Suppose you are given m (m greaterthanorequalto 2) straight lines L_i = {p_i + t_iq_i | t_i Element R}, i = 1, ..., m in R^n. Each line is defined by two n-vectors p_1, q-2. The vector p, is a point on the line; the vector q_i specifies the direction. We assume that the vectors q_i are normalized (||q_i = 1) and that at least two of them are linearly independent. (In other words, the vectors q_1 are not all scalar multiples of the same vector, so the lines are not all parallel.) We denote by d_i(y) - min_u_1 Element L_i ||y - u_i| = min_L_i ||y - p_i - t_iq_i|| the distance of a point y to the line L_i. Express the following problem as a linear least squares problem. Find the point y Element R^n that minimizes the sum of its squared distances to the m lines, i.e., find the solution of the optimization problem minimize Sigma^infinity _i = 1 d_i (y)^2 with variable y. Express the least squares problem in the standard form minimize ||Ax - b||^2 with a left invertible matrix A. Clearly state what the variables x in the least squares problem are and how A and b are defined. Formulate the following problem as a least squares problem. Find a polynomial p(t) = x_1 + x_2t + x_3t^2 + x_4t^3 that satisfies the following conditions. The values p(t_i) at 4 given points t_i, in the interval [0, 1] is approximately equal to given values y_i: p(t_i) almostequalto y_i, i = 1, ..., 4. The points t_i are given and distinct (t_i notequalto t_j for i notequalto j). The values y_i are also given. The derivatives of p at t = 0 and t = 1 are small: p'(0) almostequalto 0, p'(1) almostequalto 0. The average value of p over the interval [0, 1) is approximately equal to the value at t - 1/2: integral^1_0 p(t) dt almostequalto p(1/2). To determine coefficients x_i that satisfy these conditions, we minimize E(x) = 1/4 Sigma^infinity _n = 1 (p(t_i) - y_i)^2 + p'(0)^2 + p'' (1)^2 + (integral^1_0 p(t) dt - p(1/2))^2. Give A and 6 such that E(z) = ||Ax - b||^2. Clearly state the dimensions of A and b, and what their elements are