OER Reviewed: A First Course in Linear Algebra by K. Kutler

Reviewer: James Bailey (No current affiliation)

Some components of the OER have been used by the reviewer for teaching.

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 text presents an introduction to linear algebra appropriate for students who have an understanding of basic algebra, although students who have not taken calculus might have a little trouble with some of the mathematese (summation notation, [latex]\sum^n_{j=1}a_{ij}x_j[/latex] for example, which is discussed in Appendix A.2).

There is no glossary, but the index is excellent.

All the major topics of elementary linear algebra are presented in detail along with proofs of most theorems.

There are a variety of applications, although I would have liked to see more applications from physics. Applications which are found in the text are:

• velocity and speed (as an example of vectors and their lengths);
• force and work (an application of the dot product);
• balancing chemical reactions;
• dimensionless variables;
• Kirchhoff’s Law (only for resistor networks, no capacitance or inductance)
• Fourier expansions;
• Markov chains; and
• discrete dynamical systems.

There is a good balance between routine and challenging exercises at the end of each chapter. There are solutions to selected exercises at the end of the text.

Content Accuracy – Rating: 5

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

All content in this book is accurate and is consistent with other elementary linear algebra textbooks. Diagrams are carefully and clearly drawn. My students and I have worked with the online formative assessment and we have not found any errors. I have not used any of the additional instructor resources: slide sets and lecture notes, solutions manuals, or multiple choice question banks with an exam building tool, so I cannot comment on these but I expect that they are up to the high standard of excellence found in other Lyryx products.

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.

Linear algebra is not likely to change any time soon, but if there are special needs, then Lyryx will work with instructors who adopt their online homework to adapt the textbook and customise the supplements.

Clarity – Rating: 5

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

This text assumes that the reader is fluent in English and is comfortable with high school algebra and perhaps a little geometry. It starts with familiar, easilyunderstood ideas and builds incr ementally to more complex concepts. Well chosen examples and diagrams reinforce ideas and provide guidance on how to approach problems.

This text uses colour-coded boxes to signal to readers that the box contains something which is important and they should pay attention. The colour indicates the nature of the content in the box: a stripe of darker colour at the top of the box contains a brief description of what is in the box and there is a lighter background for the text which is in the box. The function of the boxes and the colours which are used are as follows:

• Outcomes (yellow): each section begins with a list of outcomes which a student should be able to achieve upon completing the chapter;
• Definitions (blue): these are terms which a student must be fluent with in order to understand the rest of the material; definitions are motivated by previous material and elaborated on shortly after. Not all definitions come in boxes; “small” definitions (such as for skew lines, back substitution, and parameter) are in italics or boldface in the body of the text. There is a nice balance between definitions which are in the body of the text and definitions which are in blue boxes; it would have been overwhelming if the boxes had been overused!
• Examples (green): illustrate concepts and present the sorts of questions which a student might see in homework and on tests;
• Lemmas, Propositions, and Theorems (red): contain more theoretical material.

Consistency – Rating: 5

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

This textbook is consistent in its use of terminology, notation, fonts, colours, and numbering of chapters, sections, equations, material in boxes (see above), and exercises.

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.

Most sections are no more than five pages in length, although a few (Gaussian Elimination for example), are longer. Because new material relies on what came before it, deletions and rearrangements would have to be done very carefully; proofs can be omitted with impunity Some examples and exercises are designed to illustrate key aspects of a proof, and would be enough for the student to get a feel for how a proof (and the theory behind it) works. If an instructor adopts the online formative assessment, then Lyryx will help to adapt the text to meet the instructor’s needs.

Organization/Structure/Flow – Rating: 5

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

The presentation is well organised and logical. Soon after starting to read this textbook, readers will realise that there are enough clues that they can often anticipate what is coming.

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.

There were no interface problems: all links worked as expected, diagrams were simply and cleanly drawn, and colours were used when they were appropriate. When I have this book open in Sumatra PDF there is a navigation pane which makes it exceptionally easy to move between different sections of the book. There was no navigation pane when I had it open as a Microsoft Edge PDF Document.

Grammatical/Spelling Errors – Rating: 5

The OER contains no grammatical or spelling errors.

There were a number of typographical errors, but the context always made it possible to figure out what was intended. I made a list of the ones which I found and submitted it to Lyryx. Their founder and president, Claude Laflamme, replied within a week, which is remarkable considering the stress which COVID-19 has put on everyone, and inquiries from students and instructors would have been at the top of their priorities list.

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.

All pronouns are gender neutral, and there are no references which might single out a specific person or group.

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 textbook comes with online formative assessment which is algorithmically generated and can be used for homework and testing. Students receive immediate personalised feedback; grade reports and performance statistics are also available. 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. The original source files are open to anyone. Lyryx will produce custom editions for those adopting Lyryx assessment and will work with individual instructors to provide a comprehensive system which is customised for their course. They will adapt the text and help manage multiple sections.

   There is also ancillary material, Slides and a Question Bank, although I have not used them.

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

   I feel that there is a rather heavy emphasis on theory. This book is very appropriate for students who will be continuing in Mathematics and Physics (matrix mechanics in particular), but it seems to me that this theoretical approach is not the best for beginning engineers. I have tried to identify a point in the text which divides material which is suitable for engineering students from material which is unnecessarily theoretical, and there isn’t one. For a course which is appropriate for engineering students I would probably omit Chapter 9 and all proofs, although this probably still leaves too much material for a one semester course.

   Caveat: mathematicians and physicists define spherical coordinates differently (they interchange the meanings of θ and φ , which results in no end of confusion).

4. What gaps in content have you identified?
   

   I would have liked to see more applications to physics, in particular the cross product (applied to rotational motion, torque, angular momentum, the moment of inertial tensor, for example).

   There is no need for a student to do many calculations of determinants, inverses, eigenvalues, eigenvectors, etc. by hand, so it is desirable that they be able to use a computer algebra system such as Maple or MATLAB.