[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.