Let V be a vector space over  with addition and scalar multiplication denoted by + and . respectively. Let W = V x V = {(v1, v2) | v1, v2 V}. Determine whether or not W is a vector space over...

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Let V be a vector space over  with addition and scalar multiplication denoted by + and . respectively. Let W = V x V = {(v1, v2) | v1, v2 V}. Determine whether or not W is a vector space over  with addition defined by (u1, u2) (v1, v2) = (u1+V1, u2+v2) for all (u1, u2) , (v1, v2)  W and scalar multiplication defined by (a + bi) (v1, v2) = (a.v1 – b.v2,b.v1+a.v2) for all a+bi   and (v1, v2)  W. Here i = -1  and a, b  . If so, prove that all the conditions in the definition of a vector space are satisfied. If not, show by example that at least one condition does not hold.

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