1. Determine whether or not each of the following sets V = (F,G) are valid vector spaces. Except in part e., the group operator is parallelogram vector addition: a. G is the set of points (r, y) in the Cartesian plane such that y = 3r + 2. F is the field of real numbers. b. G is the set containing only the zero element (r 0, y = 0), F is again the reals. c. G is the set of points (z, y) such that y = r - 2.r. F is the reals. d. G is the set of all points (r, y) in the Cartesian plane. F is the reals. e. G is the set of ordered pairs g = (p,q) where p and q each belong to corresponding vector spaces P and Q, which are both vector spaces over the same field F. For each pi, P2 in P and each q1, 42 in Q, define the sum of two elements in G (denoted here by the A operation) as follows: 9A92 = (P1, 41)A(p2, 42) = (P1 + P2, q1 + 2) %3D %3D Scalar multiplication is defined by a(pı, q1) = (op1, aq1), for any a in F. %3D

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