as an image because it's easier to format:Here's the question

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.