MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Every time that I've taught someone what a vector space is, I first spent some time on what a field is, including examples of finite fields. It's rather hard for someone to claim you can take the real and imaginary part of a vector if you say "Ahh, but the field I'm thinking of is $\mathbb{F}_2$, so what does that mean?" It's always seemed to help with what a vector space is to first see what a field is, and get some experience manipulating axioms that are more familiar.