Linear Algebra Self-Study: Textbook & Quantum Mechanics

[TGV320]

Hello,
I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra.

Some textbooks in my country are quite bad and only teach the students the computation. Others delve in quite deeply, with loads of proofs. How should I study that subject?
Thanks

 

[malawi_glenn]

For physics, its more important to know what the theorems says and under what circumstances they are valid than how to actually prove them.

There are many nice free online Linear Algebra mathbooks which you can use:

“A first course in linear algebra (Kuttler)”
https://batch.libretexts.org/print/Finished/math-14495/Full.pdf
more info math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)

“Linear algebra with applications (Nicholson)”
https://lila1.lyryx.com/textbooks/OPEN_LAWA_1/marketing/Nicholson-OpenLAWA-2021A.pdf
more info https://lyryx.com/linear-algebra-applications/

“A first course in linear algebra (Beezer)”
http://linear.pugetsound.edu/download/fcla-3.50-tablet.pdf
more info http://linear.pugetsound.edu/

“Linear algebra (Hefferon)”
https://joshua.smcvt.edu/linearalgebra/book.pdf
more info https://joshua.smcvt.edu/linearalgebra/

“Linear algebra done wrong”
https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-04.pdf
more info https://www.math.brown.edu/streil/papers/LADW/LADW.html
Don't let the title scare you, it is just a hint to the book called “Linear algebra done right” (which is a great book, but not a good first one)

“Linear algebra (Math online)”
http://mathonline.wikidot.com/linear-algebra

 

[TGV320]

Hi,
Thanks, I'll have a look into that.

 

[malawi_glenn]

Hi,
Thanks, I'll have a look into that.

You should also be able to find say "Elementary Linear Algebra" by Anton & Rorres cheap used. It is one of the most used books for beginner classes on Linear Algebra and have loads of different editions. Same goes for "Linear algebra" by Friedberg, Insel & Spence - a classic beginner book.

This insight post by micromass seems very helpful too https://www.physicsforums.com/insights/self-study-algebra-linear-algebra/

Note that QM deals with complex vector spaces, so it is good to cover those at some point in your Linear Algebra career.

 

[Hall]

I would like to offer my views on the subject and its available resources:

What is it about?
Upto school level mathematics, we usually deal with numbers, like $x$ in all of our equations are just the numbers; following commutativity, associativity and other usual things. For a short course we deal with functions when we study Calculus, but still functions represent numbers only.

When we come to Linear Algebra, it deals with all possible mathematical objects: numbers, vectors, functions, matrices, quaternions etc. But we restrict our study to only those objects which follow a properly laid down axioms (a total 10 of them). It’s like a whole algebraic analysis (not in the mathematical sense of the word) of all possible mathematical objects (often called vectors) which follow those axioms. However, there are some books which treat linear algebra only as a tool of solving linear equations (of which I’m a very strong opponent). In a few books, you might find the whole effort is on solving and analysing $A x =b$, I feel that’s not a very good motivation.

Then there are books which treat the subject solely from matrix point of view, which is fine, but not good if you’re beginning for the first time.

Prof. Gilbert Strang and Mr. Tom Apostol teach the subject, as far as it seems to me, for the sake of the subject only. Prof. Strang focuses more on matrices, while Mr. Apostol always consider matrix as a representation of an operator.

How to study the subject?
Accept this fact that we don’t understand Linear Algebra in the beginning, we find no motivation for defining the axioms for a space, linear span and independence shall be seemed as a made-up thing, and a set is orthogonal and the same set could be not orthogonal (depends on how we define inner product) would seem a little unusual. But just remain with it for a month, and you might realize that you belong to a different class of people in society, we care about things that do not exist at all.

Linear Algebra and Quantum Mechanics:
If your final aim is QM, and if we are teleological people, completing Linear Algebra completely before entering into QM would be a bad idea. You might complete Linear Algebra, and then try to see QM as a mathematical subject, following some given axioms, by studying, for example, Mr. Dirac or Mr. Neumann; but this approach is not advised by learned men. Another possible pathway could be to try a few lectures of Leonard Susskind’s on YouTube, or/and Feynman’s Vol III, and see how much Linear Algebra you need to study before it and how much can be learned through it.

 

[CrysPhys]

Hello,
I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra.

Some textbooks in my country are quite bad and only teach the students the computation. Others delve in quite deeply, with loads of proofs. How should I study that subject?
Thanks

You have two related threads: https://www.physicsforums.com/threa...hysics-at-the-same-time.1017340/#post-6660626 and https://www.physicsforums.com/threads/math-analysis-and-ode.1017374/. By now, you have received the following advice, which you should follow (this assumes your primary focus is on physics, not math):

* Whether it's MIT OCW or some other online university physics curriculum, look first at the physics courses you want to take.

* Each physics course will list the prerequisite and corequisite physics and math courses. If these are not listed on the online versions, check the in-class versions.

* The prerequisite and corequisite physics and math courses will depend on the specific university physics program and the specific textbooks used. Undergrad physics textbooks typically include material on the math required, but in varying degrees of depth. So a physics course using a physics textbook that covers a lot of explanatory math and incorporating the needed math as part of the physics course will have less math prerequisite or corequisite than a physics course using a physics textbook that does not cover a lot of explanatory math (or the physics textbook does cover the needed math, but the physics course does not incorporate it).

 

[Mondayman]

I used Anton as my primary and started digging into Friedberg before I left school. As someone who struggles with abstract math, I found these to be good starting points.

 

[malawi_glenn]

completing Linear Algebra completely before entering into QM would be a bad idea. You might complete Linear Algebra,

One can never "complete" a mathematical subject. There is always more to learn about it. So this advice is applicable to all math subjects. One does need to have a PhD in linear algebra to do tackle quantum mechanics ;)

Furthermore there are other mathematical subjects one should know about before QM. Differential equations, partial differential equations, functional analysis, Fourier analysis. For scattering problems etc some complex analysis is needed (contour integration).

 

[CrysPhys]

OP: Here is a response from a different thread. It is also relevant to your scenario:

There are other factors involved, such as time constaints. It is not plausible to study math for 10 years and then start with physics

 

[malawi_glenn]

OP: Here is a response from a different thread. It is also relevant to your scenario:

Yeah that was a good quote!

One just need to know "enough" to move forward, and how to gain more knowledge when one hits a wall.

 

[TGV320]

Hello,
Thanks, I will keep those recomendations in mind.

 

[Jody]

I bought the Schaum's outline book to help me practice. I also thought Gilbert Strang had very good lectures online it was a very good supplement and refresher when I needed later.