Linear Algebra Done Right · Problem 8A.19

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · null-spaces-range-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

is not nilpotent. Show that

V = \operatorname{null} T^{\dim V - 1} \oplus \operatorname{range} T^{\dim V - 1}.

For operators that are not nilpotent, this exercise improves 8.4.