[vlasev]

I believe this is largely because in the field of mathematics, Linear algebra is just the seed that sprouts the growth of other very very useful mathematical subjects like Abstract Algebra, Functional Analysis and so on. Linear Algebra is used as a stepping stone to more general theories that are also super useful.

Take Hilbert spaces for example. They are based on linear algebra. They are quite general and you might argue that there's a lot of symbol twiddling there. However, Hilbert spaces are/were essential in the study of Quantum Mechanics, which we can argue is a very important topic.

And if you only stick with matrices and numerics, you're bound to get stuck in the numbers and details and miss the big picture. A lot of results are much cleaner to obtain once you divorce yourself from the concrete world of matrix representation.

Of course, we should probably have the best of both worlds. I'm not saying applications are unimportant. Take something like signal processing, which relies heavily on both numerics and general theory.

So I'd like to add something to your point. Math departments optimize the education of math students towards the more general, and perhaps students not interested in pursuing pure math should have course-work that reflects that.