Linearly Dependent Vectors

Let , , ..., be vectors in a vector space over a field . They are linearly dependent iff there exist scalars , , ..., in , not all zero, such that

(1)

If no such scalars exist, then the vectors are said to be linearly independent.

If is -dimensional, choose a vector basis and let be the th coordinate of . The condition for linear dependence is then

(2)

(3)

This matrix equation has a nontrivial solution iff its determinant is 0. Thus, in this -dimensional setting, the vectors are linearly dependent iff

(4)

and linearly independent otherwise.

Let and be vectors in , equipped with the standard dot product. Then the following three conditions are equivalent (Gray 1997).

1. and are linearly dependent.

2. .

3. The matrix has matrix rank less than two.