Linear Dependence

linear-algebra

Definition

Linear Dependence

Let $V$ be a vector space over a field $F$. A set $S\subseteq V$ is linearly dependent if it is not linearly independent.

Equivalently, there exist distinct $v_1,\dots ,v_k\in S$ and scalars ${\alpha }_i\in F$, not all zero, such that

$$\sum _{i=1}^k{\alpha }_i{v}_i=0.$$

Equivalently, some vector in $S$ lies in the linear span of the others.