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Get Full Access to Linear Algebra With Applications - 4 Edition - Chapter 3.1 - Problem 54
Get Full Access to Linear Algebra With Applications - 4 Edition - Chapter 3.1 - Problem 54

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# Solved: In Exercises 53 and 54, we will work with the binary digits (or bits) 0 and 1 ISBN: 9780136009269 434

## Solution for problem 54 Chapter 3.1

Linear Algebra with Applications | 4th Edition

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Problem 54

In Exercises 53 and 54, we will work with the binary digits (or bits) 0 and 1, instead of the Teal numbers R. Addition and multiplication in this system are defined as usual, except for the rule 1 -1- 1 = 0. We denote this number system with F2, or simply F. The set of all vectors with n components in F is denoted by Fn; note that F consists of 2n vectors. (Why?) In information technology, a vector in F8 is called a byte. (A byte is a string of 8 binary digits.)The basic ideas of linear algebra introduced so far (for the real numbers) apply to F without modifications. A Hamming matrix with n rows is a matrix that contains all nonzero vectors in F" as its columns (in any order). Note that there are 2n 1 columns. Here is an example:3 rows 23 1 = 7 columns.a. Express the kernel of H as the span of four vectors in F7 of the form1 0 0 1 0 1 1 H = 0 1 0 1 1 0 1 0 0 1 1 1 1 0v \ =* * * ** * * ** * * *b. Form the 7 x 4 matrix II CM0 < V3 = 0 , v4 = 0 0 1 0 0 0 0 1 0 0 0 0 1M = V\ V2 V3 I V4 Explain why \m(M) = ker(//). If x is an arbitrary vector in F4, what is H(Mx)l(See Exercise 53 for some background.) When information is transmitted, there may be some errors in the communication. We present a method of adding extra information to messages so that most errors that occur during transmission can be detected and corrected. Such methods are referred to as error-correcting codes. (Compare these with codes whose purpose is to conceal information.) The pictures of mans first landing on the moon (in 1969) were televised just as they had been received and were not very clear, since they contained many errors induced during transmission. On later missions, much clearer error-corrected pictures were obtained.ln computers, information is stored and processed in the form of strings of binary digits, 0 and 1. This stream of binary digits is often broken up into blocks of eight binary digits (bytes). For the sake of simplicity, we will work with blocks of only four binary digits (i.e., with vectors in F4), for example,| 1011|1001|1010|1011| 1 0 0 0 I ... .Suppose these vectors in F4 have to be transmitted from one computer to another, say, from a satellite to ground control in Kourou, French Guiana (the station of the European Space Agency). A vector u in F4 is first transformed into a vector i; = Mu in F7, where M is the matrix you found in Exercise 53. The last four entries of v are just the entries of ; the first three entries of v are added to detect errors. The vector v is now transmitted to Kourou. We assume that at most one error will occur during transmission; that is, the vector w received in Kourou will be either v (if no error has occurred) or w = v + 5/ (if there is an error in the zth component of the vector). a. Let H be the Hamming matrix introduced in Exercise 53. How can the computer in Kourou use Hw to determine whether there was an error ifl the transmission? If there was no error, what is Hw? If there was an error, how can the computer determine in which component the error was made? b. Suppose the vector Kourou0_is received in Kourou. Determine whether an error was made in the transmission and, if so, correct it. (That is, find v and u.)

Step-by-Step Solution:
Step 1 of 3

Chapter 2: Graphs and Functions 2.1.1. Place the following points on the axis: A (2,-3) B (-3,4) C (0,5) D (-6,0) E (-4,-7). 6 C B 4 2 0 D -7 -6 -5 -4 -3 -2 -1 0 1 2 3 -2 A -4

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Step 3 of 3

##### ISBN: 9780136009269

The answer to “In Exercises 53 and 54, we will work with the binary digits (or bits) 0 and 1, instead of the Teal numbers R. Addition and multiplication in this system are defined as usual, except for the rule 1 -1- 1 = 0. We denote this number system with F2, or simply F. The set of all vectors with n components in F is denoted by Fn; note that F consists of 2n vectors. (Why?) In information technology, a vector in F8 is called a byte. (A byte is a string of 8 binary digits.)The basic ideas of linear algebra introduced so far (for the real numbers) apply to F without modifications. A Hamming matrix with n rows is a matrix that contains all nonzero vectors in F" as its columns (in any order). Note that there are 2n 1 columns. Here is an example:3 rows 23 1 = 7 columns.a. Express the kernel of H as the span of four vectors in F7 of the form1 0 0 1 0 1 1 H = 0 1 0 1 1 0 1 0 0 1 1 1 1 0v \ =* * * ** * * ** * * *b. Form the 7 x 4 matrix II CM0 < V3 = 0 , v4 = 0 0 1 0 0 0 0 1 0 0 0 0 1M = V\ V2 V3 I V4 Explain why \m(M) = ker(//). If x is an arbitrary vector in F4, what is H(Mx)l(See Exercise 53 for some background.) When information is transmitted, there may be some errors in the communication. We present a method of adding extra information to messages so that most errors that occur during transmission can be detected and corrected. Such methods are referred to as error-correcting codes. (Compare these with codes whose purpose is to conceal information.) The pictures of mans first landing on the moon (in 1969) were televised just as they had been received and were not very clear, since they contained many errors induced during transmission. On later missions, much clearer error-corrected pictures were obtained.ln computers, information is stored and processed in the form of strings of binary digits, 0 and 1. This stream of binary digits is often broken up into blocks of eight binary digits (bytes). For the sake of simplicity, we will work with blocks of only four binary digits (i.e., with vectors in F4), for example,| 1011|1001|1010|1011| 1 0 0 0 I ... .Suppose these vectors in F4 have to be transmitted from one computer to another, say, from a satellite to ground control in Kourou, French Guiana (the station of the European Space Agency). A vector u in F4 is first transformed into a vector i; = Mu in F7, where M is the matrix you found in Exercise 53. The last four entries of v are just the entries of ; the first three entries of v are added to detect errors. The vector v is now transmitted to Kourou. We assume that at most one error will occur during transmission; that is, the vector w received in Kourou will be either v (if no error has occurred) or w = v + 5/ (if there is an error in the zth component of the vector). a. Let H be the Hamming matrix introduced in Exercise 53. How can the computer in Kourou use Hw to determine whether there was an error ifl the transmission? If there was no error, what is Hw? If there was an error, how can the computer determine in which component the error was made? b. Suppose the vector Kourou0_is received in Kourou. Determine whether an error was made in the transmission and, if so, correct it. (That is, find v and u.)” is broken down into a number of easy to follow steps, and 627 words. This full solution covers the following key subjects: . This expansive textbook survival guide covers 41 chapters, and 2394 solutions. The full step-by-step solution to problem: 54 from chapter: 3.1 was answered by , our top Math solution expert on 03/15/18, 05:20PM. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since the solution to 54 from 3.1 chapter was answered, more than 402 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

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