School of Mathematics and Statistics
MAST10007 Linear Algebra, Semester 2 2023
Written Assignment 6
Submit a single pdf file of your assignment solutions via the MAST10007 website
before 12 noon on Monday 9th October.
• This assignment is worth 2.22% 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.
• 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. Let T : R3 → R3 be the linear transformation given by
T (x, y, z) = (x+ y, x+ 2y − z, 2x+ y + z).
Let S be the ordered standard basis of R3 and let
B = {(1, 0, 1), (−2, 1, 1), (1,−1, 1)}
be an ordered basis of R3.
(a) Find the transition matrices PS,B and PB,S .
(b) Using the two transition matrices from part (a), find the matrix representation of T with
respect to the basis B.
2. Consider the matrix
A =
 4 0 −10 3 0
1 0 2
 .
(a) Find the eigenvalues of A.
(b) Find a basis for the eigenspace of A corresponding to each eigenvalue.
3. Let A ∈Mn,n and let λ be an eigenvalue of A. Suppose A satisfies the equation
2A2 + 5A− 10In = 0n,n .
Show that λ satisfies the equation
2λ2 + 5λ− 10 = 0.
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