question 4

4. [16 marks] Consider the linear transformation T: 193(0) > 193(0) given by T(P(z)) = WT?) and the bases ,8 = l,(z +i),(z +i)2,(z +17)3 and '1' = (41 1)(z + 1)(z i). (3 1)(z + 1)(3 +17). (3 1)(z =7)(z +17). (7- + 1)(z ilk +17) a) Find 3[T]3 and 141"]? b) Find a matrix X that implements the similarity between 3[T]3 and -,[T]-,. (i.e. X3[T]3X'1 = 4Tb c) Compute the trace and determinant of T. d) Is 3[T]3 diagonalizable