Invariant subspaces under linear operators

[jimmypoopins]

Homework Statement

Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.

Homework Equations

U is invariant under a linear operator T if u in U implies T(u) is in U.

The Attempt at a Solution

Assume {0} does not equal U does not equal V. Let {u1,...,un} be a basis for U. Extend to a basis for V: {u1,...,un,v1,...,vm}. Since V does not equal {0}, m is greater than or equal to 1. Define a linear operator by T=v1, i=1,...n and T(vi)=v1, i=1,...,m. Then U is not invariant under T.

I think this is a counterexample to the contrapositive of the statement. does it work? (the contrapositive is If U does not equal {0} does not equal V, then U is not invariant under every operator on V, right?)

 

[morphism]

I think you have the right idea, but your write up has some typos in it so I'm not sure.