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Difference between revisions of "Linear independence"

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One of the main concepts in [[Linear-algebra(2)|linear algebra]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592901.png" /> be a [[Vector space|vector space]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592902.png" />; the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592903.png" /> are said to be linearly independent if
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592904.png" /></td> </tr></table>
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One of the main concepts in [[Linear-algebra(2)|linear algebra]]. Let $V$ be a [[vector space]] over a field $K$; the vectors $a_1,\ldots,a_n$ are said to be linearly independent if
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$$
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k_1 a_1 + \cdots + k_n a_n\neq 0
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$$
  
for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592905.png" /> except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592906.png" />. Otherwise the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592907.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592908.png" />) are said to be linearly dependent. The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l0592909.png" /> are linearly dependent if and only if at least one of them is a linear combination of the others. An infinite subset of vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929010.png" /> is said to be linearly dependent if some finite subset of it is linearly dependent, and linearly independent if any finite subset of it is linearly independent. The number of elements (the cardinality) of a maximal linearly independent subset of a space does not depend on the choice of this subset and is called the rank, or dimension, of the space, and the subset itself is called a [[Basis|basis]] (or base).
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for any set $k_i \in K$ except $k_1 = \cdots = k_n = 0$. Otherwise the vectors $a_1,\ldots,a_n$ ($n > 1$) are said to be ''linearly dependent''. The vectors $a_1,\ldots,a_n$ are linearly dependent if and only if at least one of them is a linear combination of the others. An infinite subset of vectors of $V$ is said to be linearly dependent if some finite subset of it is linearly dependent, and linearly independent if any finite subset of it is linearly independent. The number of elements (the cardinality) of a maximal linearly independent subset of a space does not depend on the choice of this subset and is called the rank, or dimension, of the space, and the subset itself is called a [[basis]] (or base).
  
In the special case when the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929011.png" /> are elements of some number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929013.png" /> is a subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929014.png" />, there arises the concept of linear independence of numbers. Linear independence of numbers over the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929015.png" /> can be regarded as a generalization of the concept of irrationality (cf. [[Irrational number|Irrational number]]). Thus, the two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929017.png" /> are linearly independent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059290/l05929018.png" /> is irrational.
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In the special case when the vectors $a_1,\ldots,a_n$ are elements of some number field $K$ and $k$ is a subfield of $K$, there arises the concept of linear independence of numbers. Linear independence of numbers over the field of rational numbers $\mathbb{Q}$ can be regarded as a generalization of the concept of irrationality (cf. [[Irrational number]]). Thus, the two numbers $\alpha$ and $1$ are linearly independent if and only if $\alpha$ is irrational.
  
 
The concepts of linear dependence and independence of elements have also been introduced for Abelian groups and modules.
 
The concepts of linear dependence and independence of elements have also been introduced for Abelian groups and modules.
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====Comments====
 
====Comments====
Abstract dependence relations are also known as matroids, cf. [[#References|[a1]]] and [[Matroid|Matroid]].
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Abstract dependence relations are also known as matroids, cf. [[#References|[a1]]] and [[Matroid]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.A. Welsh,  "Matroid theory" , Acad. Press  (1976)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.A. Welsh,  "Matroid theory" , Acad. Press  (1976)</TD></TR>
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</table>

Revision as of 19:40, 20 November 2014


One of the main concepts in linear algebra. Let $V$ be a vector space over a field $K$; the vectors $a_1,\ldots,a_n$ are said to be linearly independent if $$ k_1 a_1 + \cdots + k_n a_n\neq 0 $$

for any set $k_i \in K$ except $k_1 = \cdots = k_n = 0$. Otherwise the vectors $a_1,\ldots,a_n$ ($n > 1$) are said to be linearly dependent. The vectors $a_1,\ldots,a_n$ are linearly dependent if and only if at least one of them is a linear combination of the others. An infinite subset of vectors of $V$ is said to be linearly dependent if some finite subset of it is linearly dependent, and linearly independent if any finite subset of it is linearly independent. The number of elements (the cardinality) of a maximal linearly independent subset of a space does not depend on the choice of this subset and is called the rank, or dimension, of the space, and the subset itself is called a basis (or base).

In the special case when the vectors $a_1,\ldots,a_n$ are elements of some number field $K$ and $k$ is a subfield of $K$, there arises the concept of linear independence of numbers. Linear independence of numbers over the field of rational numbers $\mathbb{Q}$ can be regarded as a generalization of the concept of irrationality (cf. Irrational number). Thus, the two numbers $\alpha$ and $1$ are linearly independent if and only if $\alpha$ is irrational.

The concepts of linear dependence and independence of elements have also been introduced for Abelian groups and modules.

Linear dependence is a special case of a wider concept, that of an abstract dependence relation on a set.


Comments

Abstract dependence relations are also known as matroids, cf. [a1] and Matroid.

References

[a1] D.J.A. Welsh, "Matroid theory" , Acad. Press (1976)
How to Cite This Entry:
Linear independence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_independence&oldid=11370
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article