School of Mathematics and Statistics MAST10007 Linear Algebra, Semester 2 2024 Assignment 4 Submit a single pdf file of your assignment solutions via the MAST10007 website before 10am on Monday 7th October. • This assignment is worth 3% of your final MAST10007 mark. • Assignments must be neatly handwritten, but this includes digitally handwritten documents using an ipad or a tablet and stylus, which have then been saved as a pdf. • Full working must be shown in your solutions. You cannot use MATLAB to do your assignment. • Marks will be deducted in every question for incomplete working, insufficient justification of steps and incorrect mathematical notation. • You must use methods taught in MAST10007 Linear Algebra to solve the assignment questions. 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)) = xp′(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. Page 1 of 1 51作业君版权所有