Linear Algebra 1 (MATS2202)

Mathematics and Computer Science - MCS

Semester: Second Semester

Level: 200

Year: 2017

Answers (200 )
REPUBLIC OF CAMEROON REPUBLIQUE DU CAMEROUN
Peace-Work-Fatherland Paix-Travail-Patrie
MINISTRY OF HIGHER EDUCATION MINISTERE DB L’ENSEIGNEMENT SUPERIEURE
THE UNIVERSITY OF BAMENDA UNIVERSITE DE BAMENDA
P.0 Box 39 BAMBILI TEL: (+23) 22 81 63 50
SECOND SEMESTER EXAMINATION 2016/2017 SESSION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: MATS 2202 COURSE TITLE: Linear Algebra
DURATION: 3 HRS Answer all questions.
1.
E
XERCISE
1/
10
MARKS
Let S = {(x , y,z) : x — 2 y + 2 = 0} and T = {(x, y, z)
3
: x - z =0}.
(1)
Prove that S and T are subvector spaces of
3
;
(2)
Find the basis of S and T;
(3)
Determine the subvector space S T and its basis;
(4)
Prove that
3
is the unique subvector space containing S and T.
2.
E
XERCISE
2/
40
MARKS
Let f
k :




Part A : 10marks.
(1)
Find the matrix A
k
of f
k
(2)
Prove that det(A
k
) — (k — 1 )
2
(k + 2);
(3)
Find the kernel and the image of f
1
;
(4)
Find the kernel and the image of f
2
;
(5)
Suppose that k {2,1} prove that f
k
is invertible and compute

.
Part B : 10marks.
(1)
Compute the eigen values of A-1 and the corresponding eigen vectors together with the multiplicities.
(2)
Show that A
-1
is diagonalizable.
(3)
Compute for all positive integer n.
(4)
Give the matrix of 


Part C : 10marks.
(1)
Prove that (A
2
I
3
) ( A
2
4I
3
) = 0
(2)
Deduce that A
2
is diagonalizable.
(3)
Write

as a function of A
2
(4)
Compute u
n
, v
n
and w
n
, with 2u
o
= -2v
o
= w
o
= 2 and

!
!
!

!
Pat D : 10marks.
(1) Let "#$%&. Prove that " is an eigen value of a matrix M if and only if 1/" is an eigen vector of M
-1
, and the
corresponding eigen vectors are the same.
(2) Deduce the eigen values of

and the co rre spo nd i ng e i gen vecto r s
(3) Find three functions x, y, z satisfying:
'
(
)
(
*
+,
+-
+.
+-
+/
+-
+,
+-
+.
+-
+/
+-
+,
+-
+.
+-
+/
+-
and x(0) =y(0)=z(0) = 1
3.
E
XERCISE
3/
20
MARKS
(1)
Diagonalize the following form q(x, y, z) = -x
2
+ 2xy -3y
2
+ 2xy + 2yz - z
2
.
(2)
Plot the following: i) q
1
(x,y) = x
2
4y
2
= 1 ii) q
2
(x, y) = 4x2 + 9y2 = 36; iii) q
3
(x, y) = x
2
– 2x + y
2
+ 4y = 4
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