A [[Ordered Set]] of [[Vector|Vectors]] is [[Linear Dependence|Linearly Dependent]] if it is [[Logical Negation|not]] [[Linear Independence|Linearly Independent]]. $\huge L_{D}(\set{\vec{v_{1}},\cdots,\vec{v_{{m}}}}) = \neg L_{\op I} $ The [[Ordered Set]] $\set{\vec{v_{1}},\cdots,\vec{v_{{m}}}}$ is [[Linear Dependence|Linearly Dependent]] if the there exists a [[Trivial|Nontrivial]] solution (at least one $x_i\ne 0$) to the [[Homogeneous Equation]] created by the [[Linear Combination]] of the [[Ordered Set]]. For the [[Ordered Set]] of [[Vector|Vectors]], if you can express any one [[Vector]] as a [[Linear Combination]] of the rest, then that [[Ordered Set]] is [[Linear Dependence|Linearly Dependent]] $\huge \begin{align} \vec v_{i} &= x_{i}\vec v_{i} + x_{i-1}\vec v_{i-1} +x_{i+1}\vec{v_{i+1}}+ \cdots x_{m} \vec v_{m} \\ \vec 0 &= \pa{x_{i}\vec v_{i} + x_{i-1}\vec v_{i-1} +x_{i+1}\vec{v_{i+1}}+ \cdots x_{m} \vec v_{m}} - \vec v_{i} \\ &\therefore \set{\vec{v_{1}}+\dots+\vec{v_{{m}}}} \in L_{D} \end{align}$ #### Using [[Augmented Matrix|Augmented Matrices]] If ($\iff$) the [[Reduced Row Echelon Form]] of the [[Augmented Matrix]] form of the [[Homogeneous Equation]] formed by the [[Ordered Set]] of [[Vector|Vectors]] contains any rows that are all $0$ (eg. has a [[Free Parameter]]), then that [[Ordered Set]] is [[Linear Dependence|Linearly Dependent]].