Problem 1 (The positive quadrant of R is a vector space) In this problem, we will see how the defini- tion of the vector addition and scalar multiplication affects being a vector space or not. Define the positive quadrant Q of R² as follows: Q := {(x, y) = R² | x > 0 and y > 0}. 1. Suppose we use the usual addition and multiplication of vectors, i.e., (x1,y1) + (x2,y2) = (x1+x2, Y1 + y2) c(x, y) = (cx, cy). Does Qwith these operations define a vector space? If so, provide a proof; if not, justify why. 2. Now suppose we define vector addition and scalar multiplication in a different way: (x1,y1) + (x2, y2) = (x1x2, Y1Y2) c(x, y) = (x, y). Prove that Q with these modified operations form a vector space. Remark: This illustrates that the question "Is V a vector space?" doesn't entirely make sense: the operations are equally important as the set itself. A better question would be "Is (V,+,) a vector space?" or "Can V be equipped with the structure of a vector space?".

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