I need help getting started

**Sw0rDz** · December 13th 2010, 03:45 PM

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.

**Opalg** · December 14th 2010, 12:02 AM

Originally Posted by Sw0rDz

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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