Notes 12: Coding
Lecture Oct 22, 2009
We begin with an example of a behavior of matrix multiplication. We can have matrices
A, B
so that neither
A
nor
B
is the zero matrix, but
AB
is the zero matrix. This happens
when the image of
B
is contained in the kernel of
A
.
Example
1
.
Let
A
=
1
0
0
0
1
0
0
0
0
, B
=
0
0
0
0
0
0
0
0
1
.
Example
2
.
Let
A
=
0
1
0
0
,
then
A
2
is the zero matrix, but
A
is not the zero matrix.
Application 1.
We are concerned with sending information acurately form one site to
another.
We assume that the transmission is not completely accurate, that is, there is
noise in the system.
We systematically encode our information.
There are many ways
of doing this. Because of the noise we try to build redundancy into our coding process.
Redundancy allows us to detect errors, but the more redundancy we build into our system
the less efficient it is. We investigate this problem a little bit.
Our information occurs as strings of zeros and ones.
One way that is often used to
check accuracy is called a parity check. We start with a string of zeros and ones and add
a one or a zero at the end of the string. We add this extra bit in a way that insures that
the number of 1’s that occurs is even for example to the string 00101 we would add a zero
to get 001010 or to the string 10101 we would add a 1 to get 101011.
We describe this system using linear algebra. First we give an algebraic structure to
the set
{
0
,
1
}
. We set 0+
x
=
x
=
x
+0
,
1+1 = 0
,
1
·
x
=
x
=
x
·
1
,
1
·
1 = 1
.
We denote this
by
F
and call it the field with 2 elements. This satisfies all the usual rules of arithmetic.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Multiplication, Matrices, Vector Space, FN, HY, parity check

Click to edit the document details