If no such scalars exist, then we say that the vectors are linearly independent. More generally, we say that a (possibly infinite) subset is linearly independent if all finite subsets of are linearly independent.
Let be a subset of a vector space. Then, is linearly dependent if and only if there exists a such that can be expressed as a linear combination of the vectors in the set (all the vectors in other than (http://planetmath.org/SetDifference)).
Remark. Linear independence can be defined more generally for modules over rings: if is a (left) module over a ring . A subset of is linearly independent if whenever for and , then .
|Date of creation||2013-03-22 11:58:40|
|Last modified on||2013-03-22 11:58:40|
|Last modified by||rmilson (146)|