Why teach linear algebra before abstract algebra?

I have to provide a counterpoint to the rather cynical answers already present. To be fair, almost everyone seems to have interpreted the question to mean "what is the rationale for the current system of putting linear algebra first", whereas I would like to take the perspective that there is a good pedagogical and mathematical rationale for doing it this way, regardless of historical precedent or the needs of service classes.

The worst way to teach math is in historically-correct order: history is rife with epic intellectual struggles to find the correct generalization from within the context of an existing (possibly quite unfamiliar to us) perspective on math, previous partial generalizations and poorly-understood (possibly incorrect!) foundations. I had a professor once who said that he'd taken an abstract algebra class that proceeded from Lagrange's work on solvability of polynomials, and that the most he got out of it was that it's very difficult to think like Lagrange.

The second-worst way is in logically-correct order. That's not to say that there is no place for a rigorous development of mathematics; obviously, that is necessary at some point in the presentation of any of its fields. But you can't just sit people down, not even interested and intelligent people, and say "now we are going to learn the axiomatic development of mathematics". They have no personal reason to buy into this narrative and will probably become bored and confused.

And yet, mathematics was developed in a particular order historically, and later formalized in a particular order logically, so there is something to these choices. To teach a novice effectively you have to create a faux history for them and then develop the material internally consistently within the history. Sometimes this involves lying if that means getting across a useful piece of intuition; or omission, if that means simplifying some deeply technical preparations; or repetition, if that means that the first round of lessons makes more sense in the context of what came before than the second, complete round would have. I am reminded here of a dictum, possibly due to Littlewood, that in explaining math, a single triviality omitted is a trivial gap to fill, but two trivialities in a row can be deep. If to get from lecture 1 to lecture 3 requires the students keeping in mind a whole house of cards of connections before reaching the payoff, then there should be a payoff inside the house that is given in lecture 2.

Linear algebra has this exact relationship to abstract algebra. It starts with something that anyone with a little experience using math is familiar with: solving equations. It proceeds through something that, while apparently complicated, is also familiar: defining notation and some odd operations (matrices and row operations, multiplication, etc). This is, after all, part of solving equations too. Finally, it can lead to really abstract concepts such as vector spaces (= free modules), abstract linear spaces (= modules), change of basis (= conjugation, surely something to be learned before taking group theory!), and the first isomorphism theorem (the "rank-nullity theorem"). Problems in linear algebra can be written first to sound semi-physical or geometric, and then to make reference to concepts that were taught in such problems, and then to be fully abstract. By the end of a linear algebra course, students should have at least some foundation for thinking abstractly, as well as a big list of familiar references that will recur in abstract algebra.

So my response to your question is necessarily, "why not teach linear algebra before abstract algebra?" I think it's unfortunate that more algebra books do not use it as motivation.

Because the number of consumers of the linear algebra course (math majors, engineers, physics majors, economists, almost everyone else) is much much bigger than the number of consumers of abstract algebra course.

There are several aspects that contribute to the decision to invest quite a lot of time on linear algebra before introducing abstract algebra.

Firstly, there is the historical aspect. Linear algebra came first, and groups, rings and the rest of the gang came (considerably) later. University curricula change rather slowly and we still see the left-overs of past centuries historical developments.

There is also a question of practicality. Linear algebra services numerous fields and while abstract algebra is certainly of great importance, it can be argued (successfully) that linear algebra equips one with plenty of immediate tools for use in many areas.

Then there is the general fear of the abstract. Since many students find abstract material to be very difficult (for whatever psychological reasons), universities tend to cater to the 'needs' of the students by postponing the more abstract concepts till later. I know that I personally would have loved to have learned vector spaces as a special case of modules, and deduce plenty of the theory of linear spaces from the more general theorems of module theory, but it seems to not be a preferred path for most students since linear algebra is amenable to geometric visualizations, while general module theory is not.