1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    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...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Direct Proof Date
Proof regarding direct sum of the dual space of a v-space Jan 19, 2017
Direct proofs Nov 27, 2016
Direct Sum: Vector Spaces Feb 29, 2016
Discrete Math Question Nov 12, 2014
Direct proof confusion? Sep 6, 2012