Prove: Invariant Subspaces are g(T)-Invariant

[hitmeoff]

Homework Statement

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).

Homework Equations

Cayley-Hamilton Theorem?

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 \forallv\epsilonW, T(v) \epsilon W

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

 

[Fredrik]

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.