If V is a linear space, we must have that \(lv=v\),
for all v from V. Take for example \(v=(0,1)\)
Then \(lv=1(0,1)=(1 \cdot 1,1 \cdot 0)=(1,0)= v.\)
Thus 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.)