Cardinality of Finite Vector Space

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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

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 Cartesian Space:

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

Thus:

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

$\blacksquare$