- #1
*melinda*
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- 0
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?
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?