Span of a Finite Family of Vectors

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}, and let $v\in V^{n}$ be an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} in $V$, with components $v_{k}$.

The \textbf{span} of $v$ is the set

where $c_{k}$ denotes the $k$-th component of the $n$-tuple $c$ in $K$, and the sum is the \reftext{def:finite-sum-vector-space-2026a}{finite sum in $V$} of the $n$-tuple whose $k$-th component is $c_{k}v_{k}$.