\begin{definition}[Definition:Linearly Independent/Sequence/Real Vector Space]
Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space.
Let $\sequence {\mathbf v_n}$ be a sequence of vectors in $\R^n$.
Then $\sequence {\mathbf v_n}$ is '''linearly independent''' {{iff}}:
:$\ds \forall \sequence {\lambda_n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$
where $\mathbf 0 \in \R^n$ is the zero vector and $0 \in \R$ is the zero scalar.
\end{definition}