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Thread: Linear Algebra help

  1. #1
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    Linear Algebra help

    Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

    Prove the following statements
    a) S^0 is a subspace of L(v,w)
    b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.
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  2. #2
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    Quote Originally Posted by JCIR View Post
    Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

    Prove the following statements
    a) S^0 is a subspace of L(v,w)
    b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.
    What do you need to show that $\displaystyle S^0$ is a subspace?
    You need to show it is closed under vector addition and scalar multiplication.
    Can you show that?
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    What do you need to show that $\displaystyle S^0$ is a subspace?
    You need to show it is closed under vector addition and scalar multiplication.
    Can you show that?
    It will be helpful if you can show condition 2. I will be able to show that 0 is in there and the it is closed under scalar multiplication.

    Mainly part b is the one I dont get.
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  4. #4
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    Quote Originally Posted by JCIR View Post
    Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

    b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.
    You want to prove $\displaystyle S^0_2 \subseteq S_0^1$. The way we show this is by showing if $\displaystyle T\in S_2^0$ then $\displaystyle T \in S_1^0$. By definition $\displaystyle T \in S_2^0$ means $\displaystyle T(\bold{x}) = \bold{0}$ for all $\displaystyle \bold{x}\in S_2$. But $\displaystyle S_1\subseteq S_2$ so $\displaystyle T(\bold{x}) = \bold{0}$ for all $\displaystyle \bold{x} \in S_1$. Thus, $\displaystyle T\in S_1^0$.
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