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m235notes12

# m235notes12 - Notes 12 Coding Lecture We begin with an...

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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.

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