Let $V$ be a [[vector space]] over the [[field]] $K$, and let $\boldsymbol v_{1}, \ldots,\boldsymbol v_{n}$ be [[Element of a Set|elements]] of $V$. We shall say that $\boldsymbol v_{1}, \ldots, \boldsymbol v_{n}$ are **linearly independent** over $K$ if zero is the only possible value for elements $a_{1}, \ldots, a_{n}$ in $K$ $ a_{1} \boldsymbol v_{1}+\cdots+a_{n} \boldsymbol v_{n}=O \iff a_i=0\ \forall i $ --- If $S$ is a [[set]] of linearly independent [[Vector|vectors]], then $\dim \left(\operatorname{span} (S)\right)= \vert S \vert$