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Is it a real concern or are you humble bragging about the lack of advanced math in an introductory text on _linear_ algebra?

No, that's a legitimate concern. It may or may not be relevant for what you specifically care about, but it's definitely a legitimate thing to ask about.

It's not uncommon for even undergraduate linear algebra to cover abstract spaces and notions of linearity which generalize beyond R^n. For example, the function space P_n consisting of all polynomials with degree less than or equal to n. Hoffman-Kunze, Halmos, Axler and Friedberg-Insel-Spence are all examples of undergraduate textbooks which cover this material.

This isn't just theoretical. Function spaces like P_n are useful in applied mathematics. And even if you don't use function spaces, it's very common for engineers, physicists and applied mathematicians to work in the complex space C^n rather than R^n.

This is just the first semester course. In my experience teaching it at GT, it is mostly taken by freshman, and I'm not sure how many majors require it but my students have come from a huge variety of majors. I'm not sure that abstract vector spaces is that useful for, say, psychology majors. There is a second semester course that usually uses Axler.

Various function spaces were definitely brought up in my linear algebra class. That kind of material comes up even in intro differential equations (eg systems of differential equations, Lp spaces).

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