Difference between revisions of "Span"
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* <math>\text{Span}(\{v_1,\cdots,v_m\})=\left\{\lambda_1v_1+\cdots+\lambda_mv_m|\ \lambda_i\in F\right\}</math><math>=\left\{\sum^m_{i=1}\lambda_iv_i\Big|\ \lambda_i\in F\right\}</math> | * <math>\text{Span}(\{v_1,\cdots,v_m\})=\left\{\lambda_1v_1+\cdots+\lambda_mv_m|\ \lambda_i\in F\right\}</math><math>=\left\{\sum^m_{i=1}\lambda_iv_i\Big|\ \lambda_i\in F\right\}</math> | ||
| + | ==Immediate theorems== | ||
| + | {{Begin Theorem}} | ||
| + | The span is [[Subspace#Vector space|vector subspace]] of {{M|V}} | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|Proof}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
Latest revision as of 17:09, 28 May 2015
Definition
Given a set of vectors [ilmath]S[/ilmath] in a vector space [ilmath](V,F)[/ilmath] the span[1] is defined as follows:
- [math]\text{Span}(S)=\left\{\sum^n_{i=1}\lambda v_i\Big|\ n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\right\}[/math]
It is very important that only finite linear combinations are in the span.
Span of a finite set of vectors
Given a finite set [ilmath]\{v_1,\cdots,v_m\} [/ilmath] of vectors the span[2] can be more simply written:
- [math]\text{Span}(\{v_1,\cdots,v_m\})=\left\{\lambda_1v_1+\cdots+\lambda_mv_m|\ \lambda_i\in F\right\}[/math][math]=\left\{\sum^m_{i=1}\lambda_iv_i\Big|\ \lambda_i\in F\right\}[/math]
Immediate theorems
The span is vector subspace of [ilmath]V[/ilmath]
TODO: Proof
