Linear algebra vs. classical mechanics

[jbrussell93]

The time has come to schedule for next semester's classes. I will be a senior in physics and choosing some electives. I am trying to decide between taking matrix theory (linear algebra) or graduate level classical mechanics. I really WANT to take the mechanics course but I feel that maybe I should take matrix theory.

A linear algebra course is not required for my degree, but I have been introduced to bits of it in classes such as multivariable calculus, ODE's, and math methods for physics. The downside to "matrix theory" is that it is also an intro to proofs class, so I'm not sure how useful it will be for me. I'm sure I will miss out on some useful things by not taking it, but is it worth giving up the chance to take graduate classical mechanics? I figure that I have a decent enough grasp on the basics of linear algebra that I can simply pick up anything I run into during my physics classes... Any thoughts?

 

[SteamKing]

It depends on where your education stops. Will you pursue a graduate degree in physics or are you stopping with a bachelor's? IMO, linear algebra is ubiquitous enough in science and engineering that you should take the course, or at least have some introduction to the subject beyond 'bits and pieces', whether or not it involves a lot of proofs. I'm surprised that you have managed to get almost thru an entire undergraduate degree without LA.

 

[jbrussell93]

That is a good point I did not mention. I am definitely looking towards grad school either in physics or geophysics. I guess I will always have the option to take linear algebra during my final semester as a senior, but classical mechanics is only offered in the fall. Also, my university does not offer a second semester semester of E+M for undergrads so the only chance I will have for exposure to the material is through the graduate electromagnetism course, but that course has classical mechanics as a prerequisite, technically. I should also mention that I will be taking the first semester of quantum mechanics next semester which might be interesting without linear algebra.

 

[Sentin3l]

Linear algebra is a fundamental area of math that every physics student should know and understand. I would highly recommend you take the LA course.

 

[micromass]

The time has come to schedule for next semester's classes. I will be a senior in physics and choosing some electives. I am trying to decide between taking matrix theory (linear algebra) or graduate level classical mechanics. I really WANT to take the mechanics course but I feel that maybe I should take matrix theory.

A linear algebra course is not required for my degree, but I have been introduced to bits of it in classes such as multivariable calculus, ODE's, and math methods for physics. The downside to "matrix theory" is that it is also an intro to proofs class, so I'm not sure how useful it will be for me. I'm sure I will miss out on some useful things by not taking it, but is it worth giving up the chance to take graduate classical mechanics? I figure that I have a decent enough grasp on the basics of linear algebra that I can simply pick up anything I run into during my physics classes... Any thoughts?

That you call the LA course "matrix theory" is a bit weird and it indicates me that the LA course is not of a very high level. Can you perhaps post a list of the topics in the course? And can you indicate how much you're already familiar with it?

 

[jbrussell93]

The course description:
"Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs."

I have at least some exposure to everything but Jordan normal forms. Especially matrices, determinants, eigenvalues, and eigenvectors.

It is the LA course that math majors are required to take and has a prerequisite of calc III (though I'm not sure why).

 

[micromass]

The course description:
"Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs."

I have at least some exposure to everything but Jordan normal forms. Especially matrices, determinants, eigenvalues, and eigenvectors.

It is the LA course that math majors are required to take and has a prerequisite of calc III (though I'm not sure why).

The course description looks very decent. I think this course really contains a lot of topics that are worth knowing about as a physicist.

Whether you should take the course or not depends on how much you know already. If everything in the course is going to be review, then there's not much sense to it. I think it would be wise to find the syllabus/course text, open it up at some topics like "vector spaces", "linear maps" or "eigenvalues" and see how much really is familiar to you.

I'll try to help you now by asking some questions, (see if you can answer them out of the blue):
- Can you define the dual space V* and prove that V and V* are isomorphic (if V is a finite dimensional vector space).
- What can you say about eigenvectors and eigenvalues of a self-adjoint linear map?
- What is the relationship between a linear map and a matrix?
- Under what conditions is it possible to diagonalize a matrix?

Finally, I think the LA course has Calc III as prereq because of a mathematical maturity condition. I doubt very much you'll see much calculus in the LA class (if at all). It's just that people who haven't taken calculus aren't seen as "mature" enough to handle LA. Personally, I think that's a stupid prereq and I prefer LA as a prereq to Calc III.

 

[jbrussell93]

I'll try to help you now by asking some questions, (see if you can answer them out of the blue):
- Can you define the dual space V* and prove that V and V* are isomorphic (if V is a finite dimensional vector space).
- What can you say about eigenvectors and eigenvalues of a self-adjoint linear map?
- What is the relationship between a linear map and a matrix?
- Under what conditions is it possible to diagonalize a matrix?

I appreciate your help. I honestly cannot answer any of these questions from my current knowledge. Mostly, I am not familiar with the terms such as isomorphic, dual space, self-adjoint linear map. Maybe I will seriously consider taking this course.

One thing I am concerned about which I've mentioned in a previous thread, is that my intermediate mechanics course did not cover Lagrangian/Hamiltonian mechanics. I've studied it some on my own and did most of the problems from the chapter (Mechanics by Simon), but the graduate course would be my chance to formally develop these ideas. Do you think I should still prioritize LA?

My other option would be to take the grad mechanics course next semester and then LA my final semester.

 

[micromass]

I appreciate your help. I honestly cannot answer any of these questions from my current knowledge. Mostly, I am not familiar with the terms such as isomorphic, dual space, self-adjoint linear map. Maybe I will seriously consider taking this course.

In that case, I highly recommend taking the course.

One thing I am concerned about which I've mentioned in a previous thread, is that my intermediate mechanics course did not cover Lagrangian/Hamiltonian mechanics. I've studied it some on my own and did most of the problems from the chapter (Mechanics by Simon), but the graduate course would be my chance to formally develop these ideas. Do you think I should still prioritize LA?

My other option would be to take the grad mechanics course next semester and then LA my final semester.

Yes, I think taking both courses would be beneficial. Lagrangian and Hamiltonion mechanics is of great importance so a course in that would be a very good thing. I think your option of "mechanics now, LA final semester" is a good one.

 

[NATURE.M]

To be honest, I don't think your linear algebra course will actually cover the concepts that you mentioned you lack understanding for (e.g isomorphism, dual space, self-adjoint linear map). It seems you are comfortable in answering the last two questions asked by micromass pertaining to diagonalizing a matrix and linear maps/matrices (which will definitely be covered in the course).

At least in my experience, the course description for my introductory linear algebra course was fairly similar to yours. And in that class, we never touched upon isomorphisms, dual spaces, self-adjoints, vector spaces over complex numbers, etc.
So maybe self-studying these topics would be preferable rather than taking an actual course.

 

[micromass]

It seems you are comfortable in answering the last two questions asked by micromass pertaining to diagonalizing a matrix and linear maps/matrices (which will definitely be covered in the course).

He did say he could not answer any of the four questions. Maybe the OP should indicate how comfortable he is with the last two questions, but if he truly cannot answer them, taking the course would be a good idea in my opinon. What do you think?

I completely agree with you that the first question is more specialized and does not pop up in a lot of LA courses. I consider the second question to be a lot more important though, I think it's pretty sad that courses don't cover the spectral theorem and self-adjoint (or symmetric or hermitian) matrices.

 

[jbrussell93]

With regards to the last question, I know how to diagonalize a matrix but I'm not sure under which circumstances it is or is not possible to do so.

Regardless of whether or not I am able to answer these questions, you have convinced me that it would be a good idea take the course before I graduate. So that is what I'll plan to do.

 

[leroyjenkens]

The course description:
"Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs."

Sounds exactly like the LA class I took, which was very useful, except I've never even heard of Jordan normal forms. I guess we didn't get that far in the book.

 

[micromass]

With regards to the last question, I know how to diagonalize a matrix but I'm not sure under which circumstances it is or is not possible to do so.

To be honest, I don't think there really is an easy criterium which decides whether a matrix is diagonalizable or not. I was thinking more of "partial implications". For example, if you have an $n\times n$ matrix with $n$ different eigenvalues, then it's diagonalizable. Or if the matrix is real and symmetric, or hermitian, then it's diagonalizable. Do these things sound familiar?

 

[jbrussell93]

To be honest, I don't think there really is an easy criterium which decides whether a matrix is diagonalizable or not. I was thinking more of "partial implications". For example, if you have an $n\times n$ matrix with $n$ different eigenvalues, then it's diagonalizable. Or if the matrix is real and symmetric, or hermitian, then it's diagonalizable. Do these things sound familiar?

Oh, well in that case yes I am familiar with the first condition. I'm not familiar with hermitian matrices though.

 

[mpresic]

This is a tough question. First of all, my graduate study is (far) behind me so my priorities may be misplaced. Do most physics graduate schools look favorably on undergraduates coming to them with significant graduate work. Specifically, perhaps your intended graduate school will want (possibly require) you to complete graduate classical mechanics at their institution. I know all graduate schools will require a "core" graduate curriculum that will require a substantial number of credits completed with them. Will you have some advantage in taking classical mechanics at your current institution.

I applaud your efforts to shore up your classical mechanics. Symon is a good textbook for intermediate mechanics, that you are working your way through. This will definitely pay dividends when you test for your qualifying exams. I see you are concerned with not being exposed to Lagrangian and Hamiltonian formulation. Most of your colleagues at graduate school will have this background.

Bottom line: If the LA course is at the level of a second LA course (say out of Sheldon Axler's book) it is useful but graduate CM may be better (or even intermediate CM or engineering CM where Hamiltonian/Lagrangian may be introduced). If you have deficiencies in LA (first course level), it is best to address them now. My own feeling (it could be wrong) you do not have to be in a hurry for graduate coursework.