Definition:Linearly Independent/Set/Complex Vector Space

Definition

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

Let $S \subseteq \C^n$.

Then $S$ is a linearly independent set of complex vectors if and only if every finite sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

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

where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.

Also see

• Definition:Linearly Dependent Set of Complex Vectors: a subset of $\C^n$ which is not linearly independent.

Sources

• 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.2$ Vector Spaces over $C$