In Israel the two mandatory courses of first year undergrad math are real-analysis and abstract linear algebra (I think it's the same in Europe). You define fields before you define vector spaces, and you give as examples F_p, Q, R, C.$\mathbb{F}_p, \mathbb{Q}, \mathbb{R}, \mathbb{C}$.
Once you teach what a linear transformation is, you have several examples involving F_2 $\mathbb{F}_2$ coming from computer science; e.g. Hamming code.
I'm not claiming that teaching first years abstract linear algebra is good (when I was an undergrad, half the students flunked first year math), just that if you do it you must have some non R $\mathbb{R} / C \mathbb{C}$ examples.
