# Linear algebra problem: linear operators and direct sums

Tags:
1. Jan 18, 2017

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(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. Jan 18, 2017

### 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.

3. Jan 19, 2017