Definition:Linearly Independent/Sequence/Real Vector Space

From ProofWiki
Jump to navigation Jump to search


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

Let $\left\langle{\mathbf v_n}\right\rangle$ be a sequence of vectors in $\R^n$.

Then $\left\langle{\mathbf v_n}\right\rangle$ is linearly independent if and only if:

$\displaystyle \forall \left\langle{\lambda_n}\right\rangle \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$

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

Also see