# Linear Algebra - Linear Combination/Span

• Sep 13th 2008, 06:55 PM
leungsta
Linear 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, 07:06 PM
ThePerfectHacker
Quote:

Originally Posted by leungsta
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.

Let $\bold{x} \in \text{spam}(S_1)$ then it means $\bold{x} = a_1\bold{u_1}+...+a_n\bold{u}_n$ for some $\bold{u}_1,...,\bold{u}_n \in S_1 \subseteq S_2$. And thus, $\bold{x} \in \text{spam}(S_2)$.

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