Applied vs Proof Based Linear Algebra

[bentleyghioda]

Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of linear algebra, while the other is more focused on proofs. These are the descriptions of the two courses:

MATH 232: Linear equations, matrices, determinants. Introduction to vector spaces and linear transformations and bases. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner products and orthogonality; least squares problems. An emphasis on applications involving matrix and vector calculations.

MATH 240: Linear equations, matrices, determinants. Real and abstract vector spaces, subspaces and linear transformations; basis and change of basis. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner products and orthogonality; least squares problems. Applications. Subject is presented with an abstract emphasis and includes proofs of the basic theorems.

I am wondering if I would benefit more from the proof based class or the more applied class as a physics major. I do not have a lot of experience with proofs, but I will likely be taking a discrete mathematics course the same semester as linear algebra, and from what I’ve read, this course acts as an introduction to proofs. If it helps to know, I plan on going to grad school for physics, and I would also like to take several upper level math courses.

 

[jedishrfu]

I would think the applied version would be more useful especially since you don’t have much proof experience. Proofs would be useful if you planned to focus on theoretical physics but even then applied uses would be more useful overall. Mathematicians benefit the most from proof based courses.

 

[FactChecker]

The upper level math classes may be impossible if you do not have a strong background in proofs. If they are also applied math physics classes, that would be different.

 

[jedishrfu]

The upper-level math classes may be impossible if you do not have a strong background in proofs. If they are also applied math physics classes, that would be different.

So true, I blocked that traumatic memory until now. I once took an Abstract Topology course thinking I took proofs in geometry class so it should be no problem. It was definition on top of definition with no end in sight and I was a lost babe in a sea of upper-level math majors. Fortunately, my prof took pity on the hubris of a poor hapless physics major and guided me to a passing grade and implicitly warning to never tread into the ocean of no return without the proper math major background.

Thanks @FactChecker for bringing back that memory on my schooling of yesteryear.

 

[bentleyghioda]

Thank you for your replies. I’m not sure what courses I will be taking in the future, but I will keep in mind that many upper level math courses require proofs.

 

[bentleyghioda]

Do you think it would be beneficial to take some proof heavy math courses that are designed for math majors? For example, would I gain any tools from these classes that could be applied to physics?

 

[FactChecker]

If your ultimate goal is physics, I think you should learn your math from the physics perspective. Those classes should cover any of the theory that you need without getting hung up on every detail that formal proofs require.

 

[MidgetDwarf]

I would take the intro proof based linear algebra... Although hard, you will walk out out from the class with a deeper understanding. Plus Linear Algebra is very important in Physics. You have 1 year to prepare...

 

[PeroK]

@bentleyghioda the other question is whether you want to take a look at more formal mathematics. The safe, focused option is to go for the course more aligned to physics. Taking the proof-based course might be more adventurous and widen your knowledge of the maths-physics landscape.

 

[gleem]

Definitely take the application based class. It is a new subject for you and as a freshmen you should not overestimate your ability or over think your career. I think you will find it challenging for is my experience you will encounter much formal treatment of the subject while keeping you focused on why you are taking the course. It might be of value to find out which text each class uses and read the preface since it should contain the authors goals and methods