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LIA502S - LINEAR ALGEBRA 1 - 1ST OPP - NOVEMBER 2023 |
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nAm I BI A un IVERS ITV
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof Naturaland Applied
Sciences
Department of Mathematics,
Statistics and Actuarial Science
13Ja(ksonKaujeuaStreet
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
LEVEL:5
COURSE: LINEAR ALGEBRA 1
COURSECODE: LIAS02S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: QUESTION PAPER
MR GABRIELS MBOKOMA, DR NEGA CHERE
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:
Non-Programmable Calculator
This paper consists of 3 pages including this front page.
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Question 1
1.1 Given that u = (6 - :i:, 4 - y) and v = (x - 4, y + 2) arc vectors in JR2 , such that u = v,
solve for x and y?
[4]
1.2 Determine a unit vector perpendicular to both of the vectors A = c + d and B = c - d,
where c = 3i + 2j + 2k and d = i + 2j - 2k.
[7]
1.3 Consider the vectors z = (3 + 4i, 2 - i) and w = (1 + 3i, 1 - 2i) in C2 . Determine whether
z and w are orthogonal.
[6]
1.4 Prove that if x and y are orthogonal vectors in R"', then show that
[6]
Question 2
2.1 ·write down a 4 x 4 matrix whose i/h entry is given by a.,j = ij~-l, and comment on your
matrix.
[6]
2.2 Let A be a square matrix. State what is meant by each of the following statements.
(a) A is symmetric
[l]
(b) A is orthogonal
[l]
(c) A is skew-symmetric
[l]
2.3 Conside, the matcL, A - ( ~l ; ~5 ) .
a) Use the Cofactor expansion method along the second col-u,mnto evaluate the determi-
nant of A.
[7]
b) Is A invertible? If it is, Use the Gauss-.Jon.lan Elimination method to find A- 1. [14]
c) Find <let (3(2A)- 1).
[6]
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..
Question 3
Determine whether or not the vector (-1,1,5) is a linear combination of the vectors (1,2,3), (0,1,4)
and (2,3,6).
[15]
Question 4
a) Prove that a vector space cannot have more than one zero vector.
[6]
b) Let Mnn be a vector space whose elements a.re all the n x n matrices, with the usual addition
and scalar multiplication for matrices. Determine whether the following set is a subspace
of lv'lnn·
S = {A E l\\llnn Itr(A) = O}
[11]
c) Prove or disprove that if U and W are subspaces of a vector space V, then Un W is also a
subspace of V.
[9]
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