[surfgirlblondie]

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*

[Corribus]

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.

[Enthalpy]

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)