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Math Help - Proof

  1. #1
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    Proof

    Show that if U is a subspace of a vector space V that is in turn a subspace of a vector space W, then U is a subspace of W
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  2. #2
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    Quote Originally Posted by skittle View Post
    Show that if U is a subspace of a vector space V that is in turn a subspace of a vector space W, then U is a subspace of W
    U\subset V\subset W

    By transitivity, U\subset W.

    I think you can just do this.
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  3. #3
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    Just say everything in V is in W, and everything in U is in V.

    Therefore everything in U must be in W.
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  4. #4
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    What dwsmith and Prove It suggest shows that U is a subset of W. You still need to show it is a vector space by showing that it is closed under addition and scalar multiplication.
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    What dwsmith and Prove It suggest shows that U is a subset of W. You still need to show it is a vector space by showing that it is closed under addition and scalar multiplication.
    If U is a subset of W, and it is a vector space over the same field with addition and scalar multiplication identical with the restriction of the equivalent operations on W to U then it is a subspace of W. That U is a subspace of V a subspace of W is sufficient to justify this last requirement.

    CB
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