Page 17 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

The absolute value “|a|” of an integer a is defined by Notes
|a| = a when a 0
= – a when a < 0
Thus, except when a = 0, |a| Z .
N
A non-zero integer p is called a prime if it is neither 1 nor –1 and if its only divisors are 1, –1,
p, –p.
When a = bc with |b| > 1 and |c| > 1, we call a composite. Thus every integer a 0, 1 is either a
prime or composite.
The operation of division of one integer by another is carried out in accordance with the division
algorithm, which can be stated as follows.
Given two positive integers a, b there exists uniquely two non-negative integers q, r such that
a = bq + r, 0 r < b
The number q is called the quotient, and r the remainder obtained on dividing a by b.

Two other forms of the theorem, which are successive generalisations, are as follows:
(i) Given two integers a, b with b > 0, there exist unique integers q, r, such that
a = bq + r, 0 r < b
(ii) Given two integers a, b with b 0, there exist unique integers q, r, such that

a = bq + r, 0 r < |b|.
Greatest Common Divisor

A greatest common divisor (GCD) of two integers a and b is a positive integer d such that
(i) d|a and d|b, and
(ii) if for an integer c, c|a and c|b, then c|d.
We shall use the notation (a, b) for the greatest common divisor of two integers a and b. The
greatest common divisor is sometimes also called highest common factor (HCF).
Every pair of integers a and b, not both zero, has a unique greatest common divisor (a, b) which
can be expressed in the form (a, b) = ma + nb for some integers m and n.
Rational Numbers

Let (a, b) Z × Z , where Z is the set of non-zero integers. Then the equivalence class
0 0
b
a
( , ) {( , ):( , ) ~ ( , );m Z ,n Z }
n
n
m
m
b
a
0
is called a rational number.
The set of all equivalence classes of Z × Z determined by the equivalence relation ~ defined as
0
above is called the set of rational numbers to be denoted by Q.
The addition and multiplication operations in Q are defined as follows:
c
( , ) ( , ) (ad bc ,bd )
a
b
d
and ( , ) ( , ) ( ,bd )
c
b
a
ac
d
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