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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


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