Linear Algebra - Infinite fields and vector spaces with infinite vectors

[corey115]

Homework Statement

Let F be an infinite field (that is, a field with an infinite number of elements) and let V be a nontrivial vector space over F. Prove that V contains infinitely many vectors.

Homework Equations

The axioms for fields and vector spaces.

The Attempt at a Solution

I'm thinking this is easier than I'm making it. Can I say, at the very least, F is countably infinite, so then there exist an infinite amount of scalars to apply to V?

 

[HallsofIvy]

Yes, it really is that easy. Since V is a non-trivial vector space it contains a non-zero vector, v. And then for any a in F, av is in V. The "non-trivial" part of the proof is showing that if $a_1\ne a_2$ then $a_1v\ne a_2v$ but that is easy to show.