1. Eigen space and dimension

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

a) I have to set up the eigenvalue-problem and solve it :

my solution : keλt

b) Now I have to find the dimension of the single eigen spaces and plot the graph for 3 different vectors from each eigen space when λ is

-5 and 0.

I'm having some problems with this part. I I've chosen k as 2, 5 and 13 and then I just plot them with the two different λ values :

k = 2

2e-5t

2e0t = 2

k = 5

5e-5t

5e0t = 5

k = 13

13e-5t

13e0t = 13

Eigenspaces :

E-5 = ke-5t

E0=k

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