1 answer

Linear algebra question

Question:

let V denote the set of ordered pairs of real numbers.If (a1,a2) and (b1,b2) are elements of V and cR, define

(a1,a2) + (b1,b2) = (a1+b1,a2 b2)andc(a1,a2) = (ca1,a2).

Is V a vector space over R with these operations? justify your answer.


Answers

To qualify as a vector space, the set V and the operations of addition and multiplication have to adhere to a number of requirements called axioms.

Axiom
Associativity of addition----followed as (a1,a2) + (b1,b2) = (a1+b1,a2 b2)
if (a1,a2) and (b1,b2) belong to c then (a1+b1,a2 b2) belong to c

Commutativity of addition...followed as if if (a1,a2) belong to c then (a2,a1) belong to c

Identity element of addition....(0,0) identity element of addition

Inverse elements of addition.... followed as if if (a1,a2) belong to c then (-a1,-a2) belong to c and it is inverse element of addition

Distributivity of scalar multiplication with respect to vector addition:-

if u &v belong to c then a(u + v) = au + av belongs to c

Identity element of scalar multiplication:- (1,1) is identity element for scalar multiplication

as all are satisfied V is a vector space over R with these operations given.

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