# Let S be a subset of an F-vector space V. Show that Span(S) is a subspace of V.

Let S be a subset of an F-vector space V. Show that Span(S) is a subspace of V.
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smallq9

Theorem. For any set of vectors $S=\left\{v1,\dots ,vn\right\}$ in a vector space V, span (S) is a subspace of V.
Proof. Let $u,w\in \left(S\right),k\in F.$

Then there exist $c1,\dots ,cn\in R,\dots ,cn\in R\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}k1,\dots ,kn\in Fk\in F$ such that
$u=c1v1+c2v2+\dots +cnvn$
and
$w=k1v1+k2v2+\dots +knvn$
Note that
$u+w=\left(c1v1+c2v2+\dots +cnvn\right)+\left(k1v1+k2v2+\dots +knvn\right)=\left(c1+k1\right)v1+\left(c2+k2\right)v2+\dots +\left(cn+kn\right)vn\in \left(S\right)$
Thus span (S) is closed under addition. M1)
$ku=k\left({c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\cdots +{c}_{n}{v}_{n}\right)=k\left({c}_{1}{v}_{1}\right)+k\left({c}_{2}{v}_{2}\right)+\cdots +k\left({c}_{n}{v}_{n}\right)=\left(k{c}_{1}\right){v}_{1}+\left(k{c}_{2}\right){v}_{2}+\cdots +\left(k{c}_{n}\right){v}_{n}\in \left(S\right)$
This hosws that span (S) is closed under scalar multiplication. Hence , span (S) is a subspace of V.