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Suppose V and W arc finito dimcnsional with dim V =n and dim W = m. Let +y, +w and•v, 'w denote vector addition and scalar multiplication for V, W

respectively. Define the set U:=\{(v, w): v \in V, w \in W\} with addition and scalar multiplication operations on U defined by the formulas \left(v_{1}, w_{1}\right)+\left(v_{2}, w_{2}\right)=\left(v_{1}+v v_{2}, w_{1}+w w_{2}\right) c(v, w)=(c \cdot v v, c \cdot w w) \quad \text { for all } c \in F (a) Prove that U is a vector space over F with respect to these operations by explicitlyverifying that it satisfies the vector space axioms. (b) Prove that U is isomorphic to Fm,n.

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