Complex conjugate operator is linear

Click For Summary
SUMMARY

The complex conjugate operator is not linear, contrary to claims made in some textbooks. While it satisfies the condition A(f+g) = A(f) + A(g), it fails to meet the requirement A(c*f) = c*A(f) when c is a scalar, particularly when c is the imaginary unit i. This distinction is crucial for understanding the properties of linear operators in complex analysis.

PREREQUISITES
  • Understanding of linear operators in mathematics
  • Familiarity with complex numbers and their properties
  • Knowledge of scalar multiplication in vector spaces
  • Basic concepts of functional analysis
NEXT STEPS
  • Study the properties of linear operators in functional analysis
  • Explore examples of linear and non-linear operators in complex analysis
  • Learn about vector spaces and their implications on operator behavior
  • Investigate the implications of complex conjugation in various mathematical contexts
USEFUL FOR

Mathematicians, students of advanced mathematics, and anyone studying linear algebra or complex analysis will benefit from this discussion.

Jimmy25
Messages
69
Reaction score
0
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?
 
Physics news on Phys.org
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.
 

Similar threads

How Can We Prove the Conjugate Transpose Property of Complex Matrices?
  • · Replies 5 ·
Replies
5
Views
2K
Partial fractions with complex linear terms
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad Quick question regarding creation/annihilation operators
  • · Replies 1 ·
Replies
1
Views
1K
Conjugate vs Complex Conjugate
  • · Replies 9 ·
Replies
9
Views
3K
Is f(z) =(1+z)/(1-z) a real function?
  • · Replies 17 ·
Replies
17
Views
3K
Linear operator in 2x2 complex vector space
  • · Replies 18 ·
Replies
18
Views
3K
Understanding Complex Conjugates in QM (Griffiths pg. 13)
  • · Replies 2 ·
Replies
2
Views
2K
Contour integration around a square
  • · Replies 26 ·
Replies
26
Views
5K
Prove that det(A*) = [det(A)]*
  • · Replies 2 ·
Replies
2
Views
7K
Proving various properties of complex numbers
  • · Replies 3 ·
Replies
3
Views
1K