Linear Algebra Done Right · Problem 8A.2

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · null-spaces-powers

Problem

Suppose

T \in \mathcal{L}(V)

m

is a positive integer,

v \in V

, and

T^{m-1}v \neq 0

but

T^m v = 0

. Prove that

v, Tv, T^2 v, \ldots, T^{m-1}v

is linearly independent. The result in this exercise is used in the proof of 8.45.