 |
LIA601S - LINEAR ALGEBRA 2 - 2ND OPP - JULY 2022 |
|
|
1 Page 1 |
▲back to top |
6
'&
3 be
i} wy
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: JULY 2022
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
DR NEGA CHERE
MODERATOR:
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)
|
|
2 Page 2 |
▲back to top |
QUESTION 1
Write true if each of the following statements is correct and write false if it is incorrect. Justify
your answer.
1.1. IfA is an eigenvalue of matrix A, then A - Al is invertible.
[3]
1.2. Ann xn matrix with fewer than n linearly independent eigenvectors is not diagonalizable.
[2]
1.3. The characteristic polynomial and the minimal polynomial of a square matrix can have
different irreducible factors.
[2]
QUESTION 2
Show that v is an eigenvector of A and find the corresponding eigenvalue.
A=|_[f-4 -—o2h)v.=_72[sol
(5]
QUESTION 3
xX
x —y
Let T: R? > R? defined ovt([v]) = | 2z |
Z
X+Z
3.1. Show that T is linear.
[13]
3.2. Find the translation matrix A of T.
[7]
1
3.3. Use the result in (3.2) to find T{ |—2] }.
[4]
2
QUESTION 4
Let T,(X1,X2,X3) = (4X, + X3,-2x, + Xz, —-X, — 3X2) and Tz (x1, X2,X3) = (x1 + 2X2, —X3,
4X, ~~ X3).
4.1. Find the standard matrices of T, and Tp.
[12]
4.2. Use the result in (4.1) to find the standard matrices of T2 © T;.
[5]
|
|
3 Page 3 |
▲back to top |
QUESTION 5
Let T be a linear operator on R? defined by T(x, y, z) = (3x—z, 3y + 2z,x+y +z) and
B = {v4,V2,V3} bea basis of R? in which v, = (1,0,1),v2 = (0,1,2) and v3 = (1,1, 0).
5.1. Find the coordinate vector [v]g of v where v= (a, b, c) is any vector in R°.
[13]
5.2. Use the result in (5.1) to find the coordinate vector of the vector v = (1, 2, —1) with
respect to the basis B.
[4]
QUESTION 6
1
0
1
Let A = |0
1
1}.
1
1
0
6.1. Find the eigenvalues of A.
{11]
6.2. Find the eigenspace corresponding to the largest eigenvalue in (6.1).
[9]
QUESTION 7
Find the quadratic form q(X) that corresponds to the symmetric matrix
2
iL =2
A= 1
—1
zZ 1
[10]
2
Zz
0
END OF SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER