Question 418456: Determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail: (a)the set of all real numbers with the standard operations of addition and multiplication
(b) the set of all pairs of real numbers of the form (x,0), with standard operations on R^2
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! (a) Under the standard operations of addition and multiplication, the set of all real numbers follows the closure, distributive, associative, additive inverse, and identity element properties. It also follows the usual distributive laws. Therefore R is a vector space.
(b)The set of all pairs of real numbers of the form (x,0), with standard operations on R^2 is isomorphic to R, and hence is also a vector space.
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