Can I Find an Expression Relating x to x1 and x2 while Decomposing Operator A?

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In summary, the conversation discusses decomposing a linear operator A into (L-G) and solving for x in the equation Ax=y. The speaker also mentions knowing solutions to Lx1=y and Gx2=0 and wanting to find an expression relating x to x1 and x2. They clarify by writing the equations in a different way and expressing a desire for an equation for v in terms of v1 and v2.
  • #1
shaiguy6
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So I am trying to decompose a linear operator A, in the following manner. I am trying to solve Ax=y for x, and I also have that A=(L-G), so I am trying to solve (L-G)x=y. y is given, and so are L, A, and G. Now, I also know the solutions to Lx1=y and Gx2=0. I'd like to somehow find an expression relating x to x1 and x2. Any help would be appreciated.
 
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  • #2
In case this wasn't clear, I'll write it over this way:

(L-G)v = y
Lv1=y
Gv2=0

or equivalently:

(L-G)-1y=v
L-1y=v1
G-10=v2

v1 and v2 and v are not equal, they are different. I'd like to have an equation for v in terms of v1 and v2.
 

1. What is an operator in scientific terms?

An operator in scientific terms is a symbol or mathematical function used to manipulate or act on a mathematical expression or physical quantity.

2. How do you decompose an operator?

To decompose an operator, you can break it down into simpler components or functions that can be applied individually to a given expression or quantity.

3. Why is decomposing an operator important in scientific research?

Decomposing an operator allows scientists to simplify complex equations or physical processes, making it easier to understand and analyze data. It also helps in developing new mathematical methods and theories.

4. What are some common techniques used for decomposing an operator?

Some common techniques for decomposing an operator include using the properties of the operator, such as linearity or commutativity, and breaking it down into simpler components using mathematical operations or identities.

5. Can operators be decomposed in all scientific fields?

Yes, operators can be decomposed in various scientific fields, including mathematics, physics, chemistry, and engineering. Decomposing operators is a fundamental concept in many scientific disciplines and is essential for solving complex problems and developing new theories.

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