# Definition A set of [[vector|vectors]] $S = \{\pmb{x}_1, \pmb{x}_2, \ldots \pmb{x}_m\} \in V$, where $V$ is a [[vector space]] over the [[field]] $\mathbb{F}$, is said to be *linearly independent* if it is impossible to find a set of $\{a^1, a^2, \ldots a^m\} \in \mathbb{F}^*$ (i.e. nonzero) for which: $a^1 \pmb{x}_1 + a^2 \pmb{x}_2 + \ldots a^m \pmb{x}_m = \pmb{0}$ In other words, this means that if at least one vector in $S$ can be written as a linear combination of the other vectors, then the set $S$ is *linearly dependent*. Otherwise, $S$ is linearly independent.