Editing Linear independence (section)

{{short description|Vectors whose linear combinations are nonzero}}
{{For|linear dependence of random variables|Covariance}}
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{{More citations needed|date=January 2019}}[[File:Vec-indep.png|thumb|right|Linearly independent vectors in <math>\R^3</math>]]
[[File:Vec-dep.png|thumb|right|Linearly dependent vectors in a plane in <math>\R^3.</math>]]

In the theory of [[vector space]]s,  a [[set (mathematics)|set]] of [[vector (mathematics)|vector]]s is said to be '''{{visible anchor|linearly independent}}''' if there exists no nontrivial [[linear combination]] of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be '''{{visible anchor|linearly dependent}}'''. These concepts are central to the definition of [[Dimension (vector space)|dimension]].<ref>G. E. Shilov, ''[https://books.google.com/books?id=5U6loPxlvQkC&q=dependent+OR+independent+OR+dependence+OR+independence Linear Algebra]'' (Trans. R. A. Silverman), Dover Publications, New York, 1977.</ref>
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* An [[indexed family]] of [[vector space|vector]]s is a '''linearly independent family''' if none of them can be written as a [[linear combination]] of finitely many other vectors in the family. A family of vectors which is not linearly independent is called '''linearly dependent'''.
* A [[set (mathematics)|set]] of vectors is a '''linearly independent set''' if the set (regarded as a family indexed by itself) is a linearly independent family.

These two notions are not equivalent: the difference being that in a family we allow repeated elements, while in a set we do not.  For example if <math>V</math> is a vector space, then the family <math>F : \{ 1, 2 \} \to V</math> such that <math>f(1) = v</math> and <math>f(2) = v</math> is a {{em|linearly dependent family}}, but the singleton set of the images of that family is <math>\{v\}</math> which is a {{em|linearly independent set}}.

Both notions are important and used in common, and sometimes even confused in the literature.
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For instance, in the [[3 dimensional space|three-dimensional]] [[real vector space]] <math>\R^3</math> we have the following example:

:<math>
\begin{matrix}
\mbox{independent}\qquad\\
\underbrace{
  \overbrace{
    \begin{bmatrix}0\\0\\1\end{bmatrix},
    \begin{bmatrix}0\\2\\-2\end{bmatrix},
    \begin{bmatrix}1\\-2\\1\end{bmatrix}
  },
  \begin{bmatrix}4\\2\\3\end{bmatrix}
}\\
\mbox{dependent}\\
\end{matrix}
</math>--><!-- weights 9, 5, 4 
Here the first three vectors are linearly independent; but the fourth vector equals 9 times the first plus 5 times the second plus 4 times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the set of vectors, not of any particular vector.  For example in this case we could just as well write the first vector as a linear combination of the last three.
:<math>\mathbf{v}_1=\left(-\frac{5}{9}\right)\mathbf{v}_2+\left(-\frac{4}{9}\right)\mathbf{v}_3+\frac{1}{9}\mathbf{v}_4 .</math>
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<!-- In [[probability theory]] and [[statistics]] there is an unrelated measure of linear dependence between [[random variable]]s. -->

A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors.  The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.