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LIA601S - LINEAR ALGEBRA 2 - 2ND OPP - JANUARY 2024 |
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nAmlBIA UntVERSITY
OF SCIEnCE
Facultyof Health,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
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T: •264 61207291;
E: msas@nust.na
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QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL: 6
COURSE:LINEAR ALGEBRA 2
COURSECODE: LIA601S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECONDOPPORTUNITY/SUPPLEMENTARYEXAMINATION: QUESTION PAPER
EXAMINER:
DR. NEGA CHERE
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. The map T : IR3 -+ IR2, defined by T(x,y,z) = (x + y +2, y + z) is not a linear transfor-
mation.
(3)
2. If A and B are similar matrices then there exists an invertible matrix P such that AP =
BP.
(3)
3. For an n x n matrix A, the geometric multiplicity of each eigenvalue of A is less than or
equal to the algebraic multiplicity.
(3)
4. The index and signature of the quadratic form
q(x,y, z) = 3x2 - 4xy + 6y 2 + 4yz - 7z2 are respectively 3 and 2.
(3)
5. If q is a quadratic form on a vector space V, then q(-0:) = -q(0:).
(3)
Part II: Work out problems.
1. Let V and Vv be vector spaces over a field K and let T: V -+ W 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 + a3x 3 ) = 2a 1 - a2x 3 .
Then
(a) show that T is linear.
(12)
(b) find a basis for the kernel of T.
(7)
3. Let V be the vector space of functions with basis S = {sin 2t, cos 2t, e- 3l} and let
D: V -+ V be the differential operator defined by D f(t) = ftf(t). Find the matrix
representing D in the basis S.
(8)
4. Let A and B be n x n similar matrices.Then prove that A and B have the same deter-
minant.
(6)
5. Consider the bases B = {(1,0,0),(0,1,0),(0,0,1)}
of IR3.
and C = {(1,0,1),(0,1,1),(1,1,0)}
(a) Find the change of basis matrix Pc.-6 from B to C.
(10)
(b) Use the result in (a) and to compute [v]c where v = (1, 3, 5).
(5)
0
G D 6. (a) Show that >.=. 4 is an eigenvalue of the matrix A=
2
and find an eigen-
0
vector corresponding to this eigenvalue.
(17)
0
G (b) Show that v - ( 1) is an eigenvector for the matrix A -
1 ~2) and find
0
the corresponding eigenvalue of A.
(6)
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7. (a) Consider the bilinear form f on R2 defined by f((x 1 , y1), (x2, yz)) = 2x1x2 - 3x1y2+
4y1y2 . Find the matrix A off relative to the basis B = {(1, 1), (-2, 1)}.
(6)
(b) Show that q(x, y) = x2 + 2xy + y2 is a quadratic form on R2 .
(5)
END OF SECOND OPPORTUNITY /SUPPLEMENTARY EXAMINATION
QUESTION PAPER
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