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LIA502S - LINEAR ALGEBRA 1 - 1ST OPP - JUNE 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:
LIA5028
COURSE CODE: LINEAR ALGEBRA 1
SESSION:
JUNE 2022
PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
FIRST
EXAMINER:
MODERATOR:
OPPORTUNITY
EXAMINATION QUESTION
DR. DSI ITYAMBO
DR. N CHERE
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 p = i+j— 2k and q =i— 3j+12k
a) Find the unit vector in the direction of p.
b) Find the angle (in degrees) between p and q. Give you answer correct to 1 d.p.
[8]
Question 2
Consider the following matrices.
a) Given that C = AB, determine the element co.
[3]
b) Find (3A)?.
[3]
c) Is DB defined? If yes, then find it, and hence calculate tr(DB).
[7]
Question 3
Let A= ( aij ) be an n X n matrix.
a) When do we say that A is a symmetric matrix?
[2]
b) Prove that A+ A” is a symmetric matrix.
[5]
c) Prove that if A is an invertible symmetric matrix, then A~! is also symmetric.
[6]
Question 4
Consider the matrix A =
-1 1 2
3 0 —5
172
a) Use the Cofactor expansion method to evaluate the determinant of A.
b) Is A invertible? If it is, find Aq}.
[14]
c) Find det (3(2A)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+32%2-23 = 1
24, +to +273 = 4
3%, +4¢9+2073 = —-1
[10]
Question 6
a) Prove that a vector space cannot have more than one zero vector.
[6]
b) Let Myn be a vector space whose elements are all the n x n matrices, with the usual addition
and scalar multiplication for matrices. Determine whether the following set is a subspace
of Mnn.-
S = {A € Mnn|tr(A) = 0}
[11]
c) Prove or disprove that if S and T are subspaces of a vector space V, then SMT is also a
subspace of V.
[9]