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:


 * $\left\vert{V}\right\vert = \left\vert{K}\right\vert ^ {\dim\left({V}\right)}$

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

Thus:


 * $\left\vert{V}\right\vert = \left\vert{K^{\dim\left({V}\right)}}\right\vert$

By Cardinality of Cartesian Space, $\left\vert{K^{\dim\left({V}\right)}}\right\vert = \left\vert{K}\right\vert ^ {\dim\left({V}\right)}$.

Thus:


 * $\left\vert{V}\right\vert = \left\vert{K}\right\vert ^ {\dim\left({V}\right)}$