How do I approach this?

Show that the LTI system -2 = A z + CTu, z(T) =0 is controllable if and only if the LTI system i = Ax, y = Cx, x(0) = x0 is observable. a) Verify the two systems are dual in the following sense: The linear operator C : u -> z(0) for the first system is the adjoint of the linear operator C : x0 - y for the second system. b) (optional) Claim that controllability of (-A , -C ) is equivalent to that Range(CC) = Rn. c) (optional) Claim that observability of (A, C) is equivalent to that Null(CO) = {0}. d) Deduce the conclusion via duality