2.8 Some terminology
At this point, it’s probably worth introducing some important terminology. When, in mathematics, we prove a true statement, we often say we are proving atheorem, or a proposition. (Usually the word ‘proposition’ is used if the statement does not seem quite so significant as to merit the description ‘theorem’.) A theorem that is a preliminary result leading up to a theorem is often called a lemma, and a minor theorem that is a fairly direct consequence of, or special case of, a theorem is called a corollary, if it is not significant enough itself to merit the title theorem. For your purposes, it is important just to know that these words all mean true mathematical statements. You should realise that these terms are used subjectively: for instance, the person writing the mathematics has to make a decision about whether a particular result merits the title ‘theorem’ or is, instead, merely to be called a ‘proposition’.
Overview
This chapter has explored some of the basics of algebra, together with an introduction to mathematical proof. It is a preliminary chapter, and you should not let it detain you from proceeding with the rest of the course. As we mentioned earlier, if you’re not entirely comfortable with it, it is best to proceed anyway: particularly when it comes to proving things, you can pick up the key ideas in context in what follows.
Learning outcomes
At the end of this chapter and the relevant reading you should be able to:
demonstrate understanding of the key ideas and notations concerning sets demonstrate an understanding of what mathematical statements are demonstrate knowledge of standard mathematical notation
solve polynomial equations
prove whether certain mathematical statements are true or false construct truth tables for simple logical statements
demonstrate knowledge of what is meant by conjunction and disjunction
demonstrate understanding of the meaning of ‘if-then’ statements and be able to prove or disprove such statements
demonstrate understanding of the meaning of ‘if and only if’ statements and be able to prove or disprove such statements
find the converse of statements
prove results by various methods, including directly, and by the method of proof by contradiction
2 Test your knowledge and understanding
Attempt the following exercises.
Exercises
Exercise 2.1
Simplify, then solve for a:
6ab− a
b(b2−4bc) = 1.
Exercise 2.2
Given that the polynomial P(x) = x3+ 3x2+ 4x+ 4 has an integer root, find it and hence show that the polynomial can be expressed as a product P(x) = (x−r)Q(x) where Q(x) is an irreducible quadratic polynomial.
Exercise 2.3
Is the following statement about natural numbers n true or false? Justify your answer by giving a proof or a counterexample:
If n is divisible by 6 then n is divisible by 3.
What are the converse and contrapositive of this statement? Is the converse true? Is the contrapositive true?
Exercise 2.4
Is the following statement about natural numbers n true or false? Justify your answer by giving a proof or a counterexample:
If n is divisible by 2 then n is divisible by 4.
What are the converse and contrapositive of this statement? Is the converse true? Is the contrapositive true?
Exercise 2.5
Prove by contradiction that there is no largest natural number.
Feedback on selected activities
Feedback to activity 2.1
A∩B is the set of objects in both sets, and so it is {2,5}.
Feedback to activity 2.2
(x−1)(x+ 1) =x2−1. (x−1)(x+ 1)(x+ 2) =x3+ 2x2−x−2.
2
2.8. Comments on exercises
Feedback to activity 2.3
We will show the first, and leave the second to you.
aras = (a×a×a× · · · ×a)
| {z }
rtimes
(a×a×a× · · · ×a)
| {z }
stimes
.
Removing the brackets, we have the product of a times itself a total of r+s times; that is,
aras =a×a×a× · · · ×a
| {z }
r+stimes
=ar+s. Feedback to activity 2.4
49x−2
35y − 4xy2
(2xy)3 = 7
5x2y − 4xy2
8x3y3 = 7
5x2y − 1
2x2y = 1 x2y
7 5 −1
2
= 9
10 1 x2y Feedback to activity 2.5
(a) x2−4 = (x−2)(x+ 2) = 0, with solutions x=±2.
(b) x2+ 2x−8 = (x−2)(x+ 4) = 0, so x= 2 or x=−4 (c) 2x2−7x+ 3 = (2x−1)(x−3) = 0,so x= 12 or x= 3.
Feedback to activity 2.6
The converse is ‘if n divides 12 thenn divides 4’. This is false. For instance,n = 12 is a counterexample. This is because 12 divides 12, but it does not divide 4. The original statement is true, however. For, if n divides 4, then for some m∈Z, 4 =nm and hence 12 = 3×4 = 3nm=n(3m), which shows that n divides 12.
Comments on exercises
Solution to exercise 2.1 (a) 6ab− a
b(b2−4bc) = 6ab−ab+ 4ac= 5ab+ 4ac=a(5b+ 4c),so the equation becomes a(5b+ 4c) = 1, and solving for a:
a= 1
5b+ 4c, provided 5b+ 4c6= 0.
Note that it is an important part of the solution to declare that it is only valid if 5b+ 4c6= 0, otherwise there is no solution.
Solution to exercise 2.2
Because all the terms are separated by + signs, the integer root must be a negative number, so try x=−1. Substitution into the polynomial yields, −1 + 3−4 + 46= 0, so
−1 is not a root. Next try x=−2. This time it works, −8 + 3(4) + 4(−2) + 4 = 0, so x3+ 3x2 + 4x+ 4 = (x+ 2)(x2+λx+ 2).
Comparing the coefficients of either x2 orx terms, you should obtain λ= 1. The quadratic polynomial x2+x+ 2 cannot be factored over the real numbers, since its discriminant is negative. Therefore
P(x) =x3 + 3x2+ 4x+ 4 = (x+ 2)(x2+x+ 2) = (x+ 2)Q(x)
2
where Q(x) is an irreducible quadratic polynomial.
Solution to exercise 2.3
The statement is true. Because, suppose n is divisible by 6. Then for some m∈N, n= 6m, so n= 3(2m) and since 2m∈N, this proves that n is divisible by 3.
The converse is ‘Ifn is divisible by 3 then n is divisible by 6’. This is false. For example, n= 3 is a counterexample: it is divisible by 3, but not by 6.
The contrapositive is ‘If n is not divisible by 3 then n is not divisible by 6’. This is true, because it is logically equivalent to the initial statement, which we have proved to be true.
Solution to exercise 2.4
The statement is false. For example, n= 2 is a counterexample: it is divisible by 2, but not by 4.
The converse is ‘If n is divisible by 4 then n is divisible by 2’. This is true. Because, suppose n is divisible by 4. Then for some m ∈N,n = 4m, so n= 2(2m) and since 2m∈N, this proves that n is divisible by 2.
The contrapositive is ‘Ifn is not divisible by 4 then n is not divisible by 2’. This is false, because it is logically equivalent to the initial statement, which we have proved to be false. Alternatively, you can see that it’s false because 2 is a counterexample: it is not divisible by 4, but it isdivisible by 2.
Solution to exercise 2.5
Let’s prove by contradiction that there is no largest natural number. So suppose there is a largest natural number. Let us call it N. (What we want to do now is somehow show that a conclusion, or something we know for sure must be false, follows.) Well, consider the number N+ 1. This is a natural number. But since N is the largest natural number, we must have N + 1≤N, which means that 1≤0, and that’s nonsense. So it follows that we must have been wrong in supposing there is a largest natural number. (That’s the only place in this argument where we could have gone wrong.) So there is nolargest natural number. We could have argued the contradiction slightly differently. Instead of using the fact that N + 1≤N to obtain the absurd statement that 1≤0, we could have argued as follows: N + 1 is a natural number. But N + 1> N and this contradicts the fact that N is the largest natural number.
Chapter 3 3
Matrices
Introduction
Matrices will be the main tool in our study of linear algebra, so we begin by learning what they are and how to use them. This chapter contains a lot of definitions with which you should become familiar, including terminology associated with a matrix and the operations defined on matrices. All of the operations are defined purposefully to ensure matrices are a useful tool, as we shall see in later chapters. In particular, the definition of matrix multiplication may seem strange at first, but it turns out to be exactly what we need.
Aims
The aims of this chapter are to:
Define a matrix, the terminology associated with a matrix, and the operations defined on matrices.
Learn how to manipulate matrices algebraically using these operations.
Become familiar with the properties and rules of matrix operations, how they combine and interact.
Essential reading
R Anthony, M. and M. Harvey. Linear Algebra: Concepts and Methods. Chapter 1, Sections 1.1–1.7
This chapter of the subject guide closely follows the first half of Chapter 1 of the
textbook. You should read the corresponding sections of the textbook and work through all the activities there while working through the sections of this subject guide.
Further reading
If you would like to have another source containing the material covered in this chapter, you can look up the key concepts and definitions (as listed in the synopsis which
follows) in any elementary linear algebra textbook, such as the two listed in Chapter 1, using either the table of contents or the index.
3
Synopsis
We define a matrix and the basic terminology associated with a matrix (entry, row, column, size, square matrix, diagonal matrix, equality of matrices) and then look at the operations of addition, scalar multiplication and matrix multiplication. We show how to algebraically manipulate matrices using these operations, and state what is meant by a zero matrix and an identity matrix. We define the inverse of a square matrix, when it exists, and its properties. Next we define what is meant by powers of a square matrix, look at its properties and how it interacts with the inverse of a matrix. Then we define the transpose of a matrix and what is meant by a symmetric matrix, and look at the properties of the transpose of a matrix and how it interacts with the inverse of a matrix.