Is there a reason why you could not just type the problem in yourself? Many people are reluctant to open attachments for fear of viruses.

Here is what you have:

Consider the differential operator T:

.

a) Is this a linear operator? If so what is the representing matrix of this mapping? (I presume you mean "in the standard basis".)

b) Find at least one vector in the pre-image set

.

c) (Bonus) Find the pre-image set

.

(

is also called the "nullspace" of T.)

Now, what have you done on these yourself? A good method of finding the matrix representing a mapping, in a given basis, is to apply the mapping to the basis vectors in turn, writing the result in terms of the basis. The coefficients are the columns of the matrix.

The standard basis for

is {1, x,

,

}. What is T(1)? What is T(x)?

As for finding vectors in the null space, take a general polynomial, p(x), in

, apply T to it and set it equal to 0. What must the coefficients of p be so that T(p)= 0?