1. Let V be a vector space over F and define U = span(u1, u2, . . . , un), where for each i = 1, . . . , n, ui V . Now suppose v U. Prove U = span(v, u1, u2, . . . , un)

2. Let Fm[z] denote the vector space of all polynomials with degree less than or equal tom Z+ and having coefficient over F, and suppose that p0, p1, . . . , pm Fm[z] satisfy pj (2) = 0. Prove that (p0, p1, . . . , pm) is a linearly dependent list of vectors in Fm[z].

3. Show that the space V = {(x1, x2, x3) F3| x1 + 2x2 + 2x3 = 0} forms a vector space.