Vector space/Linearly independent/Finite index set/Definition

Linearly independent

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. A family of vectors ${}v_{i}$, ${}i\in I$, (where ${}I$ denotes a finite index set) is called linearly independent, if an equation of the form

$\sum _{i\in I}s_{i}v_{i}=0{\text{ with }}s_{i}\in K$

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