Page 14 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (A ) m + n = n + m, (commutative law)
3
(A ) m + p = n + p m = n (cancellation law)
4
All these properties can be established from the foregoing postulates and definitions only.

Multiplication Composition

In the set of natural numbers N, we define multiplication which shall be denoted by the symbol
‘X’ as follows:
(i) m × 1 = m m N
+
(ii) m × n = m × n + m, m, n N.
Sometimes we often find it convenient to represent m × n by m . n or simply by mn.

The following properties, which can be established from Peano’s postulates, hold for
multiplication.
(M ) m, n N, or mn N, (Closure law)
1
(M ) (m . n) . p = m . (n . p) or (m n) p = m (n p), (associative law)
2
(M ) m . n = n . m, or m n = n m (Commutative law)
3
(M ) m . p = n . p m = n, or m p = n p m = n. (Cancellation law)
4
Distributive Law

The distributive property of multiplication over addition is expressed in the following two forms:
If m, n, p N, we have

(i) m . (n + p) = m . n + n . p [Left distributive law]
(ii) (m + n) . p = m . p + n . p [Right distributive law]
The right distributive law can also be inferred from the left distributive law, since multiplication
is commutative.
Order Property

We say that a natural number m is greater than another number n(m > n), if there exists a number
u N, such that m = n+ u.
The number m is said to be less than the number n(m < n), if there exists a number v N, such that
n = m + v.
This order relation possesses the following property.
For any two natural numbers m and n, there exists one and only one of the following three
possibilities:
(i) m = n

(ii) m > n,
(iii) m < n.
This is known as the Trichotomy law of natural numbers.
It is evident that any set of natural numbers has a smallest number, i.e., if A is a non-empty
subset of N, there is a number m A, such that m n for every n A.

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