Let V be a vector space over a field F. Suppose that U and W are subspaces of V . We make U × W = {(u, w) | u ∈ U, w ∈ W} a vector space over F by defining (u1, w1) + (u2, w2) = (u1 + u2, w1 + w2), λ(u1, w1) = (λu1, λw1) for u1, u2 ∈ U, w1, w2 ∈ W and λ ∈ F. With these definitions of + and scalar multiplication prove that U×W is a vector space over F

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