A set 
of vectors in a vector
space over a field 
is linearly dependent if there are distinct vectors 
, where 
, and scalars 
, not all zero, such that
 |
Equivalently, at least one of these vectors is a linear combination of the others. A set containing the zero vector is therefore linearly dependent. A set for which no such nontrivial relation exists is linearly independent.
If the vectors are the columns of a matrix, they are linearly dependent precisely when the matrix rank is smaller than the number of columns.
