EL ENG 127 Midterm: ee127-sp2011-mt1-El Ghaoui-soln
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Midterm solutions: (6 points) find the projection z of the vector x = (2, 1) on the line that passes through x0 = (1, 2) and with direction given by the vector u = (1, 1). L = {x0 + tu : t r} =(cid:26)(cid:18) 2 + t. 1 + t (cid:19) : t r(cid:27) . The problem reads kx0 + tu xk2. Expanding the squares, we express the objective function as min z l kz xk2 = min t. 2 = t2ut u 2t(x x0)t u + (x x0)t (x x0) = ut u(t )2 + constant, kx0 + tu xk2 where = (x x0)t u/ut u = 0. The optimal t is thus t = 0, hence the projection is z = x0 + t u = x0. 1: (12 points) consider the 2 2 matrix. Express it as a = u sv t , with s the diagonal matrix of singular values ordered in decreasing fashion.
