Linear Algebra Done Right · Problem 8A.21

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-operators

Problem

Suppose

T \in \mathcal{L}(V)

is such that

\operatorname{null} T^{\dim V - 1} \neq \operatorname{null} T^{\dim V}

. Prove that

T

is nilpotent and that

\dim \operatorname{null} T^k = k

for every integer

k

with

0 \leq k \leq \dim V