Reflection is a linear transformation. $Re_y(\mathbf{x}) = Re_y(a_1\mathbf{e}_1 + a_2 \mathbf{e}_2) = a_1 Re_y(\mathbf{e}_1) + a_2 Re_y(\mathbf{e}_2) = -a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2$