Help me get some proofs in linear algebra

[dtparekh]

please can any one provide me the proofs!!!
Question:
Denote a vector space of all functions

f:R -> R
which are infinetly differentiable by C^∞(R). This space is called the space of smooth functions.

a. Show that C^∞ is infinite dimensional.
b. Show that differentiation is a linear transformation:
d/dx:C^∞(R)

c. For real number λ find an eigenvector for d/dx with eigen value λ[/b]

 

[omerl]

Re: Linear Algebra

a. all the function of the form 1 x x^2 ... are lineary independent (you can't have for example 1= a1*x^2+a2*x^3+... for any finite members of this group)
b striaght forward: let f,g be any functions and a real number then d(af+g)/dx=adf/dx+dg/dx
c.e^x is eigenvector (with eign value 1) actually e^ax is eigenvector (with the eignvalue a)