 |
LIA601S - LINEAR ALGEBRA 2 - 1ST OPP - NOVEMBER 2024 |
|
|
1 Page 1 |
▲back to top |
f
nAml BIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoolof Naturaland Applied
Sciences
Departmentof Mathematics,
StatisticsandActuariaSl cience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL:6
COURSE:LINEAR ALGEBRA 2
COURSECODE: LIA601S
DATE: NOVEMBER 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
FIRSTOPPORTUNITYEXAMINATION: QUESTIONPAPER
EXAMINER:
DR. NEGACHERE
MODERATOR:
DR. DAVID IIYAMBO
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLEMATERIALS:
1. Non-Programmable Calculator
ATTACHMENTS:
NONE
This paper consists of 3 pages including this front page.
|
|
2 Page 2 |
▲back to top |
QUESTION 1 [27)
Let T: P2 P2 be a mapping defined by:
1.1. Show that Tis linear.
[13]
1.2. Find the kernel of T and use it to determine whether Tis singular or nonsingular. [8]
= 1.3. Show that the mapping T: IR{3 IR{2 defined by T(x, y, z) (x · y, x + y + z) is not
linear.
[6]
QUESTION 2 [20)
m m. 2.1. Let T: Ill.2 Ill.3 be a mapping such that T[!1l = and Tm =
Then find T [:]
[i]· and use it to determine T
[10]
~[;+irnll = ( 2.2. Find the coordinate vector of the vector v 4, -2, 5) with respect to the ordered
basis = {[
3
for Ill. •
[10]
QUESTION 3 [8]
If A and Bare n x n similar matrices then prove that A and B have the same characteristic
polynomial.
[8]
QUESTION 4 [11)
4.1. If 11is. an eigenvalue of an invertible matrix A with corresponding eigenvector x, then
show that 11.1- is an eigenvalue of A- 1 with corresponding eigenvector x.
[6]
4.2. Let A be a 2 x 2 matrix. Show that the characteristic polynomial p(11.)of A is given by
p{11.) = 112.-tr{A)11. + det(A).
[S]
QUESTION S [23)
G1-2) Let A=
3 -4 .
1 -1
= 5.1. Verify whether 11. 1 is an eigenvalue of A. If it is, find the corresponding
eigenvector{s).
[16]
2
|
|
3 Page 3 |
▲back to top |
G) = 5.2. Verify whether the vector x
is an eigenvector for A. If it is, find the
corresponding eigenvalue.
[7]
QUESTION 6 [11]
Consider the following two bases = of Il~.3S: {e1, e2, e3} = {(1,0,0), (0,1,0), (0,0,1)} and
E = {vi, Vz, V3} = {(1,1,0), (0,1,1), (1,2,2)}.
6.1. Find the change of basis matrix from S to E, PE<-S·
[7]
6.2. Use the result in 6.1. to find [vJEwhere v = (1, 3, -2).
[4]
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
3