Linear independence and dependence - Imported Knowlpedia Content

Let $X$ be a vector space over $K$ , and let $M\subset X$ .

The set $M$ is linearly independent if for every finite subset $\{x_1,\dots,x_m\}\subset M$ and every choice of scalars $\alpha_1,\dots,\alpha_m\in K$ ,

\sum_{i=1}^m \alpha_i x_i = 0 \quad\Longrightarrow\quad \alpha_1=\cdots=\alpha_m=0.

If $M$ is not linearly independent, it is linearly dependent, i.e., there exists a finite subset and scalars, not all zero, giving a zero linear combination.

Linear independence is one of the two defining properties of a basis.

Examples

The standard unit vectors $e_1,\dots,e_n$ in $\mathbb{R}^n$ are linearly independent.
The set $\{(1,0),(2,0)\}\subset\mathbb{R}^2$ is linearly dependent since $2(1,0)-(2,0)=0$ .
Any set containing the zero vector is linearly dependent: if $0\in M$ , then $1\cdot 0=0$ is a nontrivial dependence.
The empty set is linearly independent (there is no finite nonempty subset to witness dependence).