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.