So what other stuff has the structure of a linear space but has elements that are not real or complex numbers?
In computational number theory you sometimes get people doing linear algebra on matrices made out of integers modulo a prime. Often the prime is 2, but larger ones are also used.
My guess is the elements have to be from a ring or maybe a field. Anyway something with a group operation on the whole set, another group operation on the set except for identity of the first group, distributive law between the two group operations.