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Thread: 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 $\displaystyle V$ is a subspace of $\displaystyle V$ 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 $\displaystyle X$ and $\displaystyle Y$ are subspaces of $\displaystyle V$ and$\displaystyle X \subset Y$.
    Then $\displaystyle X \bigcup Y = Y$.
    So$\displaystyle X \bigcup Y$ is a subspace of $\displaystyle V$

    But how would you the other way. Assuming first that the union of two subspaces, say $\displaystyle X$ and $\displaystyle Y$, of the vector space $\displaystyle V$. 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 $\displaystyle X\cup Y$ is a subspace and $\displaystyle a\in X\setminus Y~\&~b\in Y\setminus X$.
    We know that $\displaystyle a+b\in X\cup Y$.
    What is wrong with that?
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  5. #5
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    If $\displaystyle a \in X \backslash Y$ and $\displaystyle b \in Y \backslash X$ , then$\displaystyle X \backslash Y \neq \emptyset $ and $\displaystyle Y \backslash X \neq \emptyset$ . 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 $\displaystyle a+b\in X\text{ or }a+b\in Y?$
    Is it true that $\displaystyle -a\in X?$
    There is a contradiction there. What is it?
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