UECM2093/ UECM2043 OPERATIONS RESEARCH I
Assignment - May 2016
Instruction to Candidates:
1.
2.
3.
4.
Q1.
Answer ALL 5 questions.
Submit the assignment by 31/8/2014 (Wednesday) 5:00 pm. (Put your assignment in the
box in front of my office, level 8-zone 1
Chapter 3
Determinants, Eigenvalues,
Eigenvectors
3.1
The Determinant of a Matrix
The determinant of a matrix is just a number. We denote the determinant
of a matrix A by det(A) or |A|.
1. The determinant of a 1 1 matrix [a] is a.
a b
2. det
= ad bc.
c d
Chapter 6
Orthogonality
6.1
Least Square Approximations
Consider the following system:
x1 + x2 = 2
x1 x2 = 0
2x1 + x2 = 4.
The system is inconsistent. In this case, b Ax 6= 0.
T heorem 6.1.1. All solutions of AT Ax = AT b are least square solutions of
Ax
Chapter 2
Matrices and Linear Equations
2.1
Algebra of Matrices
Definition 8. A matrix
a11 a12
a
21 a22
A = .
.
.
.
am1 am2
is a rectangular array of numbers, denoted by
a11 a12 a1n
a1n
a
a2n
21 a22 a2n
or A = .
. . . . . .
. . . .
.
.
.
am1 am2
Chapter 5
Linear Transformations
5.1
Linear Transformations
Definition 49. A linear transformation is a function T : V W from a
vector space V to another vector space W that satisfies these two conditions:
1. T (u + v) = T (u) + T (v) and
2. T (cu) = cT (
Chapter 4
Vector Spaces
4.1
The Theory of linear equations
Definition 36. The null space of a matrix A is the set of all vectors x that
satisfy Ax = 0. We denote it as nullsp(A).
Example 4.1.1. Find the null space of A.
1 0
1. A =
.
0 1
1 0 x
0
x
0
=
UECM 1313 Fundamentals of Linear Algebra
Tutorial 2
Definition. Let A and B be square matrices with same size. If A and B commute, then
AB = BA.
1. Find x, y, a and b if A = B.
ab
1
4
1
(a) A =
,B =
;
2
b+2
a+b 1
6x y x + y
10 4
(b) A = x y 3x y , B = 4 4
UECM 1313 Fundamentals of Linear Algebra
Tutorial 4
3
1
1. Is 2 in the null space of A = 1
1
1
6
2. (a) Is 7 in the column space of
3
4
6
(b) Is
1 in the column space of
0
1 1
1
1 5? What about 0 ?
2 3
1
1 1 1
A = 0 1 1?
0 0 1
1 2
5 3
A=
4 9?
2
UECM 1313 Fundamentals of Linear Algebra
Tutorial 1
1. Determine whether the vectors u and v are parallel:
3
4
(a) u = 6 , v = 2 .
3
1
2
8
(b) u = 0 , v = 0 .
1
4
4
2
2. Let u = 7 and v = 6. Find the following
4
1
(a) a unit vector in the dire
UECM 1313 Fundamentals of Linear Algebra
Tutorial 3
1. For each of the following matrices, find
(i) the adjoint;
(ii) the determinant;
(iii) the inverse.
3 5 2
6 8
9
5 3
A=
; B = 4 8 9 ; C = 7 4 0 .
15 9
1 2 5
11 2 5
2. For what value of x are the given m
UECM 1313 Fundamentals of Linear Algebra
Tutorial 5
1. Determine whether each of the following functions is a linear transformation.
" #
"
#
x
2x
+
y
(a) T : R2 R2 is defined by T (
)=
.
y
y
" #
x+y
x
(b) T : R2 R3 is defined by T (
) = xy
y
y
2. Let T :
1
00:01:16,320 -> 00:01:18,846
MELVILLE:
How does one come to know the unknowable?
2
00:01:19,000 -> 00:01:21,890
"What faculties must a man possess?
3
00:01:22,040 -> 00:01:23,849
"Since it was discovered that whale oil
4
00:01:24,000 -> 00:01:27,322
"co
Please affix
passport-sized
photograph
here
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1
00:00:47,640 -> 00:00:51,122
<i>.want it that way</i>
2
00:00:51,320 -> 00:00:52,920
Don't look at me when
you sing that part.
3
00:00:53,120 -> 00:00:55,202
Feel it!
Ugh.
4
00:00:55,360 -> 00:00:57,931
<i>That way</i>
5
00:00:58,600 -> 00:01:01,331
Man
Impact of cachexia
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Cachexia
What is cachexia?
Cachexia, also known as wasting syndrome is actually a
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