- #1

- 1,028

- 83

## Homework Statement

"Let ##T## be a linear operator on a finite-dimensional vector space ##V## over an infinite field ##F##. Prove that ##T## is cyclic iff there are finitely many ##T##-invariant subspaces.

## Homework Equations

__T is a cyclic operator on V if:__there exists a ##v\in V## such that ##\langle T,v \rangle = V##

__A subspace W of V is T-invariant if:__for all ##w\in W##, ##T(w)\in W##

## The Attempt at a Solution

We prove the trivial case first. Suppose ##T## is a cyclic operator on ##\{0\}## the zero subspace. Then ##T(0)=0\in \{0\}##, and so, ##\langle T, 0 \rangle##. Thus, there is only one T-invariant subspace of the zero subspace.

I can't prove it the other way, though, and I'm not sure how to proceed.