1. The problem statement, all variables and given/known data Prove that there does not exist a self-adjoint operator T ∈ L(R3) such that T(1, 2, 3) = (0, 0, 0) and T(2, 5, 7) = (2, 5, 7). 2. Relevant equations 3. The attempt at a solution I'm having trouble seeing that there is an actual operator, self adjoint or not, that can do this. I mean, according to this, T maps from R^3 to R^3 but because it maps to R^3 and not to any subspace of R^3 (since it can map (2 5 7 ) to itself), its nullspace must be dim 0. The thing is, since (1 2 3) cannot be in the nullspace of T, T must be a zero transformation or its matrix might have [1 1 -1] for all three rows. But that can't happen if T(2 5 7)=(2 5 7). I mean, if T cannot exist, then neither can its adjoint.