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    Let \text{Sp}( \cdot ) denote the span of a subset of a vector space V. Let S and T be subsets of V.

    If I have a vector v\in \text{Sp}(S \cup T) and v\in \text{Sp}(S) + \text{Sp}(T) then does this show that \text{Sp}(S \cup T) = \text{Sp}(S) + \text{Sp}(T) ?
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    Its christmas soon =/ Get pissed and stop worrying about maths
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    If v \in U \implies v \in W then U \subset W, and if, further, v \in W \implies v \in U, then U=W.

    Think of what it means for v \in V to be a member of \langle S \cup T \rangle and what it means for it to be a member of \langle S \rangle + \langle T \rangle (pardon the different notation!). Do these coincide? If so, then they're equal.
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    without writing anything what would you guess?
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      (Original post by TheEd)
      If I have a vector v\in \text{Sp}(S \cup T) and v\in \text{Sp}(S) + \text{Sp}(T) then does this show that \text{Sp}(S \cup T) = \text{Sp}(S) + \text{Sp}(T) ?
      No, it's more like this has to be true for all v\in \text{Sp}(S \cup T), and even then, you would have just shown \text{Sp}(S \cup T) \subset \text{Sp}(S) + \text{Sp}(T); you then need to show the other direction to prove equality.
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      (Original post by nuodai)
      If v \in U \implies v \in W then U \subset W, and if, further, v \in W \implies v \in U, then U=W.

      Think of what it means for v \in V to be a member of \langle S \cup T \rangle and what it means for it to be a member of \langle S \rangle + \langle T \rangle (pardon the different notation!). Do these coincide? If so, then they're equal.
      So I need to show the other direction.

      A different question:

      If I have a vector space V and subspaces U and W of V and I want to show that V = U+W then is the following true?

      If v = u+w \in U+W for all v \in V and u\in U, w\in W then V= U + W ?
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        (Original post by TheEd)

        If v = u+w \in U+W for all v \in V and u\in U, w\in W then V= U + W ?
        False, you are never going to have v = u + w for all possible u and w. Rather, for each v, there simply has to exist some u and w, such that v = u + w.
       
       
       
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