For the complementary projection \(P=I_{n}-Q\) whose image is \(\mathcal{K}\), deduce that the composite linear transformation \(Y \mapsto L_{0} Y=P \hat{\mu}\) is nilpotent, i.e., that \(L_{0}^{2}=0\). What does nilpotence imply about the image and kernel of \(L_{0}\) ? Construct the multiplication table for \(L_{0}, L_{1}\).