Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where u ∈ U and w ∈ 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.)

Question
Vectors and spaces
Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where u ∈ U and w ∈ 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.)

2021-02-26
B is a vector space over K with addition in V and scalar multiplication by V

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