Hi

The complex functions z(t), t ∈ R, can be differentiated a number of times and are a vector space V :

W = span(e^{-t}, e^{-it}, e^{t}, e^{it}) and W is a subspace of V.

Which of the following 3 functions belong to W ?

z_{1}(t) = 3e^{t}, z_{2}(t) = cos(t), z_{3}(t) = sinh(t)

If the functions were vectors for example like this W= span((1,2),(2,1)) and I had to determine if the vector (-2,3) belonged to W, I would solve the inhomogeneous linear system but since it's not I have no idea how to determine this.

I hope that somebody can give me a hint.

Sorry for the bad description.