Linear Algebra

Reviewed by Daniel Drucker, Professor and Associate Chair, Wayne State University on 12/5/18

Comprehensiveness rating: 4 see less

This text does a good job of covering the theory in detail, especially in Chapters Two and Five. Uniqueness of reduced echelon form is proved in detail. (It’s in a section unhelpfully entitled “The Linear Combination Lemma”). Optional sections in Chapter Five give a more comprehensive treatment of Jordan canonical form than is found in most introductory texts, many of which omit this topic entirely. The author does not focus on the four fundamental subspaces, a point of view popularized by Gilbert Strang in his books Linear Algebra and Its Applications and Introduction to Linear Algebra. He emphasizes concepts and theory much more than calculation, and linear transformations much more than matrices. Matrix factorizations (LU, QR, etc.) are not covered. Orthogonality is viewed as an optional, not a central, topic. Least squares is an add-on end-of-chapter Topic, and only lines of best fit are discussed. The spectral theorem for symmetric matrices is not mentioned. Positive definite and positive semidefinite matrices are not covered. Neither are pseudoinverses or the singular value decomposition, which means that diagonalization of non-square matrices is never mentioned. Numerical linear algebra is mentioned only in the context of Gaussian elimination and the method of powers, which appear as Topics at the ends of Chapters One and Five. Operation counts and iterative methods (other than the method of powers for sparse matrices) are not discussed. Although complex vector spaces are used in the sections on diagonalization and Jordan form, there is no discussion of hermitian, skew-hermitian, unitary, or normal matrices. Many of these topics are not included in most introductory courses, but current introductory texts often include them in optional chapters to provide instructors with topics for projects and to provide students with additional material to read on their own.
The applications are interesting and varied, but limited in length and depth. They include: use of bases to describe the structure of several crystals, the use of linear systems of equations to analyze voting paradoxes, dimensional analysis (using linear systems), lines of best fit, the geometry of linear maps (nomographs for functions from R to R, and projections, rotations, reflections, and shears), magic squares (calculation of the dimension of the space of n x n magic squares), Markov chains (nontrivial examples analyzed with the help of Sage), orthonormal matrices (standardly called orthogonal matrices, though the term orthogonormal matrices is actually better), Cramer’s rule (which, like orthonormal matrices, is a topic that I think should be part of the regular exposition, not an end-of-chapter Topic), speed of calculating determinants (timing with Sage shows we shouldn’t use the permutation expansion formula), Chiò’s recursive method of calculating determinants using 2 x 2 minors, projective geometry (an excellent section identifying three types of central projection and including a proof of Desargue’s theorem), the method of powers for finding the largest eigenvalue of a sparse matrix, eigenvalues that yield static and dynamically stable population distributions, a greatly simplified version of Google’s Page Rank algorithm, linear recurrences (sometimes called linear difference equations—they can be viewed as discrete versions of differential equations), and coupled oscillators (partial solution of a particular pair of differential equations). Note that a number of these “applications” are considered mainstream topics in other texts. Omissions include linear programming, general systems of linear differential equations, classification of the behavior of 2 x 2 systems of linear differential equations, the fast Fourier transform, graphs and networks in general, image compression, quadratic forms and classification of second degree curves, computer graphics (except for a brief mention on pp. 315–316), and various geometrical topics (such as affine geometry; barycentric coordinates; convexity; problems about points, lines, and planes).
The author’s choice of topics is consistent with his goal of wanting to develop mathematical maturity in students early in their college programs. I suspect that many of the sophomores at my university, most of whom commute and work considerable hours at a job while in school, and who struggle with elementary linear algebra’s abstractions, would have even more trouble with a theory-oriented exposition like this one. Its suitability at a given school surely depends on the aptitude, level of preparedness, and time constraints of the students.
The author provides a preface, an index of notations, a comprehensive bibliography, and an index, but they are not listed in the table of contents. There is no glossary.

Content Accuracy rating: 5

I found a small number of errors in punctuation, spelling, and grammar, but they do not interfere with reading the text. The problem solutions that accompany the text appear to be correct, though I’ve only sampled them randomly as I’ve never taught from this text.

Relevance/Longevity rating: 5

I don’t think the content of this text will become obsolete any time soon. After all, the theory of linear algebra is not going to change. The choice and number of applications may need to change as tastes in topical coverage evolve.

Clarity rating: 4

I found the author’s writing to be clear, though some discussions seem unnecessarily long-winded. His illustrations are helpful, with the one problem that he tends not to label his axes. As for jargon/technical terminology, the author does use some nonstandard notations (e.g., RepB,B’) and terminology (e.g., inter-reducible). These are not obstacles in reading the book, but they may make it harder for students to read the same topics in other books on linear algebra.
I found the geometrical motivation for Cramer’s rule to be forced and unhelpful.

Consistency rating: 5

The style and terminology are consistent throughout the text.

Modularity rating: 4

Changing the order of coverage of the main sections would likely disrupt the careful development of theory in the first three chapters. End-of-chapter Topics can be covered in any order, or omitted. In the Preface, the author suggests going through the text (exclusive of Topics) in order and suggests the rate at which that can be done. He also marks some sections as optional, so that they can be skipped to devote time to other parts of the text.

Organization/Structure/Flow rating: 4

The material is logically ordered and divided into five chapters, but Chapter Three is much longer than any other chapter. It could be split into two chapters, the second one on orthogonality and its applications.

Interface rating: 5

Links in the .pdf file’s table of contents allow the reader to jump easily to any section of the text. There are also links to references in the bibliography. I found no display features that could distract or confuse the reader, except for figures in which the coordinate axes are not labeled. It is worth mentioning that the text is presented in an attractive and readable font.
Hyperlinks connect the exercises in the text with their solutions, provided the names of the files are not changed. I tried changing the file name of the text to something more descriptive than book.pdf and found that clicking on the number of an exercise did not get me back to the exercise in the text. Changing the name of the text’s file also changed my Mac’s numbering of pages in the pdf file.

Grammatical Errors rating: 4

There are a few grammatical errors and a few places where the author seems to omit a word or string two sentences together, but these do not seriously interfere with reading the text.

Cultural Relevance rating: 5

Examples in this text do not mention race, ethnicity, or people’s backgrounds. Thus they are not culturally insensitive or offensive in any way.

Comments

The author not only provides .pdf files of the book and solutions manual with helpful links, but also includes a lab manual that introduces the interested reader to Python and Sage. Besides those, he includes extra problems with solutions, an introduction to proofs, and an article on matrix arithmetic. Professor Hefferon tries hard to motivate every topic he covers and almost always succeeds. He is to be commended and appreciated for doing everything he could to help students and instructors to benefit from and make maximal use of his text. Even when it is not used as the text for a course, it can serve as a useful reference.

Reviewed by Iuliana Oprea, Associate Professor, Colorado State University on 1/7/16

Comprehensiveness rating: 5 see less

The book covers the standard material for an introductory course in linear algebra. The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The approach is developmental, the topics are covered in a comprehensive fashion, and the mathematical language of the book is very rigorous and proof-based. Nearly everything is proven either in the main text or in the exercises, which is helpful for readers who are trying to bring more rigor to their mathematical thinking and mathematical maturity.
The book is well balanced, substantial yet concise and has extensive exercise sets, with levels of difficulty varying from routine verifications to challenging problems, with worked answers and detailed solutions to all exercises. Each chapter closes with a selection of related special topics, usually applications to real world examples from physics, biology, economics, probability and abstract algebra that could be assigned as individual or group projects or could be presented in class. These special topics, such as crystals, stable populations, electrical networks, dimensional analysis, voting paradoxes and so on, together with the many interesting applications throughout the text, make this book more valuable than the average undergraduate linear algebra textbook.
On the web page of the author beamer slides for classroom use are available, that draw from the text source with respect to the notations, the numbering of theorems etc, but contain different examples than in the book. The web page also hosts a lab manual for computer work (using Sage) and links to a repository with the latex source files.

Content Accuracy rating: 5

I used the book as a textbook for two semesters and found the text to be accurate and error free. The book is available for download since 1995. The book has been tested over many years at a number of different schools and by a number of different instructors, and the author continuously improved the book based on their feedback, so it is ready to use today.

Relevance/Longevity rating: 5

The book content and presentation of topics have been updated and improved over the years; the content of the present edition is up-to-date.

Clarity rating: 5

The text presentation is very clear and well motivated; the proofs are rigorous, unambiguous but include plenty of details that make them accessible and easy to follow.

Consistency rating: 5

The text is consistent in terms of terminology and framework.

Modularity rating: 5

The text is easily and readily divisible into smaller reading sections that give enough flexibility to an instructor in the organization of the lectures. There are subsections, in the table of contents, marked as optional if some instructors will pass over them in favor of spending more time elsewhere. The book comes with a very useful semester’s time table, too.

Organization/Structure/Flow rating: 5

The topics are presented in a logical, clear fashion; a wealth of examples throughout the book is provided, and the author gives a lot of motivations for the study of most of the topics. A positive aspect is that – unlike many other textbooks - it starts with linear transformations rather than starting with matrices and then develops the intuition behind matrices.

Interface rating: 5

I had no problems using the interface and no navigation problems. The pdf file is easy to use. A nice feature is that if the pdf files for both the book and the solutions are saved in the same folder then clicking on an exercise sends you to its answer and clicking on an answer sends you back to the exercise.

Grammatical Errors rating: 5

I found no grammatical errors.

Cultural Relevance rating: 5

The book is neither culturally insensitive nor offensive.

Comments

This is a great free resource to be used as a textbook for an introductory course in linear algebra, or as a complementary material or for individual study.

Reviewed by Abraham Smith, Assistant Professor, University of Wisconsin-Stout on 1/7/16

Comprehensiveness rating: 5 see less

This is a complete textbook for Linear Algebra I. It proceeds through the expected material on vector and matrix arithmetic on examples, then it makes a nice transition to abstract vector spaces and linear operators. The major theorems in linear algebra are all covered, with nice proofs and clear examples and good exercises.

Content Accuracy rating: 5

After using the textbook for three courses, I have found no significant errors in the book. Perhaps a minor typo or two over hundreds of pages.

Relevance/Longevity rating: 5

This is a good contemporary book on linear algebra. It would be appropriate for any sophomore-level linear algebra course for pure math, applied math, CS, or related fields. It includes some nice sections on computing that could lead naturally into a course on numerical methods.

Clarity rating: 5

The text is very clear. It follows modern notation, deviating only when it makes sense for clarity for the students. The proofs are nicely written, and the author does a good job of mixing exercises into the body of the proofs.

Consistency rating: 5

The book is of consistently high quality throughout.

Modularity rating: 4

Being a thorough mathematics textbook, overall modularity is limited by the logical nature of the subject matter, but the later chapters are definitley re-organizable. For example, I am someone who prefers to teach eigenvectors and Jordan form before I teach determinants, and this is easy to do with this book.

Organization/Structure/Flow rating: 5

The material is logical and clear, to contemporary mathematical standards. A good student could read chapter-to-chapter and learn the subject.

Interface rating: 5

The text is a straightforward PDF e-book. It is well-typeset and easy to read. The LaTeX for the book is available, and the author has made some nice commands to typeset certain types of objects (like augmented matrices) that can be useful when writing supplementary material.

Grammatical Errors rating: 5

Overall excellent and clear to both native and non-native speakers.

Cultural Relevance rating: 5

The text is pure mathematics with few examples. There is nothing insensitive or offensive in it.

Comments

I used this textbook for two years at Fordham University for Linear Algebra I and also as a supplement for the advanced Linear Algebra II course. It was an excellent resource for myself and for the students. The problems are very good, and the logical flow of the book is easy to follow. It is now my first choice for a Linear Algebra I book. (For Linear Algebra II, I prefer the more abstract approach of Axler's "Linear Algebra Done Right", but I still use this as a supplement in case students aren't comfortable with earlier material.)

Reviewed by Aaron Wangberg, Associate Professor, Winona State University on 6/10/15

Comprehensiveness rating: 4 see less

This text provides a fairly thorough treatment of topics for an introductory linear algebra course. It builds up the theory of linear algebra in order to answer important questions about they solutions and the types of solutions associated with systems of linear equations, and transitions to utilizing those techniques to further answer questions pertinent to vector spaces and maps between vector spaces. In this build-up, the focus is placed upon interpretation of results, concepts and theory. The text can be used before an intro-to-proofs course, and it provides many applications in the form of end-of-chapter Topics. The text is lighter in topics like matrix algebra, systems of equations over fields other than the real numbers, computational linear algebra, the geometric interpretation of vectors and linear transformations, and the analysis of data sets using linear algebra.

The first chapter focuses on solving systems of equations and understanding the types of solutions associated with various types of systems. The second chapter focuses on properties of vector spaces and uses the techniques of the first chapter to build the concepts of linear independence, dependence, and basis. In the the third chapter, which focuses on maps between vector spaces, the techniques from Chapter 1 are again utilized to understand properties of the maps through studying the vector subspaces impacted by the maps. Matrix multiplication and matrix inverses are finally presented as composition of maps and inverse maps. Chapter 5 and 5 then focus on techniques appropriate for square matrices. Chapter 4 focuses on determinants and includes a section on the geometry of determinants, while Chapter 5 covers eigenvalues and eigenvectors. Many of the techniques used to answer questions in Chapter 1 are thus refined and re-used in later chapters.

Content Accuracy rating: 5

I've used the text for four semesters, and have found the text to be accurate and error-free. Homework solutions are available, and most solutions utilize algebra or theory. Occasionally solutions could be simplified had they utilized geometric meaning. The text provides a non-standard definition of linear transformation and uses it consistently throughout the text.

Relevance/Longevity rating: 5

The content and end-of-chapter topics are up-to-date. Most of the topics will withstand the test of time, a possible exception being the inclusion of the Page Rank topic pertinent to internet search.

Clarity rating: 4

The writing is clear and supported by illustrations. The illustrations and explanations nicely explain and summarize content; I've sometimes needed to provide additional introduction or explanation of the illustrations for students. Chapter 3 is long (110 pages) contains a lot of material, many of it introduced just when it is needed. For this reason, it may be helpful to split out some of the Chapter 3 topics early as a short interlude before beginning the chapter, or to frequently remind students of the end goal as they progress through the chapter.

Consistency rating: 4

The text utilizes a consistent style for definitions, theorems, and examples. End-of-section problems can be linked to their solutions, which can be a nice feature or flaw. The author consistently provides end-of-section problems which utilize a set of systems of equations but changes the underlying question tuned to the particular section; This is also done with some examples throughout the text. This is a particularly nice feature of the text.

Modularity rating: 3

Each chapter is divided into sections and subsections which are manageable. Some sections and subsections can be skipped, and the author nicely suggests when this may be done without impacting the course. The text could be re-arranged with care, but this may heavily interfere with the careful buildup of Chapters 1, 2, and 3.

Organization/Structure/Flow rating: 4

The organization is very standard. One nice feature available with the .pdf files is that the homework problems in the text can be hyper linked to the solutions in the associated .pdf solution file.

Interface rating: 4

I've used the text as both a printed and a .pdf file; The interface works well in both situations, and does not prefer one format to the detriment of the other. The .pdf file contains enough navigation and hyperlinking to be helpful.

Grammatical Errors rating: 4

I found no grammatical errors.

Cultural Relevance rating: 3

I did not find the text to be culturally insensitive or offensive. It avoids examples using race and ethnicity.