Linear Algebra 1 (MATS2202)
Mathematics and Computer Science - MCS
Semester: Second Semester
Level: 200
Year: 2017
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 2204 COURSE TITLE: Linear Algebra
DURATION: 3 HRS Answer all questions.
I. Compute u
n
, v
n
and w
n
, with u
o
= v
o
= w
o
= 1 and
II. Plot 6x
2
+ 4xy + 9y
2
= 10 and x
2
+ 10xy + y
2
= 12.
III. Let m G R and A a matrix. A =
a. For which value of m, A is diagonalizable?
b. Replace m by the obtain value in the previous question and Solve the system:
IV. Consider the linear map
f :
!
"
#$
%
and (e
1
, e
2
, e
3
) a canonical basis of
.
a)
Find the matrix A representing f.
b)
Compute f(e
1
), /(e
2
) and /(e
3
).
c)
Find the kernel of f and the image of f.
V. Consider
f :
!
"
#$
%
a)
Find the matrix A representing f.
b)
Find the kernel of f and the image of f.
c)
Compute the eigen values and the corresponding eigen vectors together with the multiplicities.
e) Show that A is diagonalizable,
d)
Compute A
n
.
VI. Diagonalize the following quadratic form q(x, y, z) = 3x
2
+ 2xy + 3y
2
+ 2xy + 2yz + 3z
2
.
VII. Prove that every bilinear form can be written uniquely as sum of symmetric bilinear form and
a skew symmetric bilinear form.
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