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Homework Help: Direct Sum Proof

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Here's the question... it was easier to format it in paint haha:
    linalg.jpg

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

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Apr 12, 2010 #2
    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...
     
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