Nilpotent Operator

1. Homework Statement

If T is a nilpotent transformation from V -> V, V - finite dimensional vector space.

show that $$a_{0}+a_{1}T+\cdots+a_{k}T^{k}$$ is invertible. $$a_{0}$$ nonzero.

Im having trouble finding the inverse, I know for 1+T+$$\cdots$$+T$$^{m-1}$$

the inverse is (1-T),where T$$^{m}$$=0. I also tried $$a_{0}^{-1}T^{m-1}$$
but this gives me $$T^{m-1}$$

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Dick
Homework Helper
Write that as -a0*I=a_1*T+a_2*T^2+...+a_k*T^k. Is that enough of a hint? If not notice the right side has a common factor.

I see that it shows T to be invertible but how does it show that the original polynomial

is invertible?

Dick
Homework Helper
I see that it shows T to be invertible but how does it show that the original polynomial

is invertible?
Good point. I was solving the wrong problem.

Hurkyl
Staff Emeritus
Gold Member
I also tried $$a_{0}^{-1}T^{k-1}$$
but this gives me $$T^{k-1}$$
Okay, so you figured to annihilate all of the terms of your polynomial except the T^{k-1} term. What else can you annihilate?

Dick