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LIA502S - LINEAR ALGEBRA 1 - 1ST OPP - NOVEMBER 2024 |
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n Am I BI A um VERs ITY
OF SCIEnCEAnDTECHnOLOGY
FacultoyfHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet T: +264612072913
PrivateBag1338B
E: msas@nust.na
Windhoek
W:www.nust.na
NAMIBIA
QUALIFICATION : BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BSAM
LEVEL:5
COURSE: LINEAR ALGEBRA 1
COURSECODE: LIA502S
DATE: NOVEMBER 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: EXAMINATION QUESTION PAPER
MR GABRIEL S MBOKOMA, MR ILENIKEMANYA NDADI
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.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
This paper consists of 3 pages including this front page.
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Question 1
1.1 Consider the vectors p = i + j - 2k and q = i - 3j + 12k
a) Find the unit vector in the direction of p.
[3]
b) Find the angle (in degrees) between p and q. Give you answer correct to 1 d.p. [8]
1.2 Find a unit vector perpendicular to both the vectors i + j and j + k.
[5]
1.3 Prove that if x and y are orthogonal vectors in !Rn, then
[6]
Question 2
2.1 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 + AT is a symmetric matrix.
[5]
c) Prove that if A is an invertible symmetric matrix, then A- 1 is also symmetric. [6]
2.2 Consider the following matrix.
A = (~ co~x si~ x ) .
0 sinx - cos.1:
a) Use the Cofactor expansion method to evaluate the determinant of A through column
one (1).
[6]
b) Is A invertible? If it is, find A- 1 using the adjoint matrix approach.
[12]
Question 3
Given that matrix
B = (~
3a
is symmetric, find the value of ab?
[7]
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Question 4
Use the Gau.ssian elimination method to find the solution of the following system of linear
equations, if it exists.
XI+ 3x2 - X3
1
2x1 + X2 + X3 = 4
3x1 + 4x2 + 2x3 = -1
[14]
Question 5
a) Prove that a vector space cannot have more than one zero vector.
[6]
b) Let M1m be a vector space whose elements are all then x n matrices, with the usual addition
and scalar multiplication for matrices. Determine whether the following set is a subspace
of M,m•
S = {A E Mnn Itr(A) = O}
[11]
c) Prove or disprove that if S and T are subspaces of a vector space V, then Sn T is also a
subspace of V.
[9)
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