Here is the question:

Let a, b, c be three distinct real constants, and consider the function T : 732(R) > R3 dened by where p' denotes the derivative of p. a. Prove that T is a linear transformation. b. Find the matrix of T with respect to the standard bases of 772(R) and R3. c. Find conditions on (1,1), 0 such that T is invertible. Can you explain your answer in terms of ideas from rstyear (or earlier) calculus? (1. Now take specically a = 1, b = 2 and c = 3. Let M be the matrix of T with respect to the bases 8 = {1 + t, 1 t, 1 + t2} and c = {(2,1,4), (0,1,2), (1,7, 3)} for PAR) and R3 respectively, i.e. M = [T]? Write down an expression for M as a product of three matrices; then see if you can evaluate M without calculating any matrix inverses. (If not, nd the required inverse and evaluate M anyway.)