Linearly Independent Finite Family

Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$, 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 $v$ is 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 finite sum in $V$.