# Linear algebra problem: linear operators and direct sums

In summary, the conversation discusses the proof of the linear operators E1 and E2 being projections and satisfying certain properties. The solution involves defining Im(E1) as U and Im(E2) as W, and showing that U∩W={0} by using the fact that E1(w)=0 and E2(u)=0. The final step is to apply E1 again to show that these are the only elements of U∩W.

## 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

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.

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.

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

## What is a linear operator?

A linear operator is a mathematical function that maps between vector spaces, preserving the operations of addition and scalar multiplication. It can be represented by a matrix or a set of equations.

## What is a direct sum?

A direct sum is a mathematical operation that combines two vector spaces into a larger vector space. It is denoted by a ⊕ symbol and is defined as the set of all possible combinations of elements from the two original vector spaces.

## How do you determine if a linear operator is invertible?

A linear operator is invertible if and only if it is bijective, meaning it has a one-to-one correspondence between its domain and its range. This can be determined by checking if the operator has a non-zero determinant.

## Can a direct sum of two vector spaces be equal to one of its subspaces?

Yes, it is possible for a direct sum of two vector spaces to be equal to one of its subspaces. This can happen if the two original vector spaces have a non-trivial intersection, meaning they share at least one non-zero vector.

## What are some real-world applications of linear algebra and direct sums?

Linear algebra and direct sums have many applications in fields such as engineering, physics, and computer science. They are used to solve systems of linear equations, model physical systems, and perform data analysis and compression. Examples include image and signal processing, optimization problems, and machine learning algorithms.

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