Definition:Linearly Dependent/Set/Complex Vector Space
< Definition:Linearly Dependent | Set(Redirected from Definition:Linearly Dependent Set of Complex Vectors)
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Definition
Let $\struct {\C^n, +, \cdot}_\C$ be a complex vector space.
Let $S \subseteq \C^n$.
Then $S$ is a linearly dependent set if and only if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.
That is, such that:
- $\ds \exists \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$
where $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0$.
Also see
- Definition:Linearly Independent Set of Complex Vectors: a subset of $\C^n$ which is not linearly dependent.