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Math Help - A U B is not a subspace

  1. #1
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    A U B is not a subspace

    Hello everyone!

    I'm after this linear algebra question:
    A \text{ and }B \text{are subspaces of a vector space }V. \text{ show that }A\cup B \text{ are a subspace of }V \Leftrightarrow A\subseteq B \text{ or } B \subseteq A.

    Proving that 0\in V is easy, howeve I'm not sure how to write down the proof for the other 2 axioms on paper though my professor cleared the entire matter up.

    Thanks!
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  2. #2
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    Quote Originally Posted by rebghb View Post
    Hello everyone!

    I'm after this linear algebra question:
    A \text{ and }B \text{are subspaces of a vector space }V. \text{ show that }A\cup B \text{ are a subspace of }V \Leftrightarrow A\subseteq B \text{ or } B \subseteq A.

    Proving that 0\in V is easy, howeve I'm not sure how to write down the proof for the other 2 axioms on paper though my professor cleared the entire matter up.

    Thanks!

    Take a\in A\,,\,\,b\in B , now: where is a+b ?

    Tonio
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  3. #3
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    a + b is neither in A nor in B, I know, it's in A U B, but how to prove that!
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  4. #4
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    Quote Originally Posted by rebghb View Post
    a + b is neither in A nor in B, I know, it's in A U B, but how to prove that!
    That makes no sense. If a+ b is in A\cup B it must be in either A or B (or both). That's how A\cup B is defined!

    I think that you mean "if A\cup B is a subspace, then, for all a in A, b in B, a+ b must be in A\cup B and so must be in either A or B."
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