Assignment 1 Deadline: 2023-02-07 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1. Let u = [u1, u2], v = [v1, v2] E R2. Show that (u, v) = 4u1v1 + 3u272 defines an inner product. 2. Let S = span {v1, V2, V3}. Check whether the vector u e St (a) v1 = [1, 0, 0, 0], v2 = [0, 3, 0, 0], v3 = [5, 2, 1, 0], and u = [0, 0, 0, 1]. (b) v1 = [1, -2, 3, 1], v2 = [2, 0, 3, 5], v3 = [0, 1, 2, 5], and u = [0, 1, 3, 0]. (c) v1 = [3, 4, 1, 7], v2 = [1, 0, 3, 1], v3 = [-1, 2, -1, 1], and u = [-1, -1, 0, 1]. 3. Show that if u is orthogonal to each of the vectors v1, V2, . .., Vn, then it is also orthogonal to span { V1, V2 , . . ., Vn). 4. Let (u, v) be the Euclidean inner product on R". Show that for any A E Rnxn, (u, Av) = (Au, v). 5. In each part, determine whether the given vectors span Rs. Consider R3 with the Euclidean inner product. For which values of k, are u and v orthogonal? (a) u = [1,3, -4], v = [-2, k, 6]. (b) u = [k, k, 1], v = [k, 4, 4]. 6. Prove Theorem 5.9 (a) and (b). 7. Find the orthogonal projection of v onto the subspace W spanned by vectors uj and u2, where v = [1, 2, 3], u1 = [2, -2, 1], and u2 = [-1, 1, 4]