Definition:Linearly Dependent/Sequence/Real Vector Space

Definition
Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space.

Let $\mathbf 0 \in \R^n$ be the zero vector.

Let $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of vectors in $\R^n$.

Then $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ is linearly dependent iff:


 * $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

where not all $\lambda_k$ are equal to $0$.

That is, it is possible to find a linear combination of $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $\mathbf 0$.

Also see

 * Linearly Dependent Sequence of Vector Space


 * Linearly Independent Sequence on a Real Vector Space