P_\mathbf{W}(P_\mathbf{W}\mathbf{v}) = P_\mathbf{W}\sum_{i=1}^m \langle \mathbf{e}_i,\mathbf{v}\rangle \mathbf{e}_i = \sum_{i=1}^m \langle \mathbf{e}_i,\mathbf{v}\rangle P_\mathbf{W}\mathbf{e}_i = \sum_{i=1}^m \langle \mathbf{e}_i,\mathbf{v}\rangle \mathbf{e}_i= \mathbf{v}. 