1. The problem statement, all variables and given/known data

2. Relevant equations
N/A

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(E_{1}) as U and Im(E_{2}) 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 E_{1}(w)=0 ∈ U∩W and that E_{2}(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.