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**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. Assume that V is finite dimensional.

**The attempt at a solution**

I really think that I should be able to produce a counterexample, however even if I find an appropriate subspace, I have NO idea how to show that U is invariant under

*every*operator on V. There are an arbitrary number of them!

One thing I was thinking was use a one dimensional subspace for my U, because then every transformation of a vector u in U would be a constant times u. In other words, T(u) = cu.

Does anyone know if I'm on the right track?