Linear Algebra Textbook Reviews

Linear algebra is often seen as both hard and boring. It does not need to be this way! I came to love this subject, AND find it highly intuitive, once I found the right way to think about and learn it. I recommend checking out more than one book.

Different students learn in different ways and are interested in different sorts of material, so by exposing yourself to more than one book, you will both improve your ability to learn and love linear algebra, but also to teach or explain it to others who may have a different set of interests and a different learning style from your own.

Be patient with each book, and read each one slowly and repeatedly. For many students, linear algebra is their first experience with a more abstract sort of math. The fact that linear algebra involves linear equations and does not involve any calculus can be deceptive. Most students find linear algebra tougher than Calc 3. If you're a student, give yourself more time than you're used to to work through sections and problems that look straightforward. If you're a teacher or professor, try to remember how hard this subject can be when you encounter it for the first time, and encourage your students to slow down and understand the basics thoroughly.

A clear introduction to abstract Linear Algebra. Easy enough for most advanced undergrads to use, but thorough enough to be useful for grad students as well. This book is distinguished from most other textbooks in that it does not introduce determinants as part of its main development. Determinants are explained at the end in a late chapter which functions more like an appendix than anything else.

Pros: Exceptionally clear exposition; prose and proofs are easy to read and understand. Book covers quite a lot of material quickly. One of the best books for self-study.

Cons: Presents only one way of developing the subject, and is very abstract; lacks emphasis on computation which may be important for people who have a need for working computational knowledge and intuition more than abstract understanding. The name seems a bit arrogant (implicitly putting down other approaches), and possibly could discourage students from exploring other approaches, so I think this text is best paired with other books which take other approaches.

Complemented by: Shilov's book below, my favorite text using a determinant-based approach.

This, together with Axler's Linear Algebra Done Right, makes up my favorite pair of intro texts in this subject. This is an old-fashioned book, and it uses the determinant-based approach that Axler shuns.

Pros: Dover edition bringing back an old book into print, is dirt cheap, making this one of the most affordable linear algebra texts out there. Quirky presentation helps readers to develop an interesting sort of intuition that is absent from most books on this subject.

Cons: This is an old-fashioned book and the presentation may be less natural and require a little additional understanding for students who mostly have experience with more modern abstract math books. This book makes heavy use of summation notation and writing out series of sets of equations, and uses less of the concise, abstract notation.

Abstract mathematics textbook authors are notorious for under-stating the difficulty level of their books and the extent of the prerequisites for understanding them. This book is an exception. Don't let the name fool you: this book is not as dense or tough to understand as you might expect.

Pros: Although this book begins with rather advanced topics and moves quite deep into the subject, it also moves at a leisurely pace and provides ample explanation. Provides a unique presentation of modules, which explores the analogies between modules and vector spaces more than any other text which I've used. Revisits difficult topics so as to help the reader to develop a thorough understanding of them.

Cons: Requires prior knowledge in linear algebra and familiarity with basic concepts from a typical introductory course in linear algebra; probably not accessible to students encountering the subject for the first time.

This tiny booklet, available for just a few dollars in Dover Edition, is a real gem. It takes patience and persistence to understand fully.

Pros: One of my professors remarked that this is one of those books where every time you read it, you discover some new insight. I completely agree! I think this book is especially valuable for people who already know this subject to a degree, and who want to understand it more deeply.

Cons: This book assumes prior background in linear algebra, and dives in without much explanation. At the same time, it begins by covering the topic of bases extensively, in a way which may bore many students who do not see the benefit or uses of this approach. The notation is at times cumbersome and messy, involving long strings of equations with an arbitrary number of terms.

If I had to point to a single book whose title makes it sounds incredibly dry and boring, but which is among the most captivating and accessible math books I've ever read, it would be Matrix Computations. This is a book on numerical linear algebra, the branch of linear algebra concerned with how to get computers to do linear algebra computations effectively and efficiently.

Pros: Very thorough. This book goes quite deep into the subject, but more importantly, it thoroughly explains the fundamentals, starting with very basic things like how a computer multiplies a vector by a matrix or a matrix by a matrix. Some of these basic steps are skipped or glossed over in many intro texts to the subject, and I find that by going slow at the beginning, this book allows people to work more quickly through the subject later on.

Cons: The book's name, cover, and presentation may scare some students. It's really fascinating, but you have to actually think about it before it becomes fun. This book, as wonderful and accessible as it is and, still does require some self-motivation to get into.

This book is nowhere near as thorough as Matrix Computations, reviewed above. I personally prefer the Golub and Van Loan book. But this book might be superior for students who are not quite as self-motivated as me, especially if the book is being used as a textbook for a course rather than a heavy-duty reference or book for self-study.

I find this book covers less material and leaves more holes, making it less suitable for self-study. It requires a teacher to explain some of the concepts if the students are to come out with a thorough understanding of the material. But the material that is covered is pretty clear, and I would still give this book a clear recommendation.

Whether you're looking to save money, or you like the idea of free, open-source (copyleft) textbooks, or just want to check out some additional supplemental texts, free online textbooks are a great option, and there are two very good Linear Algebra ones that I would point to:

• Linear Algebra by Jim Hefferon, Saint Michael's College. This book is also published in a very affordable paperback edition, but it is available for free (with copyleft GNU Free Documentation Licensing) online.
• A First Course in Linear Algebra by Robert Beezer. This is an introductory linear algebra textbook, also available for free download, and also available under the GNU Free Documentation Licence.