Span

Definition

Given a set of vectors $S$ in a vector space $(V,F)$ the span 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 $\{v_1,\cdots,v_m\}$ of vectors the span 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

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The span is vector subspace of $V$

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TODO: Proof

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