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LIA601S - LINEAR ALGEBRA 2 - 1ST OPP - NOV 2022 |
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n Am I BI A u ni VE Rs ITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURALRESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATIONCODE: 07BSOC; 07BSAM
LEVEL: 6
COURSECODE: LIA601S
SESSION: NOVEMBER 2022
COURSENAME: LINEARALGEBRA2
PAPER:THEORY
DURATION: 3 HOURS
MARKS: 100
EXAMINER
MODERATOR:
FIRSTOPPORTUNITYEXAMINATION 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.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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QUESTION 1 [42)
1.1. [9]
Let V and U be vector spaces over a field !RIa.nd let T: V U be a mapping. Then define what
does it means to say
(a) Tis linear.
[3]
(b) Kernel of T.
[2]
(c) rank of T and nullity of T.
[4]
1.2. [33)
,7. 2
Let T: IR3i. IR3i.be a mapping defined by T ([~]) = [ ;
(a) Show that Tis linear.
[11]
(b) Determine Ker (T), the nullity ofT and the rank of T and use the result together with the
rank theorem to determine whether Tis an isomorphism or not.
[11]
(c) Determine the matrix representation of Twith respect to the basis {vi, v2 , v3 } where
V1 = (1, 0, 1), Vz = (0, 1, 1), V3 = (1, 1, 0) of JR3l.
[5]
(d) Determine the determinant of T and trace of T.
[6]
QUESTION 2 [10)
[-:n Let Sand 'B be bases for JR2l.where S = {[~], and 'B = {[~][,~]}.
Find the change of basis matrix from 'B to S (P5..._23).
QUESTION 3 [11)
3.1. State what does it means to say two matrices are similar.
[2]
3.2. Let A and B be n x n similar matrices. Then show that det A= det B.
[5]
Gi] [i ~] 3.3. Show that A =
and B =
are not similar.
[4]
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QUESTION 4 (12]
Find the coordinate vector [p(x)h of p(x) = 5 + 4x - 3x2 with respect to the basis
QUESTION 5 (15]
3
Let A= [~ - o ~]- Then
0 0 -1
5.1. determine the characteristic polynomial and the eigenvalues of A.
[6]
5.2. is A diagonalizable? Justify your answer.
[3]
5.3. find the eigenspace corresponding to the largest eigenvalue of A.
[6]
QUESTION 6 (10]
[i 3
Find the quadratic form q(xi, x2, x3 ) for the symmetric matrix A=
-1
.
12
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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