Difference between revisions of "Span, linear independence, linear dependence, basis and dimension"
(Created page with " ==Span== ===Definition=== Given a set of vectors {{M|S}} in a vector space {{M|(V,F)}} <math>\text{Span}(S)=\{\sum^n_{i=1}\lambda v_i|n\in\mathbb{N},\ v_i\in S,\ \lambda_i\i...") |
(Added basis.) |
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| − | + | This article includes information on linear dependence and independence | |
==Span== | ==Span== | ||
===Definition=== | ===Definition=== | ||
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It is very important that only finite linear combinations are in the span. | It is very important that only finite linear combinations are in the span. | ||
| + | ==Linear Dependence== | ||
| + | A set {{M|E}} in a [[Vector space|vector space]] {{M|(V,F)}} is linearly dependent if for any '''finite''' collection of elements of {{M|E}} that finite collection is linearly dependent | ||
| + | |||
| + | That is, <math>\forall n\in\mathbb{N}</math> given a subset <math>\{e_1,...,e_n\}\subset E</math> | ||
| + | |||
| + | There are solutions to <math>\sum^n_{i=1}e_i\alpha_i=0</math> where the <math>\alpha_i\in F</math> are not all zero. | ||
| + | |||
| + | ==Linear Independence== | ||
| + | If a set is not linearly dependent it is linearly independent, but formally: | ||
| + | |||
| + | For all finite subsets of a set {{M|E}}, we have only <math>\alpha_i=0\forall i</math> as solutions to <math>\sum^n_{i=1}e_i\alpha_i=0</math> | ||
| + | |||
| + | ==Basis== | ||
| + | Usually a basis will be a finite set, for example, <math>\{(1,0),(0,1)\}</math> is a basis of {{M|\mathbb{R}^2}}. | ||
| + | ===Finite case=== | ||
| + | Given a finite set {{M|B\subset V}}, {{M|B}} is a basis of {{M|V}} if <math>\text{span}(B)=V</math> and {{M|B}} is linearly independent. | ||
| + | ===Infinite case=== | ||
| + | A Hamel basis is any linearly independent subset of {{M|V}} that spans {{M|V}} - where linearly independent is given as above. | ||
| + | The definition of independence varies slightly from how it is usually given (I explicitly say for all finite subsets) it is just a stronger form. | ||
{{Definition|Vector Space}} | {{Definition|Vector Space}} | ||
{{Todo}} | {{Todo}} | ||
Revision as of 02:38, 8 March 2015
This article includes information on linear dependence and independence
Contents
Span
Definition
Given a set of vectors [ilmath]S[/ilmath] in a vector space [ilmath](V,F)[/ilmath]
[math]\text{Span}(S)=\{\sum^n_{i=1}\lambda v_i|n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\}[/math]
It is very important that only finite linear combinations are in the span.
Linear Dependence
A set [ilmath]E[/ilmath] in a vector space [ilmath](V,F)[/ilmath] is linearly dependent if for any finite collection of elements of [ilmath]E[/ilmath] that finite collection is linearly dependent
That is, [math]\forall n\in\mathbb{N}[/math] given a subset [math]\{e_1,...,e_n\}\subset E[/math]
There are solutions to [math]\sum^n_{i=1}e_i\alpha_i=0[/math] where the [math]\alpha_i\in F[/math] are not all zero.
Linear Independence
If a set is not linearly dependent it is linearly independent, but formally:
For all finite subsets of a set [ilmath]E[/ilmath], we have only [math]\alpha_i=0\forall i[/math] as solutions to [math]\sum^n_{i=1}e_i\alpha_i=0[/math]
Basis
Usually a basis will be a finite set, for example, [math]\{(1,0),(0,1)\}[/math] is a basis of [ilmath]\mathbb{R}^2[/ilmath].
Finite case
Given a finite set [ilmath]B\subset V[/ilmath], [ilmath]B[/ilmath] is a basis of [ilmath]V[/ilmath] if [math]\text{span}(B)=V[/math] and [ilmath]B[/ilmath] is linearly independent.
Infinite case
A Hamel basis is any linearly independent subset of [ilmath]V[/ilmath] that spans [ilmath]V[/ilmath] - where linearly independent is given as above.
The definition of independence varies slightly from how it is usually given (I explicitly say for all finite subsets) it is just a stronger form.
TODO:
