Is a finite dimensional vector space always countable?

Question

Given a vector space of finite dimension, can we always find an injective map to the natural numbers?

z.

Answer 1

Depends over what field. If the field is finite or countable, e.g. $\mathbb Q$, then yes. If the field is uncountable, e.g. $\mathbb R$, then no.

The reason is that $|\mathbb F^n|=|\mathbb F|^n$, and if $|\mathbb F|\leq\aleph_0$ then $|\mathbb F|^n\leq\aleph_0$.