Suppose T ∈ L(V) and U is a subspace of V. Prove that U is invariant under T if and only if U⊥ is invariant under T∗. Now for reference, L(V) is the set of transformations that map v (a vector) from V to V. T* is the adjoint operator. The case where the dimension of U is less than V bugs me. How can U⊥ be invariant under T* when T* maps from U⊥ to V unless mapping to V also counts as mapping to U⊥ since U⊥ is in V itself. Now when I say map to V, I mean lets say V is 3 dimensional and U⊥ is 1 dimensional. Then u=(x3) gets mapped (by T*) to a vector with three nonzero components. Does this count as mapping from U⊥ to U⊥? I'm just a bit confused about this, and any help will be greatly appreciated. I guess that I'm confused enough that my post doesn't make much sense, so in that case, could someone nudge me in the right direction (ie give a good hint)? Thank you!