Page 60 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 14. Prove that the set of all real numbers of the form u 3 where u and are of the form
a b 2 in which a and b are rational numbers, is a field.
15. Prove that the set E of all even integers is a commutative ring but not a field.
16. Show that a finite commutative ring without zero divisors is a field.

1.5 Vector Spaces

Before giving a formal definition of an abstract vector space we define what is known as an
external composition in one set over another. We have already defined a binary composition in
a set A as a mapping of A × A to A. This may be referred to as an internal composition in A. Let now
A and B be two non-empty sets. Then a mapping

: f A B B
is called an external composition in B over A.
Definition: Let (F, + , .) be a field. Then a set V is called a vector space over the field F, if V is an
abelian group under an operation which is denoted by +, and if for every a F, u V there is
defined an element a u in V such that:

,
u
(i) ( a u ) au a for all a F , , V .
,
,
a
(ii) (a b )u au bu for all ,b F u V .
a
(iii) ( a bu ) ( ) , for all ,b F u V .
a
b
,
u
(iv) 1 . u u . 1 represents the unity element of F under multiplication.
The following notations will be constantly used in the forthcoming discussions.
(i) Generally F will be field whose elements shall often be referred to as scalars.
(ii) V will denote vector space over F whose elements shall be called as vectors.
Thus to test that V is a vector space over F, the following axioms should be satisfied.
V (V, +) is an abelian group.
1
(V ) Closure law: ,u V u V .
11
(V ) Associative law: For all , ,u w V (u ) w u ( w )
12
(V ) Existence of identity: There exists an element of zero vector.
13
(V ) Existence of Inverse: For all u V, there exists a unique vector –u V such that
14
u + (–u) = 0
(V ) Commutative Law:
15
u u for ,u V
V scalar multiplication is distributive over addition in V, i.e.,
2
( a u ) au a , a F , V
V distributivity of scalar multiplication over addition in F, i.e.,
3
a
(a b )u au bu , ,b F ,u V .

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