Question 3 and 4

3. [15 marks] In P3 (C) let & = 1, z, z2, 23 and B = 1, (z + i), (z + i)2, (z + i)3 and y = (z - 1)(z + 1)(z - 2), (2 - 1)(z + 1)(z + 2), (z - 1)(z - 2)(z + 2), (z+ 1)(z -2)(z+i) a) Find the change of basis matrices =[/]8 ,= [/], and ~]B b) Use these matrices to find [1 + z + z2 + 23], and [1 + z + 22 + 23 ], c) A certain polynomial p(z) in P3(C) has coordinates [p(z) 18 Find p(z) and [p(z)ly. 4. [16 marks] Consider the linear transformation T : P3(C) - P3(C) given by T(p(z) ) = zp' (2) and the bases B = 1, (z + 2), (z + i)2, (2 + i)3 and y = (z - 1)(z + 1)(z - 2), (2 - 1)(z + 1)(z + 2), (z -1)(z - 2)(z + 2), (z+ 1)(z -2)(z+i) a) Find s[T]s and ~[Tly b) Find a matrix X that implements the similarity between s[T]s and v[Tly. (i.e. X:[T]sX-1 = c) Compute the trace and determinant of T. d) Is s [T]s diagonalizable