OER Reviewed: Linear Algebra with Applications by W. Keith Nicholson

Reviewer: James Bailey (No current affiliation)

Rating

Each criterion asks the reviewer to rate it on a scale of 1 to 5 (1 = very poor and 5 = excellent).

Comprehensiveness – Rating: 5

The OER covers all areas and ideas of the subject appropriately and provides an effective index and/or glossary.

This OER contains all the topics which are normally covered in a first course in Linear Algebra over two semesters. There is no Glossary but there is a full Index. There is a list of applications on page xv.

This text is used at University of Calgary and on page ix there are suggested course outlines for two semesters. All science and engineering students take their first linear algebra course in the first semester. There is a flow chart on page x showing chapter dependencies.

In addition, there is algorithmically generated online formative assessment. Values are randomly generated each time the question is run, and are chosen to allow easy computation by hand. There are also open ancillary resources: a Partial Student Solution Manual, and Slides and a Question Bank for instructors.

Content Accuracy – Rating: 5

Content, including diagrams and other supplementary material, is accurate, error-free, and unbiased.

This OER is exceptionally carefully written. I spotted a few typos but it was always clear what was intended. All the diagrams are carefully drawn, informative, and attractive.

Relevance/Longevity – Rating: 5

Content is up-to-date, but not in a way that will quickly make the OER obsolete within a short period of time.  The OER is written and/or arranged in such a way that necessary updates will be relatively easy and straightforward to implement.

It would be easy to insert additional topics at appropriate points in the text. For example, I would like to see more physics applications, so I would add a section on rotational motion and angular momentum after the section on the cross product. I believe that this would be easy to do.

The source code (LATEX files) is available and the editorial staff at Lyryx will work with instructors to customise the material and help manage multiple sections.

Clarity – Rating: 5

The OER is written in lucid, accessible prose, and provides adequate context for any jargon/technical terminology used.

Many definitions are in boldface in the body of the text. More important Definitions, Examples, and Theorems are set in colour coded boxes which have a brief description in a strip of a darker colour at the top of the box. Often, after a statement is made using symbols or technical terms it will be repeated, but in “plain English.”

Also, there is a clickable link to the appropriate section when a reference is made to a technical term/definition/theorem which hadn’t been used for some time. This text is self contained and could be used for self study.

Consistency – Rating: 5

The OER is internally consistent in terms of terminology and framework.

I spotted no inconsistencies.

Modularity – Rating: 5

The OER is easily and readily divisible into smaller reading sections that can be assigned at different points within the course (i.e., enormous blocks of text without subheadings should be avoided).  The OER should not be overly self-referential, and should be easily reorganized, and realigned with various subunits of a course without presenting much disruption to the reader.

Subsections cover a single topic and are no longer than they need be (less than 15 pages). The Preface contains a section, Suggested Course Outlines starting at page ix, which indicates how the text is used to teach Linear Algebra over two semesters at University of Calgary (Engineering and Physics students take their linear algebra course in the first semester).

There is a flow chart on page x which shows chapter dependencies and would be useful when omitting or rearranging material.

Organization/Structure/Flow – Rating: 5

The topics in the OER are presented in a logical, clear fashion.

It is always clear how the present material relates to and depends on what came before it. Often there are statements such as:

This is a “bridging” chapter, easing the transition to abstract spaces. Concern about duplication with Chapter 6 is mitigated by the fact that this is the most difficult part of the course and many students welcome a repeat discussion of concepts…and;

The novel thing [in this chapter] is the abstraction. Getting used to this new conceptual level is facilitated by the work done in Chapter 5…

Interface – Rating: 5

The OER is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader.

I found no interface problems. The clickable links which I tried in the Index and between sections worked as I expected. Everything in the text (layout, typesetting, images, etc.) is of very high quality.

Grammatical/Spelling Errors – Rating: 5

The OER contains no grammatical or spelling errors.

I did not find any grammatical errors but I found a number of typographicalerrors (it was always clear what was intended) which I will submit to Lyryx for correction.

Diversity and Inclusion – Rating: 5

The OER reflects diversity and inclusion regarding culture, gender, ethnicity, national origin, age, disability, sexual orientation, education, religion. It does not include insensitive or offensive language in these areas.

There is an attempt, not always very successful, to be gender neutral. For example:

• “he or she” is used on page xi to refer to the instructor. It is also used on page 142;
• page 9, Exercise 1.1.19 “Workmen John and Joe” seems to confirm default assumptions about gender roles; androgynous names such as Alexis, Ashley, Brooke, Braxton, Clare, Casey, etc. could have been used.
• At times the author appears to try to upset default assumptions by saying things like “The prime minister says she…”

Recommendation

1. Do you recommend this resource for the specific course taught in the first-year engineering common curriculum (in place of a commercially available resource)?
   Yes.
2. If yes, please briefly summarize the reasons for recommending this resource
   

   This OER covers all the topics which are usually taught in a first course in linear algebra. This textbook includes online formative assessment which is algorithmically generated and can be used as homework and testing. Students receive immediate personalised feedback; grade reports and performance statistics are also provided.

   There is a licence fee of $40, but students need not pay when they work from computer labs located on campus. Lyryx promptly responds to both student and instructor inquiries. All content is wholly adaptable: the original LATEX source files are open to anyone. The editorial staff at Lyryx will work with instructors who adopt Lyryx assessment to produce custom editions and provide a comprehensive system which is ustomised for their course, and they will help manage multiple sections. It also has excellent ancillary material: a Partial Student Solution Manual, Slides, and a Question Bank.

3. If not, why? What improvement, if any, could be made?

   There is a conflict between the objectives of mathematics departments and engineering faculties. Historically, Linear Algebra is the first rigorous mathematics course (starting from axioms and including all proofs) which a student takes and is where aspiring mathematicians learn their trade. Engineering students don’t necessarily need to know how results are proved, but need to know how they are used. If I were teaching a course to engineering students based on this text, I would start with Appendices A–D (which contain useful background), then concentrate on Chapters 1–5 (which are less abstract) but omit all the proofs and emphasize examples which illustrate what the theorems etc. mean and how the proofs work. I would also include the later sections which contain useful applications.

4. What gaps in content have you identified?
   

   I would have like to see more applications from physics. There are some already: for example, Section 6.6.6 looks at linear ordinary differential equations with constant coefficients and gets as far as simple harmonic motion. But I would have liked to see rotational motion, torque, and angular momentum as applications of the cross product and the moment of inertia tensor as an application of spectral theory. There is no need for students to do many calculations of determinants, inverses, eigenvalues, eigenvectors, etc. by hand, so they should be able to use a computer algebra system such as Maple or MATLAB.

   I have reviewed two linear algebra OERs:

   Linear Algebra with Applications 2021A by W. Keith Nicholson

   and

   A First Course in Linear Algebra 2021A by K. Kuttler

   I am inclined to favour the text by Nicholson because it has more applications (there is a list on page xv). Remarkably, there is very little overlap in the applications between the two texts.