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**robeuler** A:V->V is a linear transformation with A^2=A

Prove that im(A) intersect ker(A) = 0

I know from a previous exercise that for B, an arbitrary linear transformation, there exists an integer m such that im(B^m) intersect ker(B^m)= 0. I think I can use this and say that A^m=A.

Prove that V=im(A) direct sum ker(A)

I know that dim(V)=dim(im(A))+dim(ker(A))

But I don't know how to use idempotent-ness of A to get set equality. Can I use that the intersection is 0?

Prove that there is a basis of V such that A with respect to this basis is a diagonal matrix with only 0s and 1s.

This one I have no idea how to start.