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.

Once you teach what a linear transformation is, you have several examples involving 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 / C examples.

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 $\mathbb{F}_p, \mathbb{Q}, \mathbb{R}, \mathbb{C}$.

Once you teach what a linear transformation is, you have several examples involving $\mathbb{F}_2$ 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 $\mathbb{R} / \mathbb{C}$ examples.