- #1
jumbogala
- 423
- 4
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 = 1v, then TS = 1w. 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.