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12.864 Inference from Data and Models 23 February 2004 Problem Set No. 2 Due: 1 March 2004 1. You have a single vector v1 = [1> 1> 1> 1]W , but need three more vectors in order to have a complete spanning set. Starting with v1 > construct a complete orthonormal basis set. 2. For the tracer balance model shown on P. 9 of the notes, there are three reservoirs with tracer concentrations Fl > 1 l 3> where F1 = 1=1045> F2 = 1=2385> F3 = 1=2900= Reservoir 0 is observed to have concentration, F0 = 1=2= The problem is to make some inference about the values of the exchange coe!cient ratios, {l = Ml @M0 , given also that mass is conserved (M0 = M1 + M2 + M3 ). (a) Find the solution of minimum norm, K = xW x in two ways, one of which uses Lagrange multipliers. (b) Find the change–by direct calculation–in K when F0 is changed from 1.2 to 1.18. Compare it to the value inferred from the Lagrange multipliers of part (a). (c) What would your conclusion be if F0 = 1=3? (d) Repeat (a), (b) except ﬁnd the solution minimizing K = ({2 {1 )2 + 4 ({3 {2 )2 = 3. In problem 2(a), the situation is the same, except that it is thought that mass balance is probably violated at an accuracy of about ±10%> so that the mass balance equation is changed to M0 = M1 + M2 + M3 + q where q represents the error. Can you ﬁnd a new solution? Can you ﬁnd more than one? 1