# linear independence

**linear independence** A fundamental concept in mathematics. Let *x*_{1}, *x*_{2},…, *x _{n}*

be

*m*-component vectors. These vectors are linearly independent if for some scalars α

_{1}, α

_{2},…, α

*,*

_{n}implies α

_{1}= α

_{2}= … = α

*= 0*

_{n}Otherwise the vectors are said to be

*linearly dependent*, i.e. at least one of the vectors can be written as a linear combination of the others. The importance of a linearly independent set of vectors is that, providing there are enough of them, any arbitrary vector can be represented uniquely in terms of them.

A similar concept applies to functions

*f*

_{1}(

*x*),

*f*

_{2}(

*x*),…,

*f*(

_{n}*x*) defined on an interval [

*a*,

*b*], which are linearly independent if for some scalars α

_{1}, α

_{2},…, α

*, the condition, for all*

_{n}*x*in [

*a*,

*b*],

implies α

_{1}= α

_{2}= … = α

*= 0*

_{n}#### More From encyclopedia.com

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**linear independence**