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LIA601S - LINEAR ALGEBRA 2 - 1ST OPP - NOVEMBER 2023 |
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nAm I 8 I A UnlVE RS ITY
OF SCIEnCEAnDTECHnOLOGY
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ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof Mathematics,
Statisticsand ActuarialScience
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Private Bag73388
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QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL: 6
COURSE: LINEAR ALGEBRA 2
COURSE CODE: LIA601S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
FIRSTOPPORTUNITYEXAMINATION: QUESTIONPAPER
EXAMINER:
DR. NEGACHERE
MODERATOR:
DR. DAVID IIYAMBO
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly with black or blue ink pen.
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
ATTACHMENTS:
NONE
This paper consists of 3 pages including this front page.
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Part I: True or false questions.
For each of the following questions, state whether it is true or false. Justify your answer.
1. If T: P3 -+ P3 is a linear transformation, then Tis an isomorphism.
(3)
2. If the characterstic equation of a matrix A is given by p(>.) = >2.(>.- l)(>- - 2)3, then
the size of matrix A is 6 x 6.
(2)
3. Let A be an n x n matrix. If A has fewer than n distinct eigenvalues then A is not
diagonalizable.
(3)
4. If q is a quadratic form on a vector space V, then q(-o:) = -q(o:).
(3)
Part II: Work out Problems.
1. Let V and W be vector spaces over a field K and let T: V -> Vi/ be a mapping. State
what it means to say T is linear transformation.
(3)
2. Let T be the mapping T: P3 -+ P2 defined by T(a 0 + a1x + a2x2 + a3x3) = 3ao + a3x2.
Then
(a) show that Tis linear.
(12)
(b) find a basis for the kernel of T.
(7)
3. Let A and B be n x n similar matrices.Then prove that A and B have the same Char-
acterstic polynomial.
( 11)
4. Find an orthonormal martix P for the symmetric matrix A = (~ ~) such that
025
pT AP is a diagonal matrix.
(26)
5. Consider the bases B = {l + x + x2 , x + x2, x2} and C = {l, x, x2 } of P2 .
(a) Find the cooordinate vector [p(x)]s of p(x) where p(x)= 1 + x2 .
(6)
(b) ] Find the change of basis matrix: Pc..-Bfrom B to C.
(5)
(c) Use the results in (a) and (b) to compute [p(x)]c where p(x)= 1 + x 2 .
(4)
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6. (a) Find the quadratic form q (;:) that corresponds to the symmetric matrix
A=( -;3).
(8)
-3 2 5
(b) Find the symmetric matrix corresponding to the quadratic form q(x 1, x2, x3 )
2xI + 2x1x2 + 4x2X3 - l0x1x3 - x~.
(7)
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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