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Math Help - Subspace

  1. #1
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    Subspace

    Suppose U_1 and U_2 be subspaces of a vector space V. Define the sum of U_1+U_2 as follows:
    U_1+U_2 = { u_1+u_2: u_1 is an element of U_1, u_2 is an element of U_2}

    1. Prove that U_1+U_2 is a subspace of V.

    2. Let W be a subspace of V such that U_1 subset or equal to W and U_2 subset or equal to W. Prove that U_1+U_2 subset or equal to W

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by apple12 View Post
    Suppose U_1 and U_2 be subspaces of a vector space V. Define the sum of U_1+U_2 as follows:
    U_1+U_2 = { u_1+u_2: u_1 is an element of U_1, u_2 is an element of U_2}

    1. Prove that U_1+U_2 is a subspace of V.
    clearly (U_1 + U_2) \subseteq V. it remains to show

    (1) U_1 + U_2 is closed under addition and (2) closed under scalar multiplication

    do you know what that means? can you show it?

    2. Let W be a subspace of V such that U_1 subset or equal to W and U_2 subset or equal to W. Prove that U_1+U_2 subset or equal to W

    Thanks
    You need to show that x \in (U_1 + U_2) \implies x \in W

    can you show this?
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