Cardinality of Finite Vector Space

Theorem

Let $K$ be a Galois field.

Let $V$ be a $K$-vector space.

Let the dimension of $V$ be finite.

Then:

$\size V = \size K^{\map \dim V}$

Proof

By Isomorphism from $\R^n$ via $n$-Term Sequence, $V$ is isomorphic to the $K$-vector space $K^{\map \dim V}$.

Thus:

$\size V = \size {K^{\map \dim V} }$

By Cardinality of Finite Cartesian Space:

$\size {K^{\map \dim V} } = \size K^{\map \dim V}$

Thus:

$\size V = \size K^{\map \dim V}$

$\blacksquare$