Is Linear Algebra Done Right too much for a beginner?

I have asked in Mathematics stackexchange, but I think asking here is more appropriate.

I am self studying the book Linear Algebra Done Right by Axler. That's how I started using the great Stack Exchange in Mathematics!

I do not have a background in mathematics and just finished my undergraduate and want to learn Maths.

I try to read every page and finish all exercises (now on Chapter 3: Linear Maps). And on average, it takes me an hour to finish a page (exercise included). I can finish 1/3, if not 1/4 of the questions on my own. For the rest, I need to look at the solutions to get some hints. Sometimes, I don't even understand the solutions...

Just want to know if I am too slow on average? Are the questions hard for beginners on average (on average, so not for IQ above 160 type)? (I will keep going and finish the book even if you tell me the questions are super easy.)

• I find textbooks often harder than a proper course. I would highly recommend Linear Algebra - Foundations to Frontiers - a free course on Edx from University of Texas Austin which is highly regarded in maths. It teaches standard LA as well as MATLAB which they use to deepen understanding of LA. Commented Mar 12, 2019 at 21:53
• In the preface, Axler writes that reading a page an hour is the right amount of pace (anything faster is too fast) Commented Apr 10, 2021 at 20:55

Unguided self-study of mathematics is difficult, and harder for someone with little experience at it. It is normal to take time to advance. One should think in terms of months not hours. A typical one semester class in linear algebra does not cover every detail in the typical linear algebra textbook, nor require a student to work every exercise in that book. When studying by oneself it is important to do exercises carefully and to be sure one is doing them right, and to really make sure that one is understanding B before progressing to studying C (for example, in general one should understand well Gaussian elimination and its consequences for the analysis of systems of linear equations before studying abstract linear maps).

If you are struggling, find someone knowledgeable to answer your questions and give you some guidance. There are many paths that lead to understanding linear algebra, but when one is lost it can be hard to find any of them.

The book by Axler has an unfortunately tendentious title and a strange (and in some respects questionable) fixation on determinants. I like Sergei Treil's freely available Linear algebra done wrong better. There are many books, and it is useful to look at several. What one person likes and can learn from is terrible for another person. You might learn well from Axler's book. To some extent the election depends on taste, to some extent on one's goals. Gilbert Strang's introductory textbook is one that seems to me that can work well for self study (although, maybe since my own inclinations are those of an algebraist, I find it too wordy and talkative for me to have learned from it). Strang's book works from the concrete to the abstract, showing a good pedagogical sense and emphasizing techniques are useful in practice. It is written for someone without much experience with more abstract mathematics (e.g. an engineering student) and such an approach has virtues for someone learning alone. It might also be of interest to look at some numerical linear algebra textbooks - the ones by Trefethen/Bau and Demmel are my favorites - as these communicate a completely different point of view than the standard textbooks.

• My experience of the 'done right' in the title is that he changes the definitions to make what he wants easier, and then says it's great to do it his way because it's easier. But the reason it's easier is he hasn't actually achieved the same thing. Commented Feb 3, 2019 at 8:35
• Isn't the whole idea of Linear Algebra Done Right to actually focus less on determinants, not more? Your comment "The book by Axler has an unfortunately tendentious title and a strange (and in some respects questionable) fixation on determinants." seems to go opposite to the little I know about the book. Can you expand a little on that? Thanks for your answer! Commented Feb 4, 2019 at 22:22
• @Chris Cunningham The book has a "fixation" on determinants in the negative sense. I agree fixation is not the best term, but the writing of the book was definitely influenced by issues around the determinant, so "fixation" is sort of understandable.
– KCd
Commented Feb 9, 2019 at 19:38
• If Strang's linear algebra book is as bad as his calculus book, I would stay away from it. Commented Feb 26, 2019 at 22:17
• Strang's Linear Algebra book is different in that sometimes geometric proofs are given where algebraic ones are traditional in a linear algebra course. This is both a pro and a con depending on how much you care for analytic geometry and its interaction with linear algebra. Also, I think there is more than one of these books, so I should be clear I'm refering to an earlier version. So take my comment with that qualification (we should remember edition drift is real) Commented Jun 16, 2019 at 12:56

I know your struggle well. I enrolled at a distance learning university and the Mathematics is just very painful and slow to grab. I also struggle with every proof and have to force myself to think it through. What helps is to have someone talking me through or discussing certain proofs or concepts, unfortunately I only have a mailing list which makes questions tedious. As DanFox mentioned Gilbert Strang is a good teacher and I really like his lectures (they are on youtube: MIT open courseware Lectures in linear algebra) those really helped me to get the concepts and understand my lecture much better. His look on the concepts was so different than what I had read. In this way I at least had someone explaining the things written in the book with a few more hints. Maybe that would also be of help for you.

But in general I think if you seem to understand whats written on a page your progress is fine, I would not measure it by the time you take. There are just people who get it faster than others. Speeding up for the sake of being better or comparable to others just is unnecessary struggle. But you will be getting faster as you progress. Its like learning a language. The first progress is very slow...you need vocabulary and grammar and need to put it together and after you know the basic sentence structures you get faster.

Linear Algebra Done Right is intended as a second encounter (US curriculum) with linear algebra (it says so in the introduction), and some of the exercises are a bit tricky. If you don't have a background in math, then it's perfectly normal to take what feels like a very long time for a single page.

It seems like you have already progressed a fair bit into the book. If you like its style, then I would recommend to just keep going until you get stuck, and then look at other material. You also don't need to solve every single exercise. You can also try to find exercises in other books or online. Many lecturers upload problem sheets for their courses online, and you can download them freely.

I would also recommend trying to do "reality checks" every now and then. Try to come up with some simple, maybe even silly examples. Compute some stuff. Doodle a bit. It will give a better feeling for what you have understood from a section of a book.

Besides the books already mentioned in the other answers, I would like to recommend taking a look at the following books:

Linear Algebra Problem Book by Paul Halmos: this is written in an accessible, conversational style. It teaches linear algebra through short explanations and lots of exercises. All exercises have hints and solutions at the end of the book. This book is great for self-studying.

Linear Algebra by Klaus Jänich. This is a solid introductory textbook and covers the content of a typical first semester course in linear algebra at a European university. It has regular exercises and a few quick tests interspersed within the text. There are also short sections introducing topics from multiple point of view.

Get an easier book and one that emphasizes calculations more than proofs. One with lots of "drill" exercises and the answers to them. Then do the drill. All of it. Struggling with too hard a book is not time efficient self study. If you can't do the exercises there is a problem. (Do a linear programming optimization on that!)

In general, you should look for "programmed instruction" texts when self studying. If you can't find that look for the closest to it. Don't listen to the Internet's Rudin lovers pushing you to difficult, even pedagogically flawed, texts. If you learn an easy text 4.0, you will be way better off than learning a hard text 3.0 (or 2.0 as sounds like you are doing now).

• For future readers, I strongly disagree with this recommendation. Getting drilled will get you skills you can't apply to anything but this drill exercises, and will not give even a glimpse of understanding. The understanding of linear algebra comes at fundamentally different level than working out numerical examples. Commented May 15, 2020 at 8:07

This book is better appreciated after a first exposure to the subject. However, one can also start reading this book. It is beautifully written.

As far as determinants are concerned, there are varying opinions on determinants: Consider this, if we have a linear transformation, then how do we define its determinant? We have to take help of a basis, fix coordinates to get a matrix representation and then define determinant of linear transformation using its matrix representation (after proving that it is well-defined!). For understanding his view point, one must read his article "Down with determinants"

This book presents results separately depending on whether vector space is over Real numbers or Complex numbers. I liked this aspect of the book. Most of the other books assume that vector space is over field of Complex numbers, and we are in dark as far as real vector spaces are concerned.

The new edition (3rd) also has exercises placed at the end of each section.