- #1

Adgorn

- 130

- 18

## Homework Statement

Suppose V =U⊕W. Let E_{1}and E_{2}be the linear operators on V defined by E_{1}(v)=u, E_{2}(v)=w, where v=w+u, u ∈ U, w ∈ W. Show that (a) E_{1}^{2}=E1 and E_{2}^{2}=E_{2}(i.e., that E_{1}and E_{2}are projections); (b) E_{1}+E_{2}= I, the identity mapping; (c) E_{1}E_{2}=0and E_{2}E_{1}=0.

Let E_{1}and E_{2}be linear operators on V satisfying parts (a), (b), (c). Prove V= I am E_{1}⊕ I am E_{2}

## 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(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.

Thanks to all the helpers.