Definition:Linearly Independent/Sequence/Real Vector Space

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Definition

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 if and only if:

$\ds \forall \sequence {\lambda_n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$

where $\mathbf 0 \in \R^n$ is the zero vector and $0 \in \R$ is the zero scalar.

Also see

• Linearly Dependent Sequence of Vector Space

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