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Math Help - Vector Subspaces Union problem

  1. #1
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    Vector Subspaces Union problem

    Hello,

    I am stuck with this problem.

    prove that the union of two subspaces of the vector space V is a subspace of V if and only if one of the subspaces is contained in the other.

    Thanks in advance for your help.
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  2. #2
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    Can you prove that if one subspace is contained in the other, the union is a subspace? That should be relatively straight-forward.
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  3. #3
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    Yes, I think I get that part. Is it right like this?
    If X and Y are subspaces of V and  X \subset Y.
    Then X \bigcup Y = Y.
    So  X \bigcup Y is a subspace of V

    But how would you the other way. Assuming first that the union of two subspaces, say X and Y, of the vector space V. Then how do you prove that one of them is contained in the other?
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  4. #4
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    Suppose that X\cup Y is a subspace and a\in X\setminus Y~\&~b\in Y\setminus X.
    We know that a+b\in X\cup Y.
    What is wrong with that?
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  5. #5
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    If a \in X \backslash Y and b \in Y \backslash X , then  X \backslash Y \neq \emptyset and Y \backslash X \neq \emptyset . That would be assuming that the subspaces are not contained.... but I am still lost on how to continue.
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  6. #6
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    Is it true that a+b\in X\text{ or }a+b\in Y?
    Is it true that -a\in X?
    There is a contradiction there. What is it?
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