[akater]

Another evidence that linear algebra concepts are terribly confusing when layed out without geometric background.

This had been criticised for decades. Competently and popularly presented criticism goes back to at least as far as 1957 (Artin's Geometric Algebra; see discussion of determinants somewhere near the beginning) but linear algebra is still often presented decoupled from geometry.

I wonder though if there's purely algebraic approach to matrices that explains as much (or more) as geometric one. Maybe approaching algebra of matrices consistently as an example of category algebra could be illuminating.

[dnautics]

I learned abstract linear algebra first, and didn't learn geometrical interpretations until I taught myself many years later so I could write an asteroids clone in svg.

I don't think it was a pedagogically a problem, except I couldn't bring myself to care about matrices when I was learning them... It was very easy to take my abstract knowledge and apply it, and for me it might have been harder the other way around.

In retrospect a hybrid applied/theoretical topic (like reed Solomon encoding and recovery) might have perked me up. But I might have also been a strange case.

[akater]

If by “abstract” linear algebra you mean “course that starts with the definition of vector space”, then it is geometric enough in the sense Artin talks about.

It is pedagogically a problem for people who ask questions like the one in topic. But it can also be a problem when one encounters bilinear form and linear operator in practice but can't distingish between the two. I can't think of a specific example but I was asked once about some problem (in electrical engineering, iirc) where the source of confusion was this; some transformations of a (square) matrix were natural while others were not.

Some people feel strongly about the topic—mostly those with “pure math” inclinations.