Direct Sum Proof Homework: Solve V = im(T) + ker(S)

[jumbogala]

Homework Statement

Here's the question... it was easier to format it in paint haha:

Please note I'll just write + to mean the plus with the circle around it (direct sum). + is just a normal addition.

Homework Equations

The Attempt at a Solution

V = im(T) + ker(S) means that im(T) ∩ ker(S) = {0} and that im(T) + ker(S) = V.

If ST = 1_v, then TS = 1_w. 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.

 

[jumbogala]

Update: I'm thinking about using

w = (w - TS(w)) + TS(w).

Then w-TS(w) is in ker(S) and TS(w) is in im(T). Although I guess this is not really helpful because this doesn't show that it's equal to the space V...