[dagw]

I found that the level of abstraction that linear algebra is taught at is really weird. On the one hand it was never really taught in connection with algebra in general, or put into a wider mathematical context. On the other hand we were never really shown any real world used of linear algebra or how the concepts maps onto practical applications.

It wasn't before years later when A) I'd taken more abstract algebra courses that introduced concepts like tensors and fields and B) I'd taken more practical computational courses where I had to do things like use linear algebra in 3D graphics and use eigenvectors to do dimensionality reduction and PCA, that I really understood the subject and its place in the world.

Just saying here's a random object (that we'll call a "matrix"), here's some random steps (that we'll call "taking a determinant"), now memorize how to apply the steps to the object and see if the result is zero, didn't lead to much deeper understanding.

[neuronic]

I think what courses often fail at is to specifically mention that all of these things are abstractions invented by people to deal with a specific issue and then WHY IT HELPS solving this issue.

Whenever I studied Linear Algebra it was simply "here's a matrix, here is some algorithm, use it to get some number seemingly coming out of nowhere and just believe us that this number is what you really need."

Note that I am not immediately interested in mathematical proof of why this method is correct. I want a plain English explanation of what is going on but instead you usually just get a bunch of notation to digest.