Question 1 (17 marks) a) Show that the function (V1, V2) = 2WW2 + X142 - y192 + 2122 defined for any vectors v1 = (T1, y1, 21, w1 ) and v2 = (12, 32, 22, W2) in vector space R4 is not an inner product. Hint. Indicate which axiom(s) of inner product space is(are) not satisfied and give an example of the vector(s) which do(es) not satisfy the axiom(s). (2 marks) b) Let S= {(x,y)|x,y e R). Suppose the vector addition and scalar multiplication in S are defined the same way as in R2. Let u = (u1, u2), V = (v1, v2) be vectors in S. We define an inner product in S according to the formula (u, v) = u1v1 - U1V2 - uzv1 + 21202. Show that S equipped with this inner product is a real inner product space. (4 marks) c) Let u = (4, 1) and v = (2,3) . i. Show that u and v form an orthogonal basis in the inner product space S defined in part b). Use this basis to find an orthonormal basis by normalizing each vector. (3 marks) ii. Use the inner product defined in part b) to express the vector W = (1, 1) as a linear combination of the orthonormal basis vectors obtained in part i. (2 marks) d) Let the vector space Pi have the inner product (p, q) = [ p(x)q(x)dx. Apply the Gram-Schmidt process to transform the set of vectors u1 = 1 +x, u2 = 1+ 3x into an orthonormal basis {e1, ez}. (6 marks)