> My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve.

I view that as "if you can find a way to express your problem in terms of linear equations, there are well-known techniques for finding solutions."