Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail. The set of all pairs of real numbers of the form (1,x) with the operations(1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky)

Question

Determine whether the following set equipped with the given operations is a vector space. If not vector spaces identify the vector space axioms that fail. The set of all pairs of real numbers of the form (1,x) with the operations(1,y)+(1,y&#039;)=(1,y+y&#039;) and k(1,y)=(1,ky)

Jeffrey Jordon · Accepted Answer

Not a vector space because the scalar multiplication is not hold.&amp;nbsp;If we Put k=0 then according to the definition of vector space&amp;nbsp;0&amp;nbsp;×&amp;nbsp;u=0but in this case0(1,y)=(1,0)&amp;nbsp;≠&amp;nbsp;(0,0)