Editing Linear independence (section)

===Infinite case===
An infinite set of vectors is ''linearly independent'' if every finite [[subset]] is linearly independent. This definition applies also to finite sets of vectors, since a finite set is a finite subset of itself, and every subset of a linearly independent set is also linearly independent. 

Conversely, an infinite set of vectors is ''linearly dependent'' if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.

An [[indexed family]] of vectors is ''linearly independent'' if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said to be ''linearly dependent''.

A set of vectors which is linearly independent and [[linear span|spans]] some vector space, forms a [[basis (linear algebra)|basis]] for that vector space. For example, the vector space of all [[polynomial]]s in {{mvar|x}} over the reals has the (infinite) subset {{math|1={1, ''x'', ''x''<sup>2</sup>, ...} }} as a basis.