Page 61 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

V Scalar multiplication is associative i.e., Notes
4
a
( a bu ) ( )u , a b F and u V
b
V Property of Unity: Let 1 F be the unity of F, then
5
1u u u | u V

A vector space V over a field F is expressed by writing V(F). Sometimes writing only V is
sufficient provided the context makes it clear that which field has been considered.

If the field is R, the set of real numbers, then V is said to be real vector space. If the field is Q, the
set of rational numbers, then V is said to be a rational vector space and if the field is C, the set of
complex numbers V is called a complex vector space.

Illustrative Examples

Example 31: Show that the set of all vectors in a plane over the field of real numbers is a
vector space.
Solution: Let V be the set of all Vectors in a plane and R be the field of real numbers.
(V ) (V, +) is an abelian group.
1
(V ) u , V u V (Closure axiom)
11
u
(V ) (u ) w u ( w ), for , ,w V (associative axiom)
12
(V ) There is a null vector O V such that
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u 0 u u V (additive identity)

(V ) If u V , u V and also u ( u ) 0
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Hence –u is inverse of u in V, i.e., inverse axiom is satisfied for each element in V.

(V ) u u for all , ,u V .
15
u
V 2 ( a u ) au a , a R , , V
V 3 (a b )u au bu , ,b R ,u V .
a
a
u
V 4 ( a bu ) ( ) , , b R ,u V .
ab
V 1 u u , u V , where 1 is unity of R.
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Hence V is a vector space over R.

Example 32: Let C be the field of complex numbers and R be the field of real numbers,
then prove that

(i) R is a vector space over R.
(ii) C is a vector space over C.

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