MAST10007 Linear Algebra, Semester 2 2024
Assignment 4
1. For each of the linear transformations in part (a), (b) and (c) below:
• Compute a basis for the kernel
• Compute a basis for the image
• Determine if they are invertible
(a) The mapping R : P1 → P3 given by R(p(x)) = (1 − x2)p(x).
(b) The mapping S : P2 → R3 given by S(p(x)) = (p(1), p(2), p(3)).
(c) The mapping T : P2 → P2 given by T(p(x)) = x p
′
(x).
2. Let V be a complex vector space with ordered basis B = {e1, e2, e3, e4}. Consider the linear transformation T such that
T(e1) = e2, T(e2) = e3, T(e3) = e4, T(e4) = e1.
(a) Find the matrix representation of T with respect to B.
(b) Find the eigenvalues of T.
(c) Find the eigenvectors of T.
(d) Show that the eigenvectors of T form. a basis C of V .
(e) Find the transition matrix PB,C.