Are the following sets vector spaces with the indicated operations? If not, why not?
(a) The set V of all polynomials of degree ≥ 3, together with 0; operations of P.
(b) The set {1, x, x2,...}; operations of P.
(c) The set V of 2 × 2 matrices with equal column sums; operations of M22.
(d) The set V of real numbers; usual operations.
(e) The set V of all ordered pairs (x, y) with the addition of R2, but scalar multiplication a(x, y) = (ax, - ay).
(f) The set V of all functions f: R → IR with point wise addition, and scalar multiplication defined by (af)(x) = f(ax).
(g) The set V of all 2 × 2 matrices with the addition of M22 but scalar multiplication * defined by a * X = aXT.