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.