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LIA502S - LINEAR ALGEBRA 1 - 2ND OPP - JANUARY 2025 |
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QUALIFICATION: BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BSAM
LEVEL:5
COURSE: LINEAR ALGEBRA 1
COURSECODE: LIA502S
DATE: JANUARY 2025
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
MR GABRIELS MBOKOMA, MR ILENIKEMANYANDADI
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 2 pages including this front page.
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Question 1
Consider the vectors v = 4i - 8k, a = 2i + 2j - k and b = 2i - j + 2k.
a) Find a vector of magnitude v'5in the direction of v.
[6]
b) Find the angle 0 (in radians) that is between a and b.
[6]
c) Find a unit vector that is perpendicular to both vectors a and b.
[8]
Question 2
Let A be a square matrix and let
and
a) Find S+P.
[4)
b) Show that S is symmetric and P is skew-symmetric.
[6)
c) Show that if A is symmetric, then S = A and P = 0.
[4]
Question 3
Consider the matrix B = (
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 B- 1.
[14)
[6]
Question 4
Show that u 1 = (1, 1, 1), u 2 = (1, 2, 3) and u3 = (1, 5, 8) span JR3 (use Gaussian).
[15)
Question 5
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 !Rn.
[13]
1