Definition – Linear (in)dependence.

Definition

Consider a linear combination of a finite set of vectors. If

$\sum\limits_{i=1}^{n} r_i e_i = r_1 e_1 + r_2 e_2 + ... + r_n e_n = 0$

Implies that $r_i = 0~\forall{i = 1, ..., n}$ , we say that the set $E$ of vectors is linearly independent. Conversely, if the above implies that $\exists{r_i} \ne 0,~~{i = 1, ..., n}$ , then we say that the set $E$ of vectors is linearly dependent.

Notes