Page 16 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Similarly, a negative integer can be identified with the number –u, and the set of negative
integers written as
Z = {–1, –2,–3, …}
–N
We define the negative of an integer x as the integer y, such that x + y = 0. It is easy to see that every
integer has its negative. For, let
x = (a, b)*. Then if y = (b, a)*, we have

x + y = (a, b)* + (b, a)* = (a + b, b + a)*
= (a + b, a + b)* = 0
The negative of the integer x, also called the additive inverse of x, is denoted by –x. We therefore
have, for any integer x,
x + (–x) = 0
and x = (a, b)* –x = (b, a)*.
We define subtraction of an integer if from an integer x as x + (–y), written as x – y. Thus if
x = (a, b)* and y = (c, d)*, we have
x – y = x + (–y) = (a, b) + (d, c)*
= (a + d, b + c)*

Order Relation in Integers

If x, y be the two integers, we define x = y if x – y is zero, x > y if x – y is positive and x < y if x – y
is negative.
The Trichotomy Law for integers holds as for natural numbers. Further,
x > y x + z > y + z,

and x > y, z > 0 x z > yz, x, y, z Z.
The cancellation law for multiplication states that
xz = yz, z 0 x = y.
The addition and multiplication operations on Z satisfy the laws of natural numbers with the
only modification in cancellation law of multiplication which requires p 0. Further, the addition
operation satisfies the following two properties in Z.

(i) There exists the additive identify 0 in the set, i.e., 0 Z such that a + 0 = 0 + a = a, for any
a Z.
(ii) There exists the additive inverse of every element in Z, i.e., a Z – a Z such that
a + (–a) = (–a) + a = 0.

Division

A non-zero a is said to be a divisor (factor) of an integer b if there exists an integer c, such that
b = ac.

When a is divisor of b, we write “a | b”. Also we say that b is an integral multiple of a. It is
obvious that division is not everywhere defined in Z.
The relation of divisibility in the set of integers Z is reflexive, since a | a, a Z. It is also
transitive, since a | b and b | c a | c. But it is not symmetric.

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