Definition:Linearly Dependent/Sequence/Real Vector Space

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Definition

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space.

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

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

Then $\sequence {\mathbf v_k}_{1 \mathop \le k \mathop \le n}$ is linearly dependent if and only if:

$\ds \exists \sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

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

Also see

• Linearly Dependent Sequence of Vector Space

• Linearly Independent Sequence on a Real Vector Space

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