linear independence

A set of vectors is linearly independent if no vector is expressible as a linear combination of other distinct vectors.

Formally,
a sequence of vectors $\vec{v_{1}}, \vec{v_{2}}, \dots, \vec{v_{k}}$ in a vector space $V$ are linearly dependent if there exist scalars $a_{1}, a_{2}, \dots, a_{k}$ not all zero such that

a_{1} \vec{v_{1}} + a_{2} \vec{v_{2}} + \dots + a_{k} \vec{v_{k}} = \vec{0}

where $\vec{0}$ is the zero vector.
Such vectors are linearly independent if the equation is only satisfied by scalars all zero.