MATH1002: Linear Algebra Semester 1, 2023 Assignment 2
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Assignment 2
MATH1002: Linear Algebra
Semester 1, 2023
This assignment is worth 10% of your final assessment for this course. Your answers should be well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all working. Present your arguments clearly using words of explanation and diagrams where relevant. After all, mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to master. The marker will give you feedback and allocate an overall mark to your assignment using the following criteria:
1. Let M = 「(l)
3(2) 
1(3)
, x0 = 「(l)
, and [M | x0] = 「(l)
3(2) 
1(3) 
. (a) Find the reduced row echelon form of the augmented matrix [M | x0].
(b) Using the result of part (a), or otherwise, solve the following system of linear
equations
to find all solutions.
(c) i). Show that if v0 and v1 are distinct solutions of equation (1) in part (b), then v = v1 − v0 satisfies
Mv = 0;
ii). Use i) and results of part (b) to find an explicit eigenvector of M with eigen- value 0.
2. Let T = 「(l)
2 
3
, and denote I3 = 「(l)
0(0)」
1
.
(a) i). Calculate det(T − xI3 ) for any x to find the characteristic polynomial of T;
ii). Determine the eigenvalues of T.
(b) Find an eigenvector of T associated with each of the eigenvalues of T.
(c) Express an arbitrary vector v = l 」b(a)
nation of eigenvectors of T.
3. Let ω = 
], and let A be any 2 × 2 matrix over R with detA = 1. (a) Show that AT ωA = ω .
(b) Use part (a), or otherwise, show that A− 1 = ω − 1 AT ω .
(c) We define u ∗ v = uT ωv for any u, v in R2 . Show that for all x, y ∈ R2 ,
i). x ∗ y = −y ∗ x; and
ii). (Ax) ∗ (Ay) = x ∗ y.
2023-05-10