Reviewed by Iuliana Oprea, Associate Professor, Colorado State University on 1/8/16

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.

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 1/8/16

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.

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 6/11/15

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.

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.

Comments