Determine whether the set equipped with the given operations is a vector space.

Question

Determine whether the set equipped with the given operations is a vector space.

For those that are not vector spaces identify the vector space axioms that fail.

The set of all pairs of real numbers of the form (11,x) with the operations

Chapter 4, Section 4.1, Question 11 Determine whether the set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail The set of all pairs of real numbers of the form (11,x) with the operations k(11,y) = (11.ky) O Vis not a vector space, and Axioms 4 and 5 fail to hold. Vis a vector space. V is not a vector space, and Axioms 1, 2, 3 fail to hold. Vis not a vector space, and Axiom 4 fails to hold. V is not a vector space, and Axioms 8 and 9 fail to hold. O

Explanation / Answer

The 10 axioms that a vector space V must satisfy are as under:

1. For all X, Y V, X+Y V( closure under vector addition).

2. For all X, Y , X+Y = Y+X ( commutativity of vector addition).     

3. For all X, Y, Z , (X+Y)+Z=X+(Y+Z) (Associativity of vector addition).

4. For all x, 0+X = X+0 = X ( Existence of Additive identity)

5. For any X, there exists a -X such that X+(-X)= 0 (Existence of additive inverse)

6. For any scalar k and the vector v V, the vector kv V( closure under scalar multiplication).

7. For all scalars r and vectors X,Y, r(X+Y)=rX+rY (Distributivity of vector addition).

8. For all scalars r,s and vectors X , (r+s)X=rX+sX (Distributivity of scalar addition).

9. For all scalars r,s and vectors X, r(sX)=(rs)X( Associativity of scalar multiplication).

10. For all vectors X, 1X=X ( Existence of Scalar multiplication identity).

It may be observed that all the axioms, including 1,2,3,4,5,8 and 9 hold. Therefore the given set , with the described operations is a vector space. The additive identity is (11,0) and the additive inverse of (121,x) is (11,-x).

The 2nd option is the correct answer.

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