Cardinality of Finite Vector Space
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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:
- $\size V = \size K^{\map \dim V}$
$\blacksquare$