Hoffman, Kunze Linear Algebra book: which topics to study QM?

[sphyrch]

I've started self-studying quantum mechanics. It's clear from google searching and online Q.Mech lectures, I'll need to understand linear algebra first. I'm starting with finite-dimensional linear algebra and Hoffman, Kunze is one of the widely recommended textbooks for that.

I need help regarding which topics I should cover to properly learn linear algebra prerequisite for quantum physics. Here's the table of contents:

https://pastebin.com/6EE4NdYA

From what little I know, I'm pretty sure that the chapters: Linear Equations, Vector Spaces, Linear Transformations, Inner Product Spaces and Operators on Inner Product Spaces are necessary. I'd be grateful if someone could tell me about the remaining chapters.

Thanks in advance!

 

[fresh_42]

It's hard to tell one or another chapter could be omitted, as linear algebra is basically part of the very basics in physics. You probably won't need to know, how some normal forms can be achieved, because you can always look it up, but that they exists and what they do is important, esp. the Jordan normal form. Similar is true for the other chapters, e.g. modules and Graßmann rings listed under determinants. Sooner or later you will likely meet all of those concepts. Polynomials looks as it could be left out, but then we find algebras and ideals in it. And bilinear forms are essential, too, as they occur at so many different occasions.

What you can do is leave out some technical proofs or exercises. I cannot really recommend it, but I wouldn't omit any theorems instead. Linear algebra is normally not very hard. The main difficulty is that our school systems are very focused on calculus and algebraic concepts often appear a bit strange at the beginning. However, it is necessary to get a feeling about algebraic objects, esp. vector spaces of all kind.

 

[sphyrch]

It's hard to tell one or another chapter could be omitted, as linear algebra is basically part of the very basics in physics. You probably won't need to know, how some normal forms can be achieved, because you can always look it up, but that they exists and what they do is important, esp. the Jordan normal form. Similar is true for the other chapters, e.g. modules and Graßmann rings listed under determinants. Sooner or later you will likely meet all of those concepts. Polynomials looks as it could be left out, but then we find algebras and ideals in it. And bilinear forms are essential, too, as they occur at so many different occasions.

What you can do is leave out some technical proofs or exercises. I cannot really recommend it, but I wouldn't omit any theorems instead. Linear algebra is normally not very hard. The main difficulty is that our school systems are very focused on calculus and algebraic concepts often appear a bit strange at the beginning. However, it is necessary to get a feeling about algebraic objects, esp. vector spaces of all kind.

Thanks! Since you've specified that none of the chapters should be omitted altogether, I will at the very least cover all the theorems - I suppose it's important to know how to at least apply them, if not some of their proofs.

One more thing I want to clarify - are there any important topics that I should know of, which not covered in the book? Or is the book sufficient as a first course in linear algebra (SPECIFICALLY as a prerequisite for quantum physics)?

 

[fresh_42]

... are there any important topics that I should know of, which not covered in the book?

Yes, as functional analysis (operators, Hilbert- and Banach spaces, function spaces), tensor algebra and differential geometry also play a major role in QM as well as some basics about representation theory. They cannot always be strictly separated.

Or is the book sufficient as a first course in linear algebra (SPECIFICALLY as a prerequisite for quantum physics)?

Also yes with emphasis on first course. I would even say, it is necessarily the starting point. The thing is to find a way such that you don't have to become a mathematician in order to become a physicist. However, the good ones are both. It might be more important to have some books (or e-prints nowadays) where you can look up topics you might have forgotten. Nobody knows everything, but to know where to look is at least as important. At the beginning the mountain to climb always looks huge, but once you've understood the concepts behind, things get easier and many facts will get more absorbed than learned.

And of course you can always return here and ask questions. And as a hint for these: the better prepared the questions are, the better the answers will be. Make use of our homework section and the (automatically inserted) template there. It can be a real good tutorial and is far better than being stuck for too long at a certain issue. We also have some insight articles on how to self-study certain fields, e.g.

https://www.physicsforums.com/insights/problems-self-studying/
https://www.physicsforums.com/insights/how-to-study-mathematics/
https://www.physicsforums.com/insights/self-study-algebra-linear-algebra/

This interview might also be of interest:
http://www.ams.org/publications/journals/notices/201707/rnoti-p718.pdf

In any case: good luck, success and don't forget the fun!