Homework Statement

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Let ##V## be a vector space, and let ##U, W## be subspaces of ##V## such that ##V = U \oplus W##. Let ##P_U## be the projection on ##U## in the direction of ##W## and ##P_W## the projection on ##W## in the direction of ##U##. Prove:

##P_U + P_W = Id##, ##P_U P_W = P_W P_U = 0##

Reciprocally, given ##P_1, P_2 : V \rightarrow V## that verify ##P_1 P_2 = P_2 P_1 = 0##, ##P_1+P_2 = Id##, prove that ##V = Im(P_1) \oplus Im(P_2)##

Homework Equations

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##P^2 = P## is my (abbreviated) definition of a projection.

The Attempt at a Solution

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I understand the basic idea I think. I know what the direct sum of subspaces implies, namely that the intersection of the two subspaces is 0 and that any vector ##v \in V## can be broken down into ##v = u \in U + w \in W##... I read something about proving that one of these subspaces represents the kernel and the other the image of ##V,## but is that necessarily the case? I suppose the second part of the exercise indicates otherwise.

I'm not sure how to go about the proof. I feel like I understand the idea but I don't know how to write it. A hint to get me on the right path before a full answer would be appreciated to help refine my intuition. Thanks.

Is it perhaps something like ##P_U(v) = P_U(u+w) = u = P(P(u)) \rightarrow P(u) \in Im(P_U)##... or along those lines?

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