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Linear algebra problem: linear operators and direct sums

  1. Jan 18, 2017 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations

    3. The attempt at a solution
    I proved the first part of the question (first quote) and got stuck in the second (second quote).
    I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at trying to prove that U∩W={0}. I showed that E1(w)=0 ∈ U∩W and that E2(u)=0 ∈ U∩W. From there, however, I don't know how to show that these are the only elements of U∩W. I'm fairly certain I'm missing something fairly obvious and would love assistance on the matter.

    Thanks to all the helpers.
  2. jcsd
  3. Jan 18, 2017 #2


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    Staff: Mentor

    I think you probably already have done it. Take a ##v \in U \cap W = im(E_1) \cap im(E_2)##, i.e. ##v=E_1(v_1)=E_2(v_2).## Now apply ##E_1## again.
  4. Jan 19, 2017 #3
    I knew it was right in front of me, thank you very much.
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