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Math Help - Linear Algebra - Linear Combination/Span

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by leungsta View Post
    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.
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