Complex conjugate operator is linear

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Homework Help Overview

The discussion revolves around the linearity of the complex conjugate operator, a concept in linear algebra and functional analysis. Participants are questioning the claims made in a textbook regarding this operator's properties.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring the definitions of linear operators and questioning whether the complex conjugate operator meets these criteria. They are particularly focused on the conditions for linearity, including the behavior under scalar multiplication.

Discussion Status

The discussion is active, with participants providing definitions and examples to clarify their understanding. There is an exploration of the conditions that define linearity, and some participants are highlighting the specific failure of the complex conjugate operator to satisfy one of these conditions.

Contextual Notes

Participants are referencing textbook definitions and are engaged in examining the implications of these definitions in the context of complex conjugation. There is an emphasis on the need for both conditions of linearity to be satisfied.

Jimmy25
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My textbook claims that the complex conjugate operator is linear. I can't see how this could be. Could someone give me an example of how it is not linear?
 
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Jimmy25 said:
My textbook claims that the complex conjugate operator is linear. I can't see how this could be. Could someone give me an example of how it is not linear?

If a function is linear then f(cx)=c*f(x). Right? Put c=i and f to be complex conjugation.
 
I thought that the definition of a linear operator was:

A(f+g)=A(f)+A(g)

(where A is a linear operator)
 
Jimmy25 said:
I thought that the definition of a linear operator was:

A(f+g)=A(f)+A(g)

(where A is a linear operator)

That's only part of it. And sure, complex conjugation satisfies that. But you also need A(c*f)=c*A(f), where c is a scalar. That's the part complex conjugation fails.
 

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