Span(S + T) ⊇ Span(S) + Span(T)

Question

I'm given the following problem to solve but I can't figure out the solution

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Let V be a vector space. Let S and T be subsets (not necessarily subspaces) of V.

(a) Is it necessarily true that Span(S + T) ⊆ Span(S) + Span(T)? Justify your answer.

(b) Is it necessarily true that Span(S + T) ⊇ Span(S) + Span(T)? Justify your answer.

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To solve part (a) I've taken the definition of span and shown that the lhs is just a particular case of the rhs, i.e. when all the coefficients of $s_i$ are equal to all the coefficients of $t_i$

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Any suggestion for how to go for part (b)?

Answer 1

Consider $S=\{(1,0)\}$ and $T=\{(0,1)\}$ and compute LHS and RHS.

Answer 2

I don't think part b is true. For a counterexample take $S = \{v\}$ and $T = \{−v\}$.

Answer 3

Part a. Use

• $S\subseteq\operatorname{Span}(S)$;
• if $S\subseteq S'$ and $T\subseteq T'$ then $S+T\subseteq S'+T'$
• if $S\subseteq U$, with $U$ a subspace, then $\operatorname{Span}(S)\subseteq U$.

Part b. Consider $S=\{x\}$ and $T=\{-x\}$.