Kunze or Axler for Linear Algebra Self-Study?

[benf.stokes]

Linear Algebra

Hi,

I'm just finishing my semester of linear algebra. And as my course is quite applied i feel like I'm missing a big part of linear algebra. And so for Christmas I pretend to order a more theoretical and challenging linear algebra book for self-study. My question is: should I buy Axler's Linear Algebra Done Right or Kunze's Linear Algebra?

Thanks

 

[fourier jr]

since it sound like you already know about matrices I would go with axler, although hoffman/kunze has more stuff in it. I think either would be good.

 

[Pinu7]

Shilov.

 

[Fredrik]

I haven't seen Hoffman & Kunze (or Shilov), but I really like Axler.

 

[Landau]

Hoffman & Kunze, or Steven Roman's Advanced Linear Algebra. Axler is ok, but it lacks content in my opinion.

 

[hitmeoff]

By first semester, do you mean Elementary LA, like the kind most people take in their second year of college, or do you mean your first semester of a junior level course?

If you mean the first, then can I suggest Friedberg? I have Axler and Shilov, and they are both certainly good books, but I also think Friedberg is the natural (and clearest) next step after elementary LA.

Once you have a good grip of LA at the level of Friedberg, I think you'll be able to get much much more out of Axler.

 

[Fredrik]

Hoffman & Kunze, or Steven Roman's Advanced Linear Algebra. Axler is ok, but it lacks content in my opinion.

I don't doubt you when you say that the other two have more content, but I would say that Axler contains everything that a theoretical/mathematical physicist needs to know about finite-dimensional vector spaces, and then some. I don't know what additional topics a mathematician might need to know.

By first semester, do you mean Elementary LA, like the kind most people take in their second year of college, or do you mean your first semester of a junior level course?

Seriously? Is that how it works in the US? We only had one linear algebra course, and it was during our first semester. (I'm not saying that this worked out well. In our third year, I think I might have been the only one who understood the relationship between linear operators and matrices).

 

[Landau]

@Fredrik: the first things that comes in my mind (that Axler does not talk about) is tensor products (multilinear algebra in general), modules, and orientation. Of course these things can be picked up from other sources or courses, e.g. algebra or manifolds, so it's not necessarily a bad thing. The topics Axler does cover are explained very well, except perhaps the Jordan Normal Form.

 

[George Jones]

@Fredrik: the first things that comes in my mind (that Axler does not talk about) is tensor products (multilinear algebra in general), modules, and orientation. Of course these things can be picked up from other sources or courses, e.g. algebra or manifolds, so it's not necessarily a bad thing. The topics Axler does cover are explained very well, except perhaps the Jordan Normal Form.

Landau, I think you're in Europe. The university system is much different in the U.S. At many U.S. (and Canadian) universities), none of the topics you listed are covered in a second linear algebra, and this is the market for which Axler wrote the book.

 

[hitmeoff]

Seriously? Is that how it works in the US? We only had one linear algebra course, and it was during our first semester. (I'm not saying that this worked out well. In our third year, I think I might have been the only one who understood the relationship between linear operators and matrices).

The first exposure a math/science/engineering student would have to LA is after a semester or two of calculus, and usually is in a combined course with elementary diff. eq. This course would cover topics like: Vectors, matrices, linear transformations, dot products, determinants, systems of linear equations, vector spaces, subspaces, dimension; But at an elementary level, not really heavy into theory. This level of LA is about the level of the Anton book.

For engineers and scientists, this will also be the last Linear Algebra class they take (except for math methods courses, which would basically be the Boas book). Math majors would go on to take an axiom/theorem/proof type of course on Linear algebra which would include: Vector spaces, linear independence, bases, dimension, linear transformations and their matrix representations, theory of determinants, canonical forms; inner products; similarity of matrices. And covers basically what would be covered in Friedberg or Lang's book.

 

[Landau]

Landau, I think you're in Europe.

That is correct :)

and this is the market for which Axler wrote the book.

Well, like I said: the things Axler does cover are explained quite well. OP is not referring to some particular linear algebra course, so I'm just saying that there is more to linear algebra than what Axler talks about. Obviously this is true for every single book or topic, but if I were to buy just one book on linear algebra, it would be a comprehensive one.

 

[Fredrik]

@hitmeoff: Thanks for explaining. We used Anton (6th ed) for that one and only course we had. We started out multiplying matrices and solving systems of linear equations, but the topics you mention as a part of a course that emphasizes definitions, theorems and proofs, were eventually covered, when we got to those parts of Anton's book. But they weren't emphasized as important. The exam we got was so easy that you could actually pass it without knowing what a linear operator was.

When I studied my second course in quantum mechanics about two years later, I felt that I had to get better at linear algebra, and I used Anton again, to refresh my memory and learn stuff I had never fully understood before, like the relationship between linear operators and matrices. I found that particular detail to be the most useful of all the things I studied.

Many years later, I needed to refresh my memory again, and this time I used Axler. His approach made so much more sense to me. Linear algebra is the mathematics of linear functions between finite-dimensional vector spaces. Axler's book starts by saying precisely that, while Anton doesn't define "linear transformation" until page 296!

 

[mathwonk]

Axler is excellent as a second or refresher course. But as a comprehensive reference Hoffman Kunze is much better, because it contains more topics, and more detail. Axler is a set of notes. Hoffman Kunze is a textbook.

I also have three or four free books covering linear algebra on my web site, ranging from a 15 page outline to a 400 page comprehensive algebra book.
(primer of linear algebra, notes for math 4050, math 8000[6], math 843-4-5, and the linear algebra is in 845.)

By the way, saying a linear algebra book does a good job on everything except jordan form, is like saying a calculus book does a good job on everything except integration, i.e. everything except the most important topic.

Shilov is both excellent and extremely cheap. Hoffman - Kunze is usually very expensive. If you like short authoritative books, Halmos' finite dimensional vector spaces has been a classic for decades.

 

[Landau]

The topics Axler does cover are explained very well, except perhaps the Jordan Normal Form.

By the way, saying a linear algebra book does a good job on everything except jordan form, is like saying a calculus book does a good job on everything except integration, i.e. everything except the most important topic.

I disagree. The Jordan form is important, but mainly of theoretical interest, sort of a definitive answer to the question what the 'best' form of a matrix is. Of course it is nice to have, and e.g. in differential equation (computing the exponential) it is very handy. However, computing the Jordan form by hand is something you don't want to do.

I don't see how it is "the most important topic" of linear algebra. Certainly not comparable to
'integration' in calculus. Personally, I see it the Jordan form as a corollary to the structure theorem for finitely generated modules over a PID.

 

[mathwonk]

I'm not sure I understand you. Jordan form tells you what all linear operators look like (over the complex numbers, or the algebraic closure of any field). What could be more important than that?

And it seems to me you almost contradict yourself when you say it is not important because it is a corollary of the basic and most important theorem of module theory over pid's.

I am also puzzled when you seem to dismiss topics of theoretical interest, since you like Axler, which is almost exclusively a theoretical book.

Maybe I'll understand better if you tell me something in linear algebra you think more important than Jordan form.

Maybe you prefer rational canonical form? (also a corollary of fund thm of modules over pid's).

Oh, I know. We don't have a definition of "important". So we are no doubt using different ones.

 

[hitmeoff]

Seems like I am the only one who likes the Friedberg text. Anyone want to tell me why Friedberg isn't even considered a good L.A. book, or rather why the others are better.

I usually tend to rate the clearer books better than those that are not as clear, but maybe have more depth; at least as far as a first exposure to the subject.

 

[mathwonk]

if you mean friedberg, insel, and spence, i agree it is an excellent book. I taiught from it recently, but was also motivated to write my own notes since that book completely avoided what I thought was the main idea underlying structure of linear operators, namely polynomial algebra.

In particular it is more elementary than the other sources listed here, with more examples and easy exercises. It is a good first place to learn the subject. you just won't learn as much nor as deeply.