vectors , , ..., are linearly dependent iff there exist scalars , , ..., , not all zero, such that
(1)

If no such scalars exist, then the vectors are said to be linearly independent. In order to satisfy the criterion for linear dependence,
(2)

(3)

In order for this matrix equation to have a nontrivial solution, the determinant must be 0, so the vectors are linearly dependent if
(4)

and linearly independent otherwise.
Let and be dimensional vectors. Then the following three conditions are equivalent (Gray 1997).
1. and are linearly dependent.
2. .
3. The matrix has rank less than two.