Hehe, you'd be suprised. I took an honors linear algebra course in my first year of university, and abstraction was the word of the day. Everything we did was for abstract vector spaces over arbitrary fields. We didn't look at the determinant as the signed volume of an n-cube, for example, but rather as a homomorphism from the group of matrices to the base field under multiplication. The cross product was never mentioned because it didn't generalize to arbitrary dimensions. It was good preparation for further abstract algebra courses, but not so much for others, like differential geometry, which rely heavily on geometric intuition.