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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.