Editing Linear independence (section)

== Definition ==
A sequence of vectors <math>\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k</math> from a [[vector space]] {{mvar|V}} is said to be ''linearly dependent'', if there exist [[Scalar (mathematics)|scalars]] <math>a_1, a_2, \dots, a_k,</math> not all zero, such that
:<math>a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_k\mathbf{v}_k = \mathbf{0},</math>
where <math>\mathbf{0}</math> denotes the zero vector.

This implies that at least one of the scalars is nonzero, say <math>a_1\ne 0</math>, and the above equation is able to be written as
:<math>\mathbf{v}_1 = \frac{-a_2}{a_1}\mathbf{v}_2 + \cdots + \frac{-a_k}{a_1} \mathbf{v}_k,</math>
if <math>k>1,</math> and <math>\mathbf{v}_1 = \mathbf{0}</math> if <math>k=1.</math>

Thus, a set of vectors is linearly dependent if and only if one of them is zero or a [[linear combination]] of the others.

A sequence of vectors <math>\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n</math> is said to be ''linearly independent'' if it is not linearly dependent, that is, if the equation
:<math>a_1\mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n\mathbf{v}_n = \mathbf{0},</math>
can only be satisfied by <math>a_i=0</math> for <math>i=1,\dots,n.</math> This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence.  In other words, a sequence of vectors is linearly independent if the only representation of <math>\mathbf 0</math> as a linear combination of its vectors is the trivial representation in which all the scalars <math display="inline">a_i</math> are zero.<ref>{{cite book|last1=Friedberg |last2=Insel |last3=Spence|first1=Stephen |first2=Arnold |first3=Lawrence|title=Linear Algebra|year=2003|publisher=Pearson, 4th Edition|isbn=0130084514|pages=48–49}}</ref> Even more concisely, a sequence of vectors is linearly independent if and only if <math>\mathbf 0</math> can be represented as a linear combination of its vectors in a unique way.

If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is ''linearly independent'' if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.

A sequence of vectors is linearly independent if  and only if it does not contain the same vector twice and the set of its vectors is linearly independent.

===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.

===Definition via span===
Let <math>V</math> be a vector space. A set <math>X \subseteq V</math> is ''linearly independent'' if and only if <math>X</math> is a [[Maximal and minimal elements|minimal element]] of
:<math>\{Y \subseteq V \mid X \subseteq \operatorname{Span}(Y)\}</math>
by the [[inclusion order]]. In contrast, <math>X</math> is ''linearly dependent'' if it has a proper subset whose span is a superset of <math>X</math>.