_1]–[sec:6_4], [sec:7_1]–[sec:7_3], [sec:8_1]–[sec:8_6], and [sec:9_1]–[sec:9_3] with a couple of applications as time permits.

• Chapter [chap:5] is a “bridging” chapter that introduces concepts like spanning, independence, and basis in the concrete setting of \(\mathbb{R}^n\), before venturing into the abstract in Chapter [chap:6]. The duplication is balanced by the value of reviewing these notions, and it enables the student to focus in Chapter [chap:6] on the new idea of an abstract system. Moreover, Chapter [chap:5] completes the discussion of rank and diagonalization from earlier chapters, and includes a brief introduction to orthogonality in \(\mathbb{R}^n\), which creates the possibility of a one-semester, matrix-oriented course covering Chapter [chap:1]–[chap:5] for students not wanting to study the abstract theory.

CHAPTER DEPENDENCIES

The following chart suggests how the material introduced in each chapter draws on concepts covered in certain earlier chapters. A solid arrow means that ready assimilation of ideas and techniques presented in the later chapter depends on familiarity with the earlier chapter. A broken arrow indicates that some reference to the earlier chapter is made but the chapter need not be covered.

HIGHLIGHTS OF THE TEXT

• Two-stage definition of matrix multiplication. First, in Section [sec:2_2] matrix-vector products are introduced naturally by viewing the left side of a system of linear equations as a product. Second, matrix-matrix products are defined in Section [sec:2_3] by taking the columns of a product \(AB\) to be \(A\) times the corresponding columns of \(B\). This is motivated by viewing the matrix product as composition of maps (see next item). This works well pedagogically and the usual dot-product definition follows easily. As a bonus, the proof of associativity of matrix multiplication now takes four lines.

• Matrices as transformations. Matrix-column multiplications are viewed (in Section [sec:2_2]) as transformations \(\mathbb{R}^n \to \mathbb{R}^m\). These maps are then used to describe simple geometric reflections and rotations in \(\mathbb{R}^2\) as well as systems of linear equations.

• Early linear transformations. It has been said that vector spaces exist so that linear transformations can act on them—consequently these maps are a recurring theme in the text. Motivated by the matrix transformations introduced earlier, linear transformations \(\mathbb{R}^n \to \mathbb{R}^m\) are defined in Section [sec:2_6], their standard matrices are derived, and they are then used to describe rotations, reflections, projections, and other operators on \(\mathbb{R}^2\).

• Early diagonalization. As requested by engineers and scientists, this important technique is presented in the first term using only determinants and matrix inverses (before defining independence and dimension). Applications to population growth and linear recurrences are given.

• Early dynamical systems. These are introduced in Chapter [chap:3], and lead (via diagonalization) to applications like the possible extinction of species. Beginning students in science and engineering can relate to this because they can see (often for the first time) the relevance of the subject to the real world.

• Bridging chapter. Chapter [chap:5] lets students deal with tough concepts (like independence, spanning, and basis) in the concrete setting of \(\mathbb{R}^n\) before having to cope with abstract vector spaces in Chapter [chap:6].

• Examples. The text contains over 375 worked examples, which present the main techniques of the subject, illustrate the central ideas, and are keyed to the exercises in each section.

• Exercises. The text contains a variety of exercises (nearly 1175, many with multiple parts), starting with computational problems and gradually progressing to more theoretical exercises. Select solutions are available at the end of the book or in the Student Solution Manual. There is a complete Solution Manual is available for instructors.

• Applications. There are optional applications at the end of most chapters (see the list below). While some are presented in the course of the text, most appear at the end of the relevant chapter to encourage students to browse.

• Appendices. Because complex numbers are needed in the text, they are described in Appendix [chap:appacomplexnumbers], which includes the polar form and roots of unity. Methods of proofs are discussed in Appendix [chap:appbproofs], followed by mathematical induction in Appendix [chap:appcinduction]. A brief discussion of polynomials is included in Appendix [chap:appdpolynomials]. All these topics are presented at the high-school level.

• Self-Study. This text is self-contained and therefore is suitable for self-study.

• Rigour. Proofs are presented as clearly as possible (some at the end of the section), but they are optional and the instructor can choose how much he or she wants to prove. However the proofs are there, so this text is more rigorous than most. Linear algebra provides one of the better venues where students begin to think logically and argue concisely. To this end, there are exercises that ask the student to “show” some simple implication, and others that ask her or him to either prove a given statement or give a counterexample. I personally present a few proofs in the first semester course and more in the second (see the Suggested Course Outlines).

• Major Theorems. Several major results are presented in the book. Examples: Uniqueness of the reduced row-echelon form; the cofactor expansion for determinants; the Cayley-Hamilton theorem; the Jordan canonical form; Schur’s theorem on block triangular form; the principal axes and spectral theorems; and others. Proofs are included because the stronger students should at least be aware of what is involved.

CHAPTER SUMMARIES

LIST OF APPLICATIONS

• Network Flow (Section [sec:1_4])

• Electrical Networks (Section [sec:1_5])

• Chemical Reactions (Section [sec:1_6])

• Directed Graphs (in Section [sec:2_3])

• Input-Output Economic Models (Section [sec:2_8])

• Markov Chains (Section [sec:2_9])

• Polynomial Interpolation (in Section [sec:3_2])

• Population Growth (Examples [exa:008923] and [exa:009389], Section [sec:3_3])

• Google PageRank (in Section [sec:3_3])

• Linear Recurrences (Section [sec:3_4]; see also Section [sec:7_5])

• Systems of Differential Equations (Section [sec:3_5])

• Computer Graphics (Section [sec:4_5])

• Least Squares Approximation (in Section [sec:5_6])

• Correlation and Variance (Section [sec:5_7])

• Polynomials (Section [sec:6_5])

• Differential Equations (Section [sec:6_6])

• Linear Recurrences (Section [sec:7_5])

• Error Correcting Codes (Section [sec:8_7])

• Quadratic Forms (Section [sec:8_8])

• Constrained Optimization (Section [sec:8_9])

• Statistical Principal Component Analysis (Section [sec:8_10])

• Fourier Approximation (Section [sec:10_5])

ACKNOWLEDGMENTS

Many colleagues have contributed to the development of this text over many years of publication, and I specially thank the following instructors for their reviews of the \(7^{th}\) edition:

Robert Andre
University of Waterloo

Dietrich Burbulla
University of Toronto

Dzung M. Ha
Ryerson University

Mark Solomonovich
Grant MacEwan

Fred Szabo
Concordia University

Edward Wang
Wilfred Laurier

Petr Zizler
Mount Royal University

It is also a pleasure to recognize the contributions of several people. Discussions with Thi Dinh and Jean Springer have been invaluable and many of their suggestions have been incorporated. Thanks are also due to Kristine Bauer and Clifton Cunningham for several conversations about the new way to look at matrix multiplication. I also wish to extend my thanks to Joanne Canape for being there when I had technical questions. Thanks also go to Jason Nicholson for his help in various aspects of the book, particularly the Solutions Manual. Finally, I want to thank my wife Kathleen, without whose understanding and cooperation, this book would not exist.

As we undertake this new publishing model with the text as an open educational resource, I would also like to thank my previous publisher. The team who supported my text greatly contributed to its success.

Now that the text has an open license, we have a much more fluid and powerful mechanism to incorporate comments and suggestions. The editorial group at Lyryx invites instructors and students to contribute to the text, and also offers to provide adaptations of the material for specific courses. Moreover the LaTeX source files are available to anyone wishing to do the adaptation and editorial work themselves!

W. Keith Nicholson
University of Calgary