Linear Algebra Done Right · Problem 8A.18

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-operators

Problem

Suppose T∈L(V)T \in \mathcal{L}(V) is nilpotent. Prove that T1+dim⁡range⁡T=0T^{1 + \dim \operatorname{range} T} = 0. If dim⁡range⁡T<dim⁡V−1\dim \operatorname{range} T < \dim V - 1, then this exercise improves 8.16.

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