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Thread: Subspace

  1. #1
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    Subspace

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

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

    2. Let W be a subspace of V such that $\displaystyle U_1$ subset or equal to W and $\displaystyle U_2$ subset or equal to W. Prove that $\displaystyle U_1+U_2$ 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 $\displaystyle U_1$ and $\displaystyle U_2$ be subspaces of a vector space V. Define the sum of $\displaystyle U_1+U_2$ as follows:
    $\displaystyle U_1+U_2$ = {$\displaystyle u_1+u_2$: $\displaystyle u_1$ is an element of $\displaystyle U_1$, $\displaystyle u_2$ is an element of $\displaystyle U_2$}

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

    (1) $\displaystyle U_1 + U_2$ 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 $\displaystyle U_1$ subset or equal to W and $\displaystyle U_2$ subset or equal to W. Prove that $\displaystyle U_1+U_2$ subset or equal to W

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

    can you show this?
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