Linear independence

Given a set of vectors $v_{1},\ldots v_{n}$, the vectors are said to be linearly independent if there does not exists a nontrivial Linear combination of them that is equal to zero. In other words, there does not exists a set of constants $\alpha_{1},\ldots,\alpha_{n}\in \mathbb{R}$, not all zero, such that

If such a set exists, the vectors are instead linearly dependent. If two vectors $v$ and $w$ are linearly dependent, one is always a multiple of the other, which means that there exists a constant $c\in \mathbb{R}$ for which $v=cw$.