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 S in a vector space (V,F) the span[1] is defined as follows:
- Span(S)={n∑i=1λvi| n∈N, vi∈S, λi∈F}
It is very important that only finite linear combinations are in the span.
Span of a finite set of vectors
Given a finite set {v1,⋯,vm} of vectors the span[2] can be more simply written:
- Span({v1,⋯,vm})={λ1v1+⋯+λmvm| λi∈F}={m∑i=1λivi| λi∈F}
Immediate theorems
[Expand]
The span is vector subspace of V
