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

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

my solution : ke^{λt}

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^{-5}^{t}

2e^{0t}= 2

k = 5

5e^{-5}^{t}

5e^{0}^{t}= 5

k = 13

13e^{-5}^{t}

13e^{0}^{t}= 13

Eigenspaces :

E_{-5}= ke^{-5t }

E_{0}=k

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