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Math Help - Cylic Subspace

  1. #1
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    Cylic Subspace

    I have a question on cyclic subspaces.
    V is a finite dimensional space over the field F. There exists f:V->V
    It's given to me that , k is minimal, such that the set $(v,f(v),f^2(v),...,f^k(v))$ is dependent.

    Now I have to prove that

    1-)$D=(v,f(v),f^2(v),...,f^(k-1))$

    2-)D spans Z(v,f)

    I proved the first one which was very easy. But I am having a problem proving that it spans the cyclic subspace... Can someone give me a direction. Thanks
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  2. #2
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    Re: Cylic Subspace

    What is $D$, and what is Z(v,f)?
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  3. #3
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    Re: Cylic Subspace

    Sorry, I didn't write it very clear.
    The first one claims that D is independent

    Z(v,f) is cyclic subspace . in other words,
    Definition: V is a vector space over the field F. f:V->V is a linear operator. $v\inV$ , then
    $Z(v,f)=Span(v,f(v),f^2(v),...,f^i(v),...)$ is cyclic subspace of v on f

    The first claim is pretty easy , since we know k is minimal, then we would have that D is independent.

    But the second one I couldn't prove.
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