Linear algebra problem: linear operators and direct sums

[Adgorn]

Homework Statement

Suppose V =U⊕W. Let E₁ and E₂ be the linear operators on V defined by E₁(v)=u, E₂(v)=w, where v=w+u, u ∈ U, w ∈ W. Show that (a) E₁²=E1 and E₂²=E₂ (i.e., that E₁ and E₂ are projections); (b) E₁+E₂= I, the identity mapping; (c) E₁E₂ = 0 and E₂E₁ =0.

Let E₁ and E₂ be linear operators on V satisfying parts (a), (b), (c). Prove V= I am E₁ ⊕ I am E₂

Homework Equations

N/A

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.

 

[fresh_42]

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.

 

[Adgorn]

I knew it was right in front of me, thank you very much.