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

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

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