I'm not quite sure if I'll butcher the wording of this question, but here goes...

How would I prove that given a linear map T:V-->V, such that the composition T^2 = I, and also given P and Q s.t. P = (I+T)/2 and Q = (I-T)/2, that Ker P = Im Q, and Im P = Ker Q?

I think I'm just a little fuzzy about the whole concept of how to prove that something is in the image of P or Q. I understand Ker P is when (I+T)/2 = 0, and when I solve for T and plugin to Q, I get Q = I. Does this mean that it is in the image of Q? Or am I missing something here?

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Also, as an aside, what would P and Q be called in this question? Not a "composition," right?

Thanks in advance!