Linear algebra problem: linear operators and direct sums

  • #1
127
12

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= Im E1 ⊕ Im 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.
 

Answers and Replies

  • #2
15,369
13,397
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
127
12
I knew it was right in front of me, thank you very much.
 

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