A basic question self-adjoint operator

Question

Why can't there exist a self-adjoint operator $T \in L(\Bbb R^3)$ such that $T(1,2,3) = (0,0,0)$ and $T(2,5,7) = (2,5,7)$. Hint is enough.

Answer 1

Hint: let $x=(1,2,3)$ and $y=(2,5,7)$. Suppose $T$ is self-adjoint and $Tx=(0,0,0)$ and $Ty=(2,5,7)$. Consider what happens to $\langle x,Ty\rangle$. (You'll want to make use of self-adjointness.)

Note: We could have picked $y$ to be (almost) anything and let $Ty$ be any vector that is not orthogonal to $x$ and this still would have worked.