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,
![c_1[x_(11); x_(21); |; x_(n1)]+c_2[x_(12); x_(22); |; x_(n2)]+...+c_n[x_(1n); x_(2n); |; x_(nn)]=[0; 0; |; 0]](/images/equations/LinearlyDependentVectors/NumberedEquation2.svg) |
(2)
|
![[x_(11) x_(12) ... x_(1n); x_(21) x_(22) ... x_(2n); | | ... |; x_(n1) x_(n2) ... x_(nn)][c_1; c_2; |; c_n]=[0; 0; |; 0].](/images/equations/LinearlyDependentVectors/NumberedEquation3.svg) |
(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 ![[p; q]](/images/equations/LinearlyDependentVectors/Inline15.svg)
has rank less than two.
