# Cardinality of Finite Vector Space

## Theorem

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

Let $K$ be finite.

Let the dimension of $V$ be finite.

Then:

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

## Proof

Thus:

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

Thus:

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

$\blacksquare$