Starting with a certain quantum state | ψ ψ 0 ⟩ ⟩ , consider the set of states R ( n ^ ^ , ϕ ϕ ) | ψ ψ 0 ⟩ for all possible n ^ ^ and ϕ ϕ , i.e. the set of states that come about from rotating the starting state in every possible way. The linear span of that set is a vector space, and therefore the manner in which the rotation operators map one state onto another is a representation of the group of rotation operators.