Linear Mathematics | Part IB, 2003

Let α\alpha be an endomorphism of a finite-dimensional real vector space UU such that α2=α\alpha^{2}=\alpha. Show that UU can be written as the direct sum of the kernel of α\alpha and the image of α\alpha. Hence or otherwise, find the characteristic polynomial of α\alpha in terms of the dimension of UU and the rank of α\alpha. Is α\alpha diagonalizable? Justify your answer.

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