question 3 and 4

linear algebra 2

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