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Nilpotent endomorphism/Characterization on basis/Fact/Proof

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Proof

From (1) to (2) is clear. From (2) to (3). Let be a basis (or a finite generating system), and let be such that

Then

fulfills the property for every generator. From (3) to (4) is clear. From (4) to (1). For , we have

Due to the linearity of , we have

therefore,