Def Linear Combination and Span

If $V$ is a vector space and $S=\{v_1,\dots ,v_n\}$ is a a finite collection of vectors:

A linear combination is any form: $\sum _{i=1}^n{c}_i{v}_i$ with $c\in \mathbb{R}$

Given and arbitrary $T\subseteq V$, span($T$) is the set of all finite linear combinations of elements in T. By convention, span($\mathrm{\varnothing }$)=

$T$ is a spanning set in $V$ if $V$ = span($T$)

Span Theorems

If $V$ is a vector space and $S\subseteq V$ then:

1. span($S$) is a subspace of $V$
2. $S\subseteq span(S)$
3. span($S$) is the minimal subspace containing $S$

by 3. we have: $S\subseteq span(W)⟹span(S)\subseteq span(W)$