Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, 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 $v:[n]\to V$ be a map with values $v_{k}$.

The family $v$ is \textbf{linearly independent} if the only map $c:[n]\to K$ satisfying

is the map with $c_{k}=0$ for every $k\in[n]$, the sum being the \reftext{def:finite-sum-vector-space-2026a}{finite sum in $V$}.