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Thread: 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 $\displaystyle S^0$ as
    $\displaystyle S^0 = \{f \in V*: f(x) = 0 \forall x \in S\}$, where $\displaystyle S$ is a subset of finite-dimensional vector space $\displaystyle V$.

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

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

    Question 2: Let $\displaystyle T$ be a linear operator on $\displaystyle V$ and let $\displaystyle W$ be a subspace of $\displaystyle V$. Prove that $\displaystyle W$ is $\displaystyle T$-invariant if and only if $\displaystyle W^0$ is $\displaystyle T^t$-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 $\displaystyle S^0$ as
    $\displaystyle S^0 = \{f \in V*: f(x) = 0 \forall x \in S\}$, where $\displaystyle S$ is a subset of finite-dimensional vector space $\displaystyle V$.

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

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

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