# Span

From Maths

## 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