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Notes For Optimization Theory &
Analysis
(ESE304)
Michael A. Carchidi
September 14, 2010
Chapter 2  The Algebra of Scalars, Matrices and Vectors
The following notes are based on the text entitled:
Operations Research
by
Wayne L. Winston (4th edition), and these can be viewed at
https://courseweb.library.upenn.edu/
after you log in using your PennKey user name and Password.
This chapter is provided as a review of the algebra of scalars, matrices and
vectors. It should be read by those students who require this brief review. There
are very few examples provided in this chapter of the notes. The student should
refer to Chapter 2 of the text for more examples, if needed.
2.1 Some De
f
nitions
A
scalar
is a real number and the algebra of scalars is the same algebra that
you learned in high school. An
m
×
n
matrix
A
is an array of elements having
m
rows and
n
columns. The
size
of
A
is said to be
m
×
n
(read
m
n
), and
the elements of
A
are scalars. The element of
A
in the
i
th row and
j
th column
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a
ij
. All of this information is summarized by writing
A
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a
11
a
12
a
13
···
a
1
j
a
1
n
a
21
a
22
a
23
a
2
j
a
2
n
a
31
a
32
a
33
a
3
j
a
3
n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a
i
1
a
i
2
a
i
3
a
ij
a
in
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
a
m
3
a
mj
a
mn
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=[
a
ij
]
m
×
n
where each
a
ij
is a real number (i.e., scalar). A matrix having only one column is
called a
column vector
and is represented as
u
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u
1
u
2
u
3
.
.
.
u
i
.
.
.
u
m
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
u
i
]
m
×
1
where each
u
i
is a scalar. A matrix having only one row is called a
row vector
and is represented as
v
=
h
v
1
v
2
v
3
v
j
v
n
i
v
j
]
1
×
n
,
where each
v
j
is a scalar. In these notes we shall represent scalars using
italic
type. We shall represent matrices using UPPERCASE
bold
type and we shall
represent vectors using lowercase
bold
type.
2.2 Operations on Matrices
Equality
: Two matrices
A
and
B
areequa
liftheyhavethesames
izeandthe
same corresponding elements. In other words, if
A
a
ij
]
m
×
n
and
B
b
ij
]
m
×
n
,
then
A
=
B
,i
fandon
lyi
f
a
ij
=
b
ij
for all
i
=1
,
2
,
3
,...,m
and
j
,
2
,
3
,...,n
.
Note that only matrices of the same size can be compared in this way.
Example
:
Solve for
x
,
y
and
z
if
"
12
x
3
4
x
+
y
6
#
=
"
1
y
−
2
xy
−
x
4
z
6
#
.
2
In order for these two matrices to be equal, we must have
2
x
=
y
−
2
x,
3=
y
−
x
and
x
+
y
=
z
which leads to
x
=1
,
y
=4
,and
z
=5
.
Addition
:I
f
A
and
B
havethesames
izeand
A
=[
a
ij
]
m
×
n
and
B
b
ij
]
m
×
n
,
then
C
=
A
+
B
c
ij
]
m
×
n
provided that
c
ij
=
a
ij
+
b
ij
for all
i
,
2
,
3
,...,m
and
j
,
2
,
3
,...,n
. In other words,
[
a
ij
]
m
×
n
+[
b
ij
]
m
×
n
a
ij
+
b
ij
]
m
×
n
.
In words, when you add two matrices of the same size, you add corresponding
elements. Note that only matrices of the same size can be added.
Example
:
"
123
456
#
+
"
789
10 11 12
#
=
"
1+7
2+8
3+9
4+10 5+11 6+12
#
=
"
81
01
2
14 16 18
#
.
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.
 Winter '11
 MichaelA.Carchidi

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