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need your help on linear transform

Irin
in Linear Algebra and Group Theory

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Given a linear transform T:V->V

It has to be proven that if ker(T^2) = ket(T)than Im(T^2) = Im(T)

and (the opposite) if Im(T^2) = Im(T) than ker(T^2) = ket(T)

Thank you in advance for the help

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