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Assume $T$ is a normal operator on the complex finite dimensional inner product space $(V,\langle\,\cdot ,\,\cdot\rangle)$. Prove that $Range(T^k)=Range(T)$ and $Ker(T^k)=Ker(T)$ for all natural numbers k.

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    First try showing these statements to be true assuming that $T$ is diagonal.2012-12-01

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