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 $i$.