(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose V is a vector space. Let E,F be subsets of V. Show [itex]E \subseteq F \Leftrightarrow L(E) \subseteq L(F)[/itex]

3. The attempt at a solution

Let [itex] x \in L(E)[/itex], there are scalars [itex]q_i[/itex] such that [itex]x = \sum_{i} q_i p_i[/itex] where [itex]p_i \in E[/itex]. [itex]p_i \in F[/itex] because [itex]E \subseteq F[/itex]. Thus it is shown that [itex]x \in L(F)[/itex]. From this result, L(E) is a subset of L(F).

First of all, I'm not sure if this is a convincing proof or if I have notated it correctly. Second, I don't think I understand the proof very well. For example, why is it (if this proof is true) that [itex]p_i \in E[/itex]? Similarly, is [itex]q_i \in E[/itex]? If so, how is this known? Mostly, what I think I'd like to see is a less mathy and more wordy explanation of what's going on here.

Thanks!

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# Spans in Linear Algebra

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