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7 Computer Lab 7: The Vector Space F n 2
7.1 Introduction
In this class you will learn about a new vector space, F n 2. As part of this lab you will need to do modulo 2 arithmetic. The vector space F n 2 is used in applications such as coding data to detect and correct errors in transmission.
You will need to use the m-files that you downloaded into MATLAB during Computer Lab 3. If you do not have the m-files, then follow the instructions on the Computer Lab Class Files page of the MAST10007 website to load the Linear Algebra m-files into MATLAB.
7.2 The Field F 2
Let F 2 = { 0 , 1 }.
This set containing just two elements, equipped with the operations of addition and multiplication given below, is called the integers modulo 2. Addition and multiplication are defined as follows:
Addition 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 0
Multiplication 0 × 0 = 0 1 × 0 = 0 0 × 1 = 0 1 × 1 = 1
The main difference between F 2 and Z is that 1 + 1 = 0. We call the arithmetic in F 2 , modular or mod 2 arithmetic.
Try entering mod(1+1,2) into MATLAB. What is the result?
With the above definition of addition and multiplication, F 2 is a field , that is, a system of numbers that shares many of the properties as R (the real numbers) and C (the complex numbers).
Exercise 1: Exploring F 2
- Use hand calculations to find the following:
(a) (1 + 0) × 1 = (b) 1 + 0 × 1 = (c) 1 + 1 + 1 =
- Use hand calculations to answer the following: (a) If a + b = b + a = 0 then b is called the additive inverse of a. What are the additive inverses of the elements 0 and 1 in F 2?
(b) If a × b = b × a = 1 then b is called the multiplicative inverse of a. What is the multiplicative inverse of the element 1 in F 2?
Does 0 ∈ F 2 have a multiplicative inverse?
Exercise 2: Row Reducing and Multiplying Matrices using Modulo 2 Arithmetic
In the same way that we have matrices with real entries, we can have matrices with entries in F 2. The usual operations apply, except that addition and multiplication is done using mod 2 arithmetic.
In MATLAB enter the matrices
A =
and x =
(a) Using hand calculations, find A x using mod 2 arithmetic.
A x =
^ =
(b) In the box below, use hand calculations to find the reduced row echelon form for the matrix A using mod 2 arithmetic. Remember to use 1 + 1 = 0 to obtain zeros where needed.
A =
∼
(c) Use the commands mod(A∗x,2) and rrefmod2(A) to check your answers.
mod(A∗x,2)= rrefmod2(A)=
Compare the results with the answers in (a) and (b).
The second command finds the row reduced form of the matrix A using modulo 2 arithmetic. It is a special function created for these classes. The usual command rref will not give the right answer.
Put the vectors u 1 , u 2 , u 3 , u 5 , u 6 in the columns of a matrix called C and u 1 , u 2 , u 4 , u 5 , u 6 in the columns of a matrix called D.
(d) Decide if either of the following sets of vectors spans F^52. Give a brief explanation. You may assume that dim(F^52 ) = 5. Do keep using rrefmod2.
(i) { u 1 , u 2 , u 3 , u 5 , u 6 }
(ii) { u 1 , u 2 , u 4 , u 5 , u 6 }
(e) How can you test for linear dependence and independence of the vectors v 1 ,... , v k ∈ F n 2?
(f) Are the following sets of vectors linearly dependent or linearly independent? Why?
(i) { u 1 , u 2 , u 3 , u 5 , u 6 }
(ii) { u 1 , u 2 , u 4 , u 5 , u 6 }
(g) Are either of the following sets of vectors a basis for F^52? Give a brief explanation.
(i) { u 1 , u 2 , u 3 , u 5 , u 6 }
(ii) { u 1 , u 2 , u 4 , u 5 , u 6 }
(h) Why does dim(F^52 ) = 5?
Exercise 4: Solution Space
This exercise will refer to the matrix M and the vectors v1, v2, v3 and v4 that were loaded when you ran zed2input.
(a) Let M be a m × n matrix and x be a n × 1 coordinate vector. How can you check whether or not x is in the solution space of M?
(b) Let
M =
Using MATLAB decide whether or not the following are in the solution space of M.
(i) v 1 =
(ii) v 2 =
(iii) v 3 =
Remark: In coding theory, this idea is used to detect errors in transmitted messages. The messages sent are coded using vectors in the solution space of a “check matrix” M with entries in F 2. If we receive a vector v with M v 6 = 0 , then we know that a transmission error has occurred. By choosing the matrix M cleverly (e.g. using a Hamming matrix) it is also possible to correct errors and recover the original message.
The following exercise gives an idea of how this works.
Exercise 5: Column Space
This exercise will refer to the matrix M and the vectors w1 and w2 that were loaded when you ran zed2input.
(a) Let M be a m × n matrix and w be a m × 1 coordinate vector. How can you check whether or not w is in the column space of M?
(b) Let M be as in Exercise 4.
Using MATLAB decide whether or not the following are in the column space of M.
(i) w 1 =
(ii) w 2 =
