Is Linear Algebra Necessary for Physics Beyond Quantum and Classical Mechanics?

[torquemada]

Hi I just learned that at my school the LA course is all theory/proofs and virtually no applications, oriented for pure math majors.


Although LA is important for physics, in light of this do i still stand to benefit from taking it (i.e. from a pure math approach) or can i learn whatever LA i need from the intermediate methods of mathematical physics classes that the physics dept offers. They use the book advanced engineering mathematics by erwin kreyszig and i know it has some chapters on LA, and I believe they are application oriented. Should that suffice or do i still go for pure LA? thanks :)

 

[romsofia]

I would take the LA course, just because I bet it will have some applications in physics (even if it is all proofs/theory). I don't know how far you're in your physics education, but Quantum mechanics deals a lot with linear algebra, so the course WILL help there.

 

[Floid]

I had to take an undergratuate LA course that was similar and really a waste of time.

I have Advanced Engineering Mathematics... it has a pretty basic treatment of linear algebra. I also have Elementary Linear Algebra by Spence Insel and Friedberg. It is a basic text that really doesn't cover much more than Kreyszig's book, it just takes 200 more pages to do so. It does have many more examples, but practical application is limited to circuits and coordinate rotation.

I am not a physicst so not sure what applications of linear algebra you would typically want to know, but I usually refer to Modern Control Theory by Brogan for my linear algebra reference.

Even though it is a state space controls book it has 6-7 chapters of pure linear algebra including QR decomposition, Cayley-Hamilton theorem, canonical forms, Gram-Schmidt process, vector basis... all of which I don't remember being included in Kreyszig's text. It has plenty of application, but it is mainly focused on control problems. But, in the end what you are doing is breaking higher order differential equations into sets of first order differential equations and solving using linear algebra which I am sure is one of the primary applications in physics as well.

 

[R.P.F.]

Take the class. It is a great way to train your mind as an undergrad whether or not you are going to do pure math. My philosophy can be pretty much summarized as: always go for the hard stuff.

 

[Monocles]

To offer a dissenting opinion to Floid, I also took a proof-based linear algebra course and found at least as useful as my calculus courses for physics coursework - mostly in quantum mechanics. Quantum mechanics is all linear algebra. It is often infinite-dimensional, but a good proof-based first linear algebra course should cover a little bit on infinite-dimensional spaces, and the oddities of infinite dimensional space rarely arise at the undergraduate level anyways.

You basically cannot take too much linear algebra as a physics major. I've taken two basic courses on it, and two courses on its infinite-dimensional cousins, functional analysis and the theory of operators, and I'm still learning new methods from it to apply to physics.

 

[id the sloth]

Whether or not you plan to take the class is up to you. I've met physics majors who have taken it reasoning it would be very useful, and physics majors who have avoided it because they see no point. What really matters is that you learn your Linear Algebra at some point. Some people feel like the course gives them a very solid understanding of the material and some people feel like they have a solid understanding from simply taking their quantum mechanics course or their basic linear algebra course.

While it is proof based, the linear algebra course will teach you how to work with abstract vectors, which is very important for quantum mechanics. Instead of simple column vectors, vectors can be anything that satisfy certain properties. Solutions to homogeneous differential equations happen to fulfill these properties so linear algebra has huge applications to differential equations (which is important to all of physics).

I personally felt like my quantum mechanics course gave me a solid understanding of linear algebra. My professor taught us what we needed to know and I saw how it was applied in the homework. And while my understanding is good enough to get by, I would like to take more advanced courses later on because I feel like it would be useful (I'm really interested in differential equations). I'm trying to see what grad schools I get into first, I see no point in taking a very theoretical course too so I'm going to see if the grad school I get into will have an applied version.

 

[torquemada]

Thx for the replies all.


Besides Quantum and Classical Mechanics, what other undergrad courses does LA show up in? Any optics, thermo, or E&M?

 

[Number Nine]

Thx for the replies all.Besides Quantum and Classical Mechanics, what other undergrad courses does LA show up in? Any optics, thermo, or E&M?

You should keep in mind that only only is the subject matter of LA itself useful (vectors and vector spaces), but that LA is a natural language for plenty of the other math you'll be using. Linear algebra comes up all the time when dealing with differential equations, and it makes multivariable calculus just plain easier. There is absolutely no reason not to take it (really, I recommend taking it with or before calc III).