Linearly_dependent
Definition
A list of vectors in $V$ is called linearly dependent if it is not linearly_independent.
In other words a list $v_{1},\ldots,v_{m}$ of vectors in $V$ is linearly dependent if there exist $a_{1},\ldots,a_{m} \in \mathbb{F}$, not all 0, such that $a_{1}v_{1} + \cdots + a_{m}v_{m} = 0$.
Linear Dependence Lemma (Axler — Theorem 2.21)
Suppose $v_{1},\ldots,v_{m}$ is a linearly dependent list in $V$. Then there exists $j \in \{ 1,2,\ldots,m\}$ such that the following hold:
$v_{j} \in \text{ span }\left. \left( v_{1},\ldots,v_{j - 1} \right) \right.$;
if the $j^{\mathrm{\text{th}}}$ term is removed from $v_{1},\ldots,v_{m}$, the span of the remaining list equals $\text{span }\left. \left( v_{1},\ldots,v_{m} \right) \right.$.
For (1), this means that for $j \in \{ 1,2,\ldots,m\}$, there exists at least one vector which belongs to the span of the rest of the vectors.