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**1. Homework Statement**

Prove that if T in L(V) is normal, then Ker(T

^{k}) = Ker(T) and Im(T

^{k}) = Im(T) for every positive integer k.

**2. Homework Equations**

**3. The Attempt at a Solution**

Since T is normal, I know that TT* = T*T, and also that ||Tv|| = ||T*v|| and <Tv, Tv> = <T*v, T*v>.

Ker(T) is the set of v in V such that Tv = 0. Ker(T

^{k}) is the set of v in V such that T

^{k}v = 0. To prove the first part, I want to show that T

^{k}v = 0 iff Tv = 0.

I'm not really sure where the fact that T is normal plays into this. I assume that proving the second part about Im(T) is done in much the same way. Thanks, as always, for any help.