Let V be the vector space spanned by the functions {1; t ; exp(t); exp(-t)} .
[Note that 1 represents the constant function f (t) = 1]
Let D : V !V be the linear transformation D(f) (t) = f \'(t) , i.e. the derivative operator.

(a) Show that the functions {1; t; exp(t); exp(-t)} form a basis for V. 

(b) Is D : V !V one-to-one? Justify your answer.

(c) Is D : V !V onto? Justify your answer. [Hint: it may help to use the basis you found in part (a).]

(d) Write the matrix for D with respect to the basis B = {1; t; exp(t); exp(-t )} .

(e) Identify an eigenvalue of D, justifying your answer.