1 Let S be a set of all pairs of real numbers of the form 1x

1) Let S be a set of all pairs of real numbers of the form (1,x) with the operations defined as follows:

(1, x) + (1, y) = (1, x + y) and c(1, x) = (1, cx)

Verify that S together with these operations satisfy at least any 7 vector space axioms of your choice.

Solution

Axioms of real vector spaces

A real vector space is a set X with a special element 0, and three operations:

These operations must satisfy the following axioms:

Hence all axioms are satisfied so a vector space.