Question: 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).

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 $L_{0}, L_{1}$.

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