Linear Algebra (MATS2104)
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 DE L’ENSEIGNEMENT SUPERIEURE
THE UNIVERSITY OF BAMENDA UNIVERSITE DE BAMENDA
……. …….
P.O Box 39 BAMBILI
TEL: (+23) 22 81 63 50
SECOND SEMESTER EXAMINATION 2016/2017 SESSION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: MATS 2104 COURSE TITLE: Linear Algebra
DURATION: 3HRS Answer all questions.
I. Compute
and
, with
and
II. Plot
and
.
III. Let and ! a matrix. !"
#
a. For which value of , ! is diagonalizable?
b. Replace by the obtain value in the previous question and solve the
system:
$
%
&
%
'
()
(*
+
,
+
,
-
(.
(*
-
(/
(*
+
,
IV. Consider the linear map
01
2
3
2
4
-
5
6"
-
-
#
And (7
7
7
2
) a canonical basis of
2
.
a) Find the matrix ! representing 0.
b) Compute 0
+
7
,
0
+
7
,
and 0+7
2
,.
c) Find the kernel of 0 and the image of 0.
V. Consider
01
2
3
2
4
-
5
6"
-
-
#
a) Find the matrix ! representing 08
b) find the kernel of 0 and the image of 08
c) Compute the eigen values and the corresponding eigen vectors together with the multiplicities.
e) Show that ! is diagonalizable.
d) Compute !
.
VI. Diagonalize the following quadratic form
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9
+
-
,
--
.
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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