Span

Definition

Given a set of vectors [ilmath]S[/ilmath] in a vector space [ilmath](V,F)[/ilmath] the span[1] is defined as follows:

• $\text{Span}(S)=\left\{\sum^n_{i=1}\lambda v_i\Big|\ n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\right\}$

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:

• $\text{Span}(\{v_1,\cdots,v_m\})=\left\{\lambda_1v_1+\cdots+\lambda_mv_m|\ \lambda_i\in F\right\}$$=\left\{\sum^m_{i=1}\lambda_iv_i\Big|\ \lambda_i\in F\right\}$

Immediate theorems

The span is vector subspace of [ilmath]V[/ilmath]

TODO: Proof

References

1. Advanced Linear Algebra - Roman - Springer GTM (CHECK THIS REF)
2. Linear Algebra via Exterior Products - Sergei Winitzki