Show that the function defined by
is a linear map
Would I begin the question like the following:
That seems more tedious than necessary. Just use the definition of "linear map".
Suppose p and q are two polynomials in and a is a real number.
What is T(p+ q)? Can you show that is the same as T(p)+ T(q)? What is T(ap)? Can you show that is the same as aT(p)?
Use basic properties of the derivative.