What is the dimension of eigenspaces in a function-based linear transformation?

[Tala.S]

Linear transformation f:C^∞(R) -> C^∞(R)

f(x(t)) = x'(t) a) I have to set up the eigenvalue-problem and solve it :

My solution : ke^λtb) Now I have to find the dimension of the single eigen spaces when λ is

-5 and 0. My solution :

Eigenspaces :

E-5 = ke^-5t

E0=k (because ke^0t = k)

But I don't know how to find the dimension of the single eigen spaces ?

I'm used to working with vectors but now it's functions and I'm not sure about the dimension.

 

[Tala.S]

Solved !

 

[HallsofIvy]

Good! I presume that you realized that since every solution, $Ce^{\lambda x}$ is a constant, C, times the single function $e^{\lambda x}$, the space is one dimensional.