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Math Help - I need help getting started

  1. #1
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    I need help getting started

    I'm trying to figure out if I need to use traits of Characteristic and/or Minimal Polynomials.

    Suppose that dim_F(V) = n and  \theta \epsilon Hom(V,V). Prove that if ker(\theta^{n-1}) \neq ker(\theta^n), then \theta is nilpotent, and dim(ker(\theta^j)) = j for all j \epsilon \{1, 2, ..., n\}

    I'm not looking for a solution, I simply want a head start.
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  2. #2
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    Quote Originally Posted by Sw0rDz View Post
    I'm trying to figure out if I need to use traits of Characteristic and/or Minimal Polynomials.

    Suppose that dim_F(V) = n and  \theta \epsilon Hom(V,V). Prove that if ker(\theta^{n-1}) \neq ker(\theta^n), then \theta is nilpotent, and dim(ker(\theta^j)) = j for all j \epsilon \{1, 2, ..., n\}

    I'm not looking for a solution, I simply want a head start.
    For k=0,1,2,..., let d_k = \dim\ker(\theta^k). Show that the given condition (which is equivalent to d_{n-1}<d_n) implies that the sequence (d_k) is strictly increasing for k\leqslant n. You should be able to do this directly, without needing to consider characteristic or minimal polynomials.
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