Linear Algebra Which Linear Algebra Book is Best for Self-Study: A Comparison of Lang and Axler

[blue_leaf77]

I decide to self-study linear algebra. I have heard words about some good books on this subject such as Sheldon Axler's. Unfortunately his book is only loanable for 4 days in my university library. There is this book from S. Lang that I can borrow for one month, so what do you think about this book? If you have your own recommendation please feel free to mention this out. Books which are more aligned to its use in physics such as QM are very welcome.

 

[micromass]

Please tell us a bit more about your current knowledge of LA. For example, what do you know about matrices, systems of linear equations, dot products, coordinate geometry?

In any case, my favorite book is a free one: http://www.math.brown.edu/~treil/papers/LADW/LADW.html But it is nothing for somebody who truly knows nothing about LA.

Lang is very good, but depending on your knowledge, you might be better off with the more easy-going "Introduction of linear algebra" by Lang, as opposed to his "Linear Algebra".

 

[blue_leaf77]

I'm currently in a graduate level of physics, so I have already had those basics of matrix operations (multiplication, inverse), eigenvalues problem, etc. Now I'm more into understanding of vector spaces, because it will greatly help me with my graduate QM.

 

[micromass]

OK, then I think Treil's algebra done wrong book would really be a perfect match for you. It really does contain everything of linear algebra that you need for QM and relativity. But Lang's Linear Algebra book should suit you well too, if you prefer that, although Treil is better :)

 

[blue_leaf77]

Just checked in the library website, seems like Lang's introduction to linear algebra is also available for short loan period. I guess I will go for his Linear Algebra and resort to Treil's free book when I get stuck in a certain topic. If only Treil's book is available in print version I will go for it instead, you are right his book seems to be more suited to physics student, but I don't like that I always have to turn my computer on first in order to start reading.

 

[micromass]

You could always print the book :D But yeah, I see your point.

Anyway, Lang's "introduction to linear algebra" will not be useful for you, go for his "Linear Algebra" instead.

 

[blue_leaf77]

Ok anyway thanks for all your recommendations.

 

[Fredrik]

Axler's book "Linear algebra done right" is also very good for physics students. micromass doesn't like it, I think mainly because it avoids determinants, but I think the selection of topics is very nice for quantum mechanics. You could supplement your Axler by reading the excellent chapter on determinants in Treil. (I'm sure you can study one easy chapter on a screen).

I'm not familiar with Lang's book. I think almost any linear algebra book will do, but I would avoid books like Anton, which focus too much on real vector spaces and delay the introduction of linear operators far too long.

 

[micromass]

Axler's book "Linear algebra done right" is also very good for physics students. micromass doesn't like it, I think mainly because it avoids talking about determinants

Yes, I recommend to avoid it for a first exposure to the topic. But since the OP already knows about determinants, maybe Axler would be a very good choice for him!

 

[blue_leaf77]

My situation is that, as I mentioned in my first post, I can borrow Axler's book for only four days. That's unfortunate indeed.

 

[blue_leaf77]

Just one last request, could somebody give a list of typical topics covered in linear algebra course taught in physics department, at least in your own department? I have searched for syllabuses in some physics department webpages but apparently many of them do not offer a standalone linear algebra course. I think it is packaged in another course. The thing is I already started experimenting in the lab so I won't have enough time to go through all the chapters in the book.

 

[Fredrik]

I'll list the topics that I think are the most important.

Prerequisites:

Complex numbers
Polynomials (Use Axler for this)

The basics:

Vector spaces over $\mathbb C$
Subspaces
Linear independence
Span
Bases

Inner products and norms:

Orthogonality
The norm associated with an inner product
Orthonormal bases

Linear transformations:

Components of a linear transformation with respect to a pair of ordered bases. (I wrote a https://www.physicsforums.com/threads/matrix-representations-of-linear-transformations.694922/ about this).
Matrix multiplication (Prove that $[T\circ S]_{E,G}=[T]_{E,F}[ S]_{F,G}$).
Change of basis (Prove that $[T]_F$ is similar to $[T]_E$).
Kernel and range (the definitions and the rank-nullity theorem)
The adjoint of a linear transformation
Self-adjoint and unitary linear operatorsBijective linear transformations:

Determinants (Use Treil for this)
The theorem that lists conditions equivalent to $\det T\neq 0$.

Spectral theory:

Eigenvalues and eigenvectors
The spectral theorem for self-adjoint linear operators (Surprisingly easy. See this post).

Two more things:

Positive-semidefinite linear operators (Use Treil for this)
Projection operators (Prove the finite-dimensional versions of the theorems in section 6.3 of Friedman's "Foundations of modern analysis").

 

[blue_leaf77]

Thanks a lot Fredrik.