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Math Help - Annihilators and dual spaces

  1. #1
    Member Last_Singularity's Avatar
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    Annihilators and dual spaces

    For these problems:
    1. Define the annihilator of S^0 as
    S^0 = \{f \in V*: f(x) = 0 \forall x \in S\}, where S is a subset of finite-dimensional vector space V.

    2. Let V,W be finite-dimensional vector spaces with ordered bases \beta, \gamma respectively. Then for any linear mapping T: V \rightarrow W, the mapping T^t: W* \rightarrow V* is defined by T^t(g)=gT for all g \in W* with the property that [T^t]_{\gamma *}^{\beta *} = ([T]_\beta^gamma)^t.

    Question 1: Suppose that W is a finite-dimensional vector space and T: V \rightarrow W is linear. Prove that N(T^t) = (R(T))^0.

    Question 2: Let T be a linear operator on V and let W be a subspace of V. Prove that W is T-invariant if and only if W^0 is T^t-invariant.
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  2. #2
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    Quote Originally Posted by Last_Singularity View Post
    For these problems:
    1. Define the annihilator of S^0 as
    S^0 = \{f \in V*: f(x) = 0 \forall x \in S\}, where S is a subset of finite-dimensional vector space V.

    2. Let V,W be finite-dimensional vector spaces with ordered bases \beta, \gamma respectively. Then for any linear mapping T: V \rightarrow W, the mapping T^t: W* \rightarrow V* is defined by T^t(g)=gT for all g \in W* with the property that [T^t]_{\gamma *}^{\beta *} = ([T]_\beta^gamma)^t.

    Question 1: Suppose that W is a finite-dimensional vector space and T: V \rightarrow W is linear. Prove that N(T^t) = (R(T))^0.

    Question 2: Let T be a linear operator on V and let W be a subspace of V. Prove that W is T-invariant if and only if W^0 is T^t-invariant.


    All this doesn't make much sense: you're going to define the annihilator of S^o OR you're going to define the annihilator S^o nof some set S?
    I'm almost sure it must be the latter. Now, what is V*? the v.s. of all linear functionals on V?

    later on you talk of N(T^t)...what is this? The nullity or kernel of T^t?

    Be more careful and if you want write again your question.

    Tonio
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  3. #3
    Member Last_Singularity's Avatar
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    I got it solved - thanks!
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