edit: disregard me, I'm an idiot
Spoiler:
Here's the question as an image because it's easier to format:
Please note I'll just write + to mean the plus with the circle around it (direct sum). + is just a normal addition.
My work so far
V = im(T) + ker(S) means that im(T) ∩ ker(S) = {0} and that im(T) + ker(S) = V.
If ST = 1v, then TS = 1w (Identity transformations). Thus w = T(v) and v = S(w).
S[w - TS(w)] = S(w) - STS(w) = v - ST(v) = v - S(w) = v -v = 0, therefore it's in ker(S).
Now I'm stuck. I don't know how to use this to do the proof... I think showing the intersection might go:
im(T) ∩ ker(S) = T(v) ∩ w - TS(w) = w ∩ 0 = 0. But I'm not sure.
I have no idea about the im(T) + ker(S) part though.
Hmm, I'm not sure I got all of that. I'll repeat to make sure.
So we get to the point v = (v - TS(v)) + T(S(v)). The first element is in ker(S), and the second element is in im(T).
Then ker(S) + im(T) = v.
I'm confused though. Does this hold for all v in V? Part of what was tripping me up before was that w-TS(w) is a specific element, so aren't you just proving one case?
Thanks! It definitely shows how w-TS(w) will be useful.
I looked up "complement" in the index of my book and it's not there... but based on what I remember from stats, it would be everything that is NOT in ker(S). Which I'm guessing should turn out to be equal to im(T).
So I understand how this works in theory, just not sure how to apply it.