Linear Algebra 1 (MATS2202)
Mathematics and Computer Science - MCS
Semester: Second Semester
Level: 200
Year: 2018
REPUBLIC OF CAMEROON REPUBLIQUE DU CAMEROUN
Peace-Work-Fatherland Paix-Travail-Patrie
MINISTER OF HIGHER EDUCATION MINISTERE DE L’ENSEIGNMENT
THE UNIVERSITY OF BAMENDA
UNIVERSITE DE BAMENDA
P.O. Box 39 Bambili
,
Tell: (+237) 22 81 63 50
second semester examination-2017/2018 session
Department of mathematics and computer science. course code: mats 2202;
course title: linear algebra; duration: 2 hrs. answer all questions.
1.
E
XERCISE
1/ 10
MARKS
Let S = {(z, y, z) ∈ R
3
: x — 2y + z = 0} and T = {(x, y, z)R
3
: x — z = 0}.
(1)
Prove that S and T are subvector spaces of R
3
;
(2)
Find the basis of S and T;
(3)
Determine the subvector space S n\T and its basis;
(4)
Prove that R
3
is the unique subvector space containing S and T.
2.
E
XERCISE
2/ 40
MARKS
ℝ
→ℝ
Consider f
k
:
→
++
++
++
Part A: 10 marks
(1)
Find the matrix A
k
of f
k
.
(2)
Prove that det(A
k
) = (k — l)
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 fk is invertible and compute
Part B : 10 marks.
(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 f
.
Part C : 10 marks.
(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
=2
+
+
=
+2
+
=
+
+2
Part D : 10 marks.
(1) Let ⋋∈ℝ∖
0
!
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 corresponding eigen vectors
(3) Find three functions x,y,z satisfying:
⎩
⎪
⎨
⎪
⎧
=2
&'
&(
+
&)
&(
+
&*
&(
=
&'
&(
+2
&)
&(
+
&*
&(
=
&'
&(
+
&)
&(
+2
&*
&(
and x(0)=y(0)=z(0)=1.
3.
E
XERCICE
3/ 20 MARKS
(1) Diagonalize the following quadratic form : q(x
t
y, z) =- x
2
+ 2xy –y
2
+ 2xy +2yz — x
2
.
(2) Plot the following : i) q
1
(x,y) = x
2
– 4y
2
- 1; ii) q
2
(x,y) = 4x
2
+ 9y
2
= 36; Hi) q
3
(x,y) = x
2
- 2x + y
2
+ 4y =
4
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