Finite Family Spanning a Vector Space

Let $K$ be a field, let $V$ be a vector space over $K$, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $v:[n]\to V$ be a map with values $v_{k}$.

The family $v$ spans $V$ if for every $u\in V$ there is a map $c:[n]\to K$ with

the sum being the finite sum in $V$.