# I need help getting started

• Dec 13th 2010, 03:45 PM
Sw0rDz
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.
• Dec 14th 2010, 12:02 AM
Opalg
Quote:

Originally Posted by Sw0rDz
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.