Linear Algebra Done Right · Problem 8A.5

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · null-spaces-powers

Problem

Suppose

T \in \mathcal{L}(V)

and

m

is a positive integer. Prove that

\dim \operatorname{null} T^m \leq m \dim \operatorname{null} T.

Hint: Exercise 21 in Section 3B may be useful.