Prove or give a counter example:

Let V be a finite dimensional vector space over F.

Let T be a linear operator from V to V.

Let W be a T invariant subspace of V.

Then there exists a subspace U that is also T invariant and V = W (+) U (direct sum).

I know there always exists a subspace U such that the direct sum of W + U = V but I don't know if there exists a T invariant U.

Thanks