Linear Algebra Done Right · Problem 8A.3

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · null-spaces-range

Problem

Suppose T∈L(V)T \in \mathcal{L}(V). Prove that V=null⁡T⊕range⁡T  ⟺  null⁡T2=null⁡T.V = \operatorname{null} T \oplus \operatorname{range} T \iff \operatorname{null} T^2 = \operatorname{null} T.

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