# Homework Help: Invariant Subspaces

1. Apr 13, 2010

### hitmeoff

1. The problem statement, all variables and given/known data
Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t).

2. Relevant equations
Cayley-Hamilton Theorem?

3. The attempt at a solution
Im not sure how to begin. Ok so g(t) is the characteristic polynomial of T. If W is a T-Invariant subspace of V, then $$\forall$$v$$\epsilon$$W, T(v) $$\epsilon$$ W

So for any T with a characteristic polynomial g(t), W is still T-Invariant...not sure if im even leading into the right direction. Any help on getting going with this proof?

2. Apr 13, 2010

### Fredrik

Staff Emeritus
Just have g(T) act on an arbitrary x in W, and show that the result is in W. This is much easier than you seem to be expecting.

Why are you saying that g is the characteristic polynomial of T? You said that g was arbitrary in the problem statement.