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Topic: HW Help: Linear Operators  (Read 2213 times)

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Offline surfgirlblondie

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HW Help: Linear Operators
« on: May 22, 2013, 10:55:23 PM »
A liner operator, Oˆ,is defined as having the following properties:
Oˆ (f + g) = Oˆ (f) + Oˆ (g)
Oˆ (c f) = c Oˆ (f)

where f and g are functions, and c is a (complex) constant. By applying the rules above,
determine if each the following operators are linear?
a.
Aˆ f = k f, k = constant
b.
Aˆ f = f2
c.
Aˆ f = df/dx
d.
Aˆ f = 1/f
e.
Aˆ f = f*, where * indicates complex conjugate

Can somebody please help me work through this problem, I don't understand where to start :/ Thank you all for your *delete me*

Offline Corribus

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Re: HW Help: Linear Operators
« Reply #1 on: May 23, 2013, 09:42:39 AM »
First, it will help if you format your math expression better.  For example, what is "Aˆ f = f2"?  Is that f2, or 2f, or what?

Second, you should show your work or at least what your thought process is.  That way help can be better directed.
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?  - Richard P. Feynman

Offline Enthalpy

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Re: HW Help: Linear Operators
« Reply #2 on: May 26, 2013, 10:59:51 AM »
The linearity of most operators you cite is easy to check because they "operate" on the result of the function, so the answer to "is the operator linear" is pretty much the same as "is the operation applied after the function f linear".

That is:
v -> a*v+b is linear, and so is
f -> a*f+b

The exception in your examples is
f -> df/dx
which is not a transformation applied on individual output values of f. Though, you can check what happens if you scale f or if you add g to it.

Operators are much more general and a vast set than the above examples suggest. In the quest of distribution theories, Radon (a person I guess, not the element) measures were introduced to add local infinite values (=Dirac pulses) to functions, but they would not include an operator like
f(x, y) -> f(y, x) which swaps the arguments
this operator being perfectly legitimate in distribution theories
http://en.wikipedia.org/wiki/Distribution_(mathematics)

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