If the x-coordinates of each ordered pair are unique, that is, each input of the set corresponds to only one output, then the set can be more specifically called a function.
v is a set of ordered pairs (a, b) of real numbers. Sum and scalar multiplication are defined by: \((a, b) + (c, d) = (a + c, b + d) k (a, b) = (kb, ka)\) (attention in this part) show that V is not linear space.
Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where \(u \in U\) and \(w \in W\). Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)
(This space V is called the external direct product of U and W.)