Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$}, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $e:[n]\to V$ be a map with values $e_{k}$.

The family $e$ is a \textbf{basis} of $V$ if it is \reftext{def:linear-independence-finite-family-2026a}{linearly independent} and \reftext{def:spanning-finite-family-2026a}{spans} $V$.