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

Yasmin 2021-01-05 Answered
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
Answered 2021-01-06 Author has 106 answers

Theorem. For any set of vectors S={v1,,vn} in a vector space V, span (S) is a subspace of V.
Proof. Let u,w(S),kF.

Then there exist c1,,cnR,,cnRandk1,,knFkF such that
u=c1v1+c2v2++cnvn
and
w=k1v1+k2v2++knvn
Note that
u+w=(c1v1+c2v2++cnvn)+(k1v1+k2v2++knvn)=(c1+k1)v1+(c2+k2)v2++(cn+kn)vn(S)
Thus span (S) is closed under addition. M1)
ku=k(c1v1+c2v2++cnvn)=k(c1v1)+k(c2v2)++k(cnvn)=(kc1)v1+(kc2)v2++(kcn)vn(S)
This hosws that span (S) is closed under scalar multiplication. Hence , span (S) is a subspace of V.

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