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LIA502S - LINEAR ALGEBRA 1 - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science; Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BAMS | LEVEL:
5
COURSE CODE:
LIA5025
COURSE CODE: LINEAR ALGEBRA 1
SESSION:
JULY 2022
PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
SUPPLEMENTARY
EXAMINER:
MODERATOR:
/ SECOND
OPPORTUNITY EXAMINATION
DR. DSI ITYAMBO
DR. N CHERE
QUESTION
PAPER
INSTRUCTIONS
1. Attempt all the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in black or blue inked, and sketches must be done in
pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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Question 1
Consider the vectors v = 4i — 8k, a = 2i+ 2] —k and b = 2i—j + 2k.
a) Find a vector of magnitude V5 in the direction of v.
[6]
b) Find the angle @ (in radians) that is between a and b.
[5]
c) Find a unit vector that is perpendicular to both vectors a and b.
[7]
Question 2
Consider the following matrices.
1 -2
A={4 2 1],
0
—2
14
123
B=|3 -1|, andD=
.
—2 2
214
a) Given that C = AB, determine the element c39.
[3]
b) Find (3A)?.
[3]
c) Is DB defined? If yes, then find it, and hence calculate tr(DB).
[6]
Question 3
Let A be a square matrix.
a) What does it mean to say that A is a skew-symmetric matrix?
(2]
b) Prove that AA? is a symmetric matrix.
[4]
c) Prove that A — A? is a skew-symmetric matrix.
[5]
Question 4
Consider the matrix A= B=]
12
1
3 -2 -4
2 3 -1
a) Use the Cofactor expansion method, expanding along the first row, to evaluate the determi-
nant of B.
[9]
b) Is B invertible? If it is, find Bot.
[14]
c) Find det (((2B)~1)7).
[6]
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Question 5
Use the Gaussian elimination method to find the solution of the following system of linear
equations, if it exists.
@1+%2+3%3
@j+2%0+4%3
2%, +22+623
=6
=9
= 11
[8]
Question 6
a) Prove that in a vector space, the negative of a vector is unique.
[9]
b) Determine whether the following set is a subspace of R”.
S = {(a, b,c) € R"|a+b+c=0}
[13]