Suppose that V and W are vector spaces. The Cartesian product space, denoted by V × W is defined as the set of all ordered pairs (v. w) where v ∈ V, w ∈ W, with vector addition (v, w) + ( , ) = (v +  ,w + ) and scalar multiplication c(v, w) = (cv, cw).
(a) Prove that V × W is a vector space.
(b) Explain why R × R is the same as R2.
(c) More generally, explain why Rm × Rn is the same as Rm+n.