Finite-Dimensional Vector Space

Let $K$ be a \reftext{def:field-c54-2026b}{field} and let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$.

The space $V$ is \textbf{finite-dimensional} if $V=\{0_{V}\}$, or if there exist a \reftext{def:natural-numbers-2026a}{natural number} $n$ and an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} $e\in V^{n}$ that is a \reftext{def:finite-basis-2026a}{basis} of $V$.