Vector space/Generating system/Definition

Spanning system

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. A family ${}v_{i}\in V$, ${}i\in I$, is called a generating system (or spanning system) of ${}V$, if every vector ${}v\in V$ can be written as

${}v=\sum _{j\in J}s_{j}v_{j}\,,$

with a finite subfamily ${}J\subseteq I$, and with ${}s_{j}\in K$.