What Are the Best Linear Algebra Topics for a Physicist?

[loonychune]

I've been working from a book called 'Linear Algebra' by Georgi E. Shilov and while it is nice to take the more hardcore-mathematical approach, I just don't have the time (nor think it necessary) to get what I need from such a book for physics.

So what I've done so far is looked at systems of linear equations (N equations and N unknowns) and Cramer's rules (which is a nice method for such equations).

However, I'm finding it difficult trying to organise some sort of textbook learning for everything... Riley,Hobson, Bence's book seems to cover lots in detail then side-steps the question of solving homogeneous sets of equations and M x N equations by introducing the method of SINGULAR VALUE DECOMPOSITION and Mary Boas is really too brief in covering things like eigenvalues of hermitian and anti-hermitian matrices, and probably actually too brief in her coverage of eigenvectors/values full-stop.

At risk of going on for too long, let me ask: what would be the best topics to cover?

Would the SVD method be pretty much sufficient, since if i cover the entire chapter in RHB's book, i'll get a good look at things like vector subspaces also?

At the end of the day I have to be able to formulate matrices and solve sets of simultaneous equations and could spend far too long in theory and not actually earn that ability! (course whatever extra could be useful for this year's coming course in quantum mechanics but at the same time this project has come out of me wanting to cover things I've skimmed through in the past)

Damian

 

[tbradshaw]

Linear algebra will help you solve problems faster. Knowing how to find eigenvalues is nice when solving PDE's. This will help you out tremendously. Taking determinants of a matrix comes in hand too when finding cross-products of complicated vectors.

 

[Mene]

,

As a physicist, linear algebra is a crucial tool for understanding and solving problems in the field. It is great that you are taking the time to study it in depth, but it is important to also prioritize and focus on the topics that are most relevant to your work in physics.

From your description, it seems like you have already covered some important basics such as systems of linear equations and Cramer's rule. These are fundamental concepts that are applicable in various areas of physics, so it is important to have a good understanding of them.

In terms of further topics to cover, I would recommend focusing on vector spaces, matrices, and eigenvalues/eigenvectors. These concepts are essential for understanding and solving problems in quantum mechanics, as well as many other areas of physics. The SVD method is also a useful tool to know, as it can help with solving systems of equations and understanding vector subspaces.

Additionally, it may be helpful to look into applications of linear algebra in physics, such as in quantum mechanics, electromagnetism, and statistical mechanics. This will give you a better understanding of how these concepts are used in real-world problems.

Overall, I would suggest prioritizing the topics that are most relevant to your current and future work in physics. It is important to have a solid understanding of the fundamentals before delving into more advanced topics. Good luck with your studies!