Monday
January 12
−
Lecture 4 :
Matrices
(Refers to section 3.1)
Expectations:
1.
Define the operation of addition, multiplication, scalar multiplication and
transposition on matrices.
2.
Use the rules of matrix algebra. (Hand-out)
3.
Define identity matrix, zero matrix
4.
Express the dot-product of two vectors as a product of two matrices.
We have previously spoken of
"matrix" but only as related to a system of linear
equations (coefficient matrix of a system and augmented coefficient matrix of a system) .
In this lecture we define the notion of a matrix (simply a rectangular array of numbers)
independently of a system of linear equations. We define operations on matrices and
then, once again, find a new way of representing a system of linear equations, namely a
matrix equation
.
4.1
Definition
−
Matrices
. If
m
and
n
are
positive integers, a
matrix of size
m
×
n
(or of
dimension m
×
n
), is a rectangular array of real numbers, arranged in
m rows
and
n
columns
and is represented as
A
= [
a
ij
]
m
×
n
The symbol
i
is called the
row index
and
j
is called the
column index
. The
a
ij
's are called
the
entries
or
components
of the matrix.
Example :
This is a 3 by 3 matrix
A
:
⎡
a
11
a
12
a
13
⎤
⏐
a
21
a
22
a
23
⏐
⎣
a
31
a
32
a
33
⎦
4.1.1
Two matrices are said to be
equal
matrices
if their corresponding entries are
equal. Clearly matrices of different dimensions cannot be equal.
4.1.2
Remark
−
Distinction between
vectors
and
matrices
:

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When we think in terms of "
vectors
" then
(2, 1, 7) and the numbers 2, 1, 7 written
in a column are viewed as being equal in
R
3
.
o

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