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LIA601S - LINEAR ALGEBRA 2 - 2ND OPP - JANUARY 2025 |
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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 2025
SESSION: 2
DURATION: 3 HOURS
MARKS: 100
SECONDOPPORTUNITY/ SUPPLEMENTARYE: XAMINATION QUESTIONPAPER
EXAMINER:
MODERATOR:
DR. NEGACHERE
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.
PERMISSIBLEMATERIALS:
1. Non-Programmable Calculator
ATTACHMENTS:
NONE
This paper consists of 3 pages including this front page.
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QUESTION 1 [21]
= 1.1. Let T: Mnn IR be a mapping defined by T(A) tr(A). Determine whether Tis linear
or not where Mnn is the set of all n x n matrices.
[12]
-v)z. 1.2. Let f: IR3 IR2 be a mapping defined by f(x, y, z) = (x + z, y - z,
Determine
whether f is linear or not.
[9]
QUESTION 2 (9)
Find the coordinate vector [p(x)h of p(x) = 2 - x + x2 with respect to the ordered basis
'B ={l+x, l+x 2 , x+x 2 } of P2 •
[9]
QUESTION 3 (23)
Let T: JR3
IR3 be mapping
defined
X1)
by
T
(
Xz
X3
=
(X1 + 2Xz - X3)
x2 + x3 .
X1+ Xz - 2x3
(JJ. = 3.1. Find the standard matrix for T and use it to determine T(x) where x
[8]
3.2. Find a basis and the dimension of the image of T. Use rank-nullity theorem to
determine the nullity of T and use it to determine whether Tis singular or nonsingular. [15]
QUESTION 4 (12)
G~] [i _~]. 4.1. Let A =
and B =
Show that A and Bare not similar.
[4]
4.2. If A is an eigenvalue of an invertible matrix A with corresponding eigenvector x, then
show that An is an eigenvalue of An with corresponding eigenvector x.
[8]
QUESTION 5 (7)
= = Consider the following two bases of IR3 : S {e 1 , e2, e3 } {(1,0,0), (0,1,0), (0,0,1)} and
= = E {vi, v2, v3 } {(1,1,0), (0,1,1), (1,2,2)}. Find the change of basis matrix from Sto E,
PE<-S·
[7]
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QUESTION 6 (28]
1
0
1
6.1. Determine the eigenvalues and the corresponding eigenvectors of A.
[25]
6.2. Is matrix A diagonalizable? If it is, find an invertible matrix P that diagonalizes A. [3]
END OF SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
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