Span, linear independence, linear dependence, basis and dimension
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:
