# Linear Algebra Textbooks

1. May 9, 2013

### NATURE.M

So I wanted to begin studying linear algebra prior to the start of my first academic year at university. One textbook I want to use is "Linear Algebra Done Right" by sheldon axler. Any feedback on this text would be nice.

2. May 9, 2013

### micromass

Staff Emeritus
Do you have any experience with linear algebra already? If not, I recommend not to use Axler. Axler's text is wonderful, but only as a second text on the subject. Additionally, he doesn't use determinants untill the last chapters. Determinants are very important though, so you need experience with them.

So, if you don't already know linear algebra like matrices, finding eigenvalues of matrices, diagonalization, etc., then you shouldn't use Axler.

3. May 9, 2013

### Fredrik

Staff Emeritus
It's great. It defines linear transformations as early as possible, and walks a pretty narrow path through the theory of linear algebra. So it's not as complete as some other books. In particular, it doesn't include a lot of the stuff that's covered at the start of a course that focuses on matrix multiplication, solving linear equations, applications to geometry etc. For this reason, some consider it a bad choice as a first book. But it depends on whether you want/need to study those things in depth or not.

One thing that's a bit weird about it is that it avoids using determinants most of the time.

4. May 9, 2013

### NATURE.M

Okay thanks for the advice, and I ask mainly because its a recommended text for my course.

My background is fairly minimal when it comes to matrices, vector spaces, etc. (most of it is based on the Leonard susskind lectures on QM).
So before I begin the text, I should most likely obtain a solid understanding of linear equations, matrices and determinants, right?

5. May 9, 2013

### Fredrik

Staff Emeritus
It would certainly help, yes. But I wouldn't say that it's necessary. Axler doesn't leave anything undefined. Also, I think Axler's selection of topics is great for people who want to learn linear algebra to use it in QM.

6. May 9, 2013

### micromass

Staff Emeritus
I think you'll struggle a bit with Axler. This isn't a bad thing however. If you're struggling then the course is too easy.

But I do recommend getting a text other than Axler that is easier. For example, Lang's "introduction to linear algebra" (and not "linear algebra") is a good text that is not too hard. It covers matrix manipulations in more detail than Axler, so it's worth to go through it.

So if you learn from both Lang and Axler, then you should be fine.

7. May 9, 2013

### NATURE.M

Okay thanks, and does the Linear Algebra Done Right textbook serve as a more theoretical approach to Linear Algebra, in contrast to many other texts. (Because that's what I'm aiming for).

8. May 9, 2013

### micromass

Staff Emeritus
Yes, that's for sure. The book is very theoretical. If you want a theoretical text, then Axler is one of the best choices you can make. Other choice are Roman (this is very advanced though, so I don't recommend it), Hoffman & Kunze, Shilov and Lang (both books).

9. May 9, 2013

### NATURE.M

Okay thanks, I wanted to confirm that. And i'll look into your other recommendations.

10. May 9, 2013

### mathwonk

I like this book by Sergei Treil better than Axler's book. And it's free.

Actually I agree Axler's book is well written, but I think I agree with micromass that it may not be that helpful. Some books were written to prove a point, and others are actually useful to learn from. It is more important to get a feel for a topic at first rather than see slick proofs of the main theorems.

http://www.math.uga.edu/~roy/rev.lin.alg.pdf

Here is a book i wrote as an exercise over xmas break a while back, making the whole subject essentially a set of exercises for the reader. It is also free.

http://www.math.uga.edu/~roy/rev.lin.alg.pdf

(The same criticisms of Axler probably apply to mine too.)

I second the recommendation of Shilov, as both more thorough and cheaper than Axler.

https://www.amazon.com/Linear-Algeb...8157073&sr=1-1&keywords=shilov+linear+algebra

There is no shame in starting on an easier book, also free, to see if it is helpful.

http://joshua.smcvt.edu/linearalgebra/

Here is another nice gentle introduction by a Stanford professor:

http://www.abebooks.com/servlet/SearchResults?an=paul+shields&sts=t

If you like to challenge yourself you might try to read and provide the proofs in my 15 page book above, and when stuck on a page, read the 20-30 pages of one of the more standard books where the same result is discussed. I made up many of my proofs though, so they probably won't resemble the ones in other books.

Lang is very clear on the main ideas but he essentially never gives enough examples in his books for the reader to really master the subject well, so books by Lang always need supplementation.

Last edited by a moderator: May 6, 2017
11. May 10, 2013

### dustbin

I like Halmos' "Finite Dimensional Vector Spaces." However, this also should not be used as a first text. Hoffman & Kunze is also very nice in this respect. The earlier editions can be had for pretty cheap. I don't really like Axler; his exposition annoyed me and I thought the majority of the problems were too easy.

For first texts, you might try:

Anton - "Elementary Linear Algebra"
Strang - "Linear Algebra and its Applications"

12. May 10, 2013

### Fredrik

Staff Emeritus
Anton is exceptionally well written, but I don't like the order of the subjects. Somehow he manages to postpone the introduction of linear transformations for 300 pages(!), and in the 6th edition at least (the only one I'm familiar with), complex vector spaces aren't introduced until the very end. As if they are either "advanced" topics, or not very important. They are of course extremely important and can be introduced very early in the book if the author chooses to.