>Second, the math department teaches a class in linear algebra. This is a collection of topics that you wouldn't ordinarily find in undergraduate math classes.
x math department doesn't have classes on graph theory, combinatorics, or prob & stats?
Discrete mathematics is a grab bag of topics, and it's more fair to say that a discrete math course pulls in selected topics from other areas than to say it provides a focused coverage of any of them. At the depth each topic is taught, you couldn't pull it out into its own semester's worth of lectures. On the other hand, it would be very difficult to compact linear algebra into a discrete math course. Learning linear algebra already requires you to introduce a bunch of new concepts, like fields and vector spaces, linear equations, linear transformations, matrix representation and matrix operations, determinants, inner products...
Equally importantly, linear algebra doesn't really "fit" with the topics of discrete mathematics, pedagogically speaking. It fits well with a treatment of multivariable calculus, but that's only a small part of what's taught in discrete math courses. If you wanted to teach linear algebra in a course covering a bunch of topics, you'd probably want to do a "topics in abstract algebra" course that works through groups, rings, modules, linear algebra and multilinear algebra. But then you're going well beyond what a non-math major should encounter in an undergraduate setting.
Really, linear algebra stands on its own very well. In my opinion it should receive focused coverage and not be mixed with anything else. While graph theory and combinatorics can easily receive the same coverage, their treatment is usually pushed back to advanced undergraduate or graduate level courses because engineers don't usually work with them beyond what's covered in discrete math (which is exactly why we have discrete math as a standalone course). In fact, MIT actually lists linear algebra and multivariable calculus as prerequisites for the focused courses on combinatorics. The same applies for probability theory: you can't really learn probability theory (beyond discrete) without first learning analysis.
These were our —typically— freshman/sophomore year CS classes (although I took a couple in my junior year) so I guess every school defines their requirements differently.