Thread: Direct sum question

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

Originally Posted by durrrrrrrr

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.

Hint: What would happen if T maps U into W? – and can you construct an example of a T which does that?