4.I.6G

Let $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ such that $\alpha^{2}=\alpha$. Show that $U$ can be written as the direct sum of the kernel of $\alpha$ and the image of $\alpha$. Hence or otherwise, find the characteristic polynomial of $\alpha$ in terms of the dimension of $U$ and the rank of $\alpha$. Is $\alpha$ diagonalizable? Justify your answer.

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