Vector space/Linearly independent/Definition

Linearly independent

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

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

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