1. The problem statement, all variables and given/known data Determine if the following set is a vector space under the the given operations. The set V of all pairs of real numbers of the form (1,x) with the operations: (1, y) + (1, y') = (1, y + y') k(1, y) = (1, ky) 2. Relevant equations 3. The attempt at a solution Axiom 4: There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u. u = (1, y) (1, y) + (1, 0) = (1, y+ 0) = (1, u) = u May I consider (1,0) as the zero vector??? I have a doubt if the zero vector 0 has to be (0,0) always or the zero can be defined as any vector (in this case (1,0) ) that remains the vector untouched under addition. Axim 5: for each u in V, there is an object -u in V, called a negative of u, such that u + (-u) = (-u) + u = 0 u = (1,x) -u = (-1)(1,x) = (1, -x) by definition of scalar multiplication given for the set V. (Is this correct?) then u + (-u) = (1,x) + (-1)(1,x) = (1,x) + (1,-x) = (1,x+(-x)) = (1,0) = 0 in this case (1,0) is the zero vector.