Page 18 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes The associative and commutative laws of addition and multiplication hold as for integers, and so also the
distributive law of multiplication over addition. The cancellation laws hold for addition and
multiplication, except as for integers.

The additive identity is the number (0,1) . For
a
b
a
b
( , ) (0,1) ( .1 b .1) ( , )
a
The multiplicative identity is the number (1,1) . For,
b
a
b
b
( . ) (1,1) ( .1, .1) ( , )
a
a
b
The additive inverse of ( , ) is (a b a , ) . For,
2
,
a
( , ) ( a , ) ( .b ba b 2 ) (0, ) (0,1)
b
b
a
b
The multiplicative inverse of ( , ) is ( , )a b b a if a 0. For,
a
a
b
b
( , ) ( , ) (ab ,ba ) (1,1)
The additive identity (0,1) , is defined as the rational number zero and is written as 0.
The non-zero rational number ( , )a b which is such that a 0, is said to be positive or negative
according as a a b is positive or negative.
The negative of a rational number z is its additive inverse; it is written as –x. Thus if x = ( , )a b
b
then –x = ( a , ) .
We define subtraction of a rational number y from a rational number x as x + (–y), written x – y.
Thus, if x = ( , )a b and y = ( , )c d , we have

x y x ( y ) ( , ) ( c , ) (ad bc ,bd )
b
d
a
The reciprocal of a non-zero rational number x is its multiplicative inverse, and is written as 1/x.
Thus if x = ( , )a b , then

1/x = ( , )b a , a 0, b 0.
The division of a rational number x by a non-zero rational number y, written as x y or x|y, is
defined as x. (1/y). Thus if x = ( , )a b , then

y ( , ),c 0, we have
d
c
a
x y ( , ) ( , ) (ad bc ),b 0, c 0 .
d
c
b
It can be shown that subtraction is a binary composition in Q, and division is also a binary
composition, except for division by zero.
Order Relation

Let x, y be two rational numbers. We say that x is greater than, less than or equal to y, if x – y is
positive, negative or zero, and we use the usual signs to denote these relations.

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