Show that if S1 (S subscript 1) and S2 are subsets of a vector space V such that S1 ⊆ S2, then span (S1) ⊆ span (S2). In particular, if S1 ⊆ S2 and span (S1) = V, deduce that span (S2) = V.

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- Sep 13th 2008, 05:55 PMleungstaLinear Algebra - Linear Combination/Span
Show that if S1 (S subscript 1) and S2 are subsets of a vector space V such that S1 ⊆ S2, then span (S1) ⊆ span (S2). In particular, if S1 ⊆ S2 and span (S1) = V, deduce that span (S2) = V.

- Sep 13th 2008, 06:06 PMThePerfectHacker
Let $\displaystyle \bold{x} \in \text{spam}(S_1)$ then it means $\displaystyle \bold{x} = a_1\bold{u_1}+...+a_n\bold{u}_n$ for some $\displaystyle \bold{u}_1,...,\bold{u}_n \in S_1 \subseteq S_2$. And thus, $\displaystyle \bold{x} \in \text{spam}(S_2)$.

If $\displaystyle S_1\subseteq S_2 \subseteq V \implies \text{spam}(S_1) = \boxed{V\subseteq \text{spam}(S_2) \subseteq V} \implies \text{spam}(S_2) = V$.