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

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Answer 1

Theorem. For any set of vectors $S = {v 1, \dots, v n}$ in a vector space V, span (S) is a subspace of V.
Proof. Let $u, w \in (S), k \in F .$

Then there exist $c 1, \dots, c n \in R, \dots, c n \in R and k 1, \dots, k n \in F k \in F$ such that
$u = c 1 v 1 + c 2 v 2 + \dots + c n v n$
and
$w = k 1 v 1 + k 2 v 2 + \dots + k n v n$
Note that
$u + w = (c 1 v 1 + c 2 v 2 + \dots + c n v n) + (k 1 v 1 + k 2 v 2 + \dots + k n v n) = (c 1 + k 1) v 1 + (c 2 + k 2) v 2 + \dots + (c n + k n) v n \in (S)$
Thus span (S) is closed under addition. M1)
$k u = k (c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n}) = k (c_{1} v_{1}) + k (c_{2} v_{2}) + \dots + k (c_{n} v_{n}) = (k c_{1}) v_{1} + (k c_{2}) v_{2} + \dots + (k c_{n}) v_{n} \in (S)$
This hosws that span (S) is closed under scalar multiplication. Hence , 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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