Linear independence and dependence

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 $\alpha_1,\dots,\alpha_m\in K$,

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).