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ORIE 3300/5300 – Optimization I
Fall 2009
Linear Algebra Review
Linear algebra is one of the most important keys to a solid understanding of linear programming. The
summary below, written by Prof. Todd, lists some concepts from linear algebra that will be fundamentally
important in this course. This does not include all of the topics you will need, but it is a start. The second
recitation meeting will include a guided tour of some of the linear algbra you are expected to have mastered.
By the third recitation meeting you should be prepared for a quiz on this material.
Vectors and Matrices
Def:
A
matrix
is a rectangular array of numbers. We will denote matrices by capital letters. If a matrix
A
has
m
rows and
n
columns, it is called an
m
×
n
matrix, or a matrix of dimension
m
by
n
. Generally,
a
ij
is used to denote the entry in the
i
th row and the
j
th column of the matrix
A
.
Def:
An
m
×
1 matrix is called an
m
dimensional
column vector
. A 1
×
m
matrix is called an
m

dimensional
row vector
. We will use lowercase letters to denote vectors. By convention, we will use the
term
vector
for column vectors and row vectors will explicitly be stated.
Def:
The
transpose
of a matrix
A
, denoted
A
T
, is formed by interchanging the rows and columns of
A
, that is, the
i, j
entry of
A
T
is
a
ji
. If
A
is an
m
×
n
matrix, then
A
T
is an
n
×
m
matrix. Moreover,
(
A
T
)
T
=
A
. If
u
is a column vector, then
u
T
is a row vector.
Def:
The
product
of an
m
×
p
matrix
A
and a
p
×
n
matrix
B
is deﬁned to be a new
m
×
n
matrix
C
,
written
C
=
AB
, whose elements
c
ij
are given by
c
ij
=
p
X
k
=1
a
ik
b
kj
Def:
The
identity matrix
of order
m
is a square
m
×
m
matrix, denoted
I
m
or
I
if clear from the context,
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This note was uploaded on 10/07/2010 for the course ORIE 3300 taught by Professor Todd during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 TODD

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