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LIA601S - LINEAR ALGEBRA 2 - 1ST OPP - JUNE 2022 |
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TSi.t
.J
4
NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BAMS
LEVEL: 6
COURSE CODE: LIA601S
COURSE NAME: LINEAR ALGEBRA 2
SESSION: JUNE 2022
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
EXAMINER
MODERATOR:
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
DR NEGA CHERE
DR DAVID IIYAMBO
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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QUESTION 1
Write true if each of the following statements is correct and write false if it is incorrect. Justify
your answer.
1.1. If there is a nonzero vector in the kernel of the matrix operator T, : IR" — R", then this
operator is one to one.
[2]
1.2. If the Characteristics polynomial of a 3 - square matrix Ais given by
p(a) = A? — 442 + 3A—1, then tr(A) = —4.
[2]
1.3. IfA is anon-zero eigenvalue of an invertible matrix A and v is a corresponding
eigenvector, then 1/A is an eigenvalue of A~? and v is a corresponding eigenvector.
[3]
QUESTION 2
Consider the bases E = {e;, €2,e3} = {(1,0, 0), (0,1, 0), (0, 0, 1)} and
S={u,,U,,u3}= {(1,2, 1), (2,5, 0), (3, 3, 8} of R%. Then find the change of basis matrix
Pgs from Sto E.
[6]
QUESTION 3
1 OO 2
LetA=|—1 1 1]. Showthat A = —1 is an eigenvalue of A and find one eigenvector
2 01
correspondent to this eigenvalue.
{12]
QUESTION 4
Let T: P; — P, defined by T(ap + ax + apx*) = ag +ay(x +1) +a,(x+4+ 1).
4.1. Determine whether T is a linear transformation, if so, find ker (T).
[17]
4.2. Determine rank ofT and nullity ofT and use the result to determine whether T is an
isomorphism.
[7]
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QUESTION 5
Find the coordinate vector [p(x)]g of p(x) = 5 + 4x — 3x? with respect to the basis
B={1-—x,1+x+x1—?x,?} of Py.
[12]
QUESTION 6
6.1. State the Cayley- Hamilton Theorem.
[2]
6.2. Verify the Cayley- Hamilton Theorem for the matrix.
{12]
1
A={]-1
—2
1O
0 1|.
1O
QUESTION 7
1
1
1
Find a3 x 3 matrix A that has eigenvalues 1, —1, 0 for which [| i } [| are their
1
0
0
corresponding eigenvectors.
[16]
QUESTION 8
Find the symmetric matrix that corresponds to the quadratic form
f(x, y,z) = x* + 4xy — 2y? + 8xz — 6yz.
[9]
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER