Linear Independence

linear-algebra

Definition

Linear Independence

Let $V$ be a vector space over a field $F$. A set $S\subseteq V$ is linearly independent if the only way to express $0$ as a finite linear combination of distinct vectors from $S$ is the trivial combination:

$${\alpha }_1{v}_1+\cdots +{\alpha }_k{v}_k=0,v_i\in S\text{ distinct}⟹{\alpha }_1=\cdots ={\alpha }_k=0.$$

Otherwise $S$ is linearly dependent.

Equivalently, no vector in $S$ lies in the span of the others.