Vector space/Linearly independent/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Then a family of vectors ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, in ${\displaystyle {}V}$ is called linearly independent if an equation

${\displaystyle \sum _{i\in J}s_{i}v_{i}=0{\text{ with }}s_{i}\in K{\text{ for a finite subset }}J\subseteq I}$

is only possible if ${\displaystyle {}s_{i}=0}$ for all ${\displaystyle {}i}$.