Checking if a subspace is a submodule.

Question

Consider the regular representation of the cyclic group $\{ e,a,a^2 \} $ with the standard basis. I would like to figure out if the subspace $W$ spanned by $u=e_1-e_2, v=-e_1+e_3$ is a submodule. Consider the action of the three group elements on each basis vector: $$ \begin{array}{lll} e u=u, & e v=v \\ a u=e_2-e_3=-u-v, & a v=-e_2+e_1=u \\ a^2 u=e_3-e_1=v, & a^2 v=-e_3+e_2=-u-v \end{array} $$ Since the action of the group on each of the basis vectors is in the subspace, we can conclude that it is submodule. Is this correct?