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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Put the following three-by-three linear system into the proper form for conversion to matrix notation: x=y z 7 y = 2x + 2z + 2 z = 3x 5y + 4 2 Write the set of equations from the solution to Prob 1 in the form of a matrix 3 Write the set of equations represented by the following matrix:

0 5 1 4 3/2 1 1 8 1 2 1 1

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cost as the budget Fine, but how do we nd the optimum cost Easy: By binary search! (See Exercise 81) Incidentally, there is a subtlety here: Why do we have to introduce a budget Isn t any optimization problem also a search problem in the sense that we are searching for a solution that has the property of being optimal The catch is that the solution to a search problem should be easy to recognize, or as we put it earlier, polynomial-time checkable Given a potential solution to the TSP, it is easy to check the properties is a tour (just check that each vertex is visited exactly once) and has total length b But how could one check the property is optimal As with SAT, there are no known polynomial-time algorithms for the TSP, despite much effort by researchers over nearly a century Of course, there is an exponential algorithm for solving it, by trying all (n 1)! tours, and in Section 66 we saw a faster, yet still exponential, dynamic programming algorithm The minimum spanning tree ( MST ) problem, for which we do have ef cient algorithms, provides a stark contrast here To phrase it as a search problem, we are again given a distance matrix and a bound b, and are asked to nd a tree T with total weight (i,j) T dij b The TSP can be thought of as a tough cousin of the MST problem, in which the tree is not allowed to branch and is therefore a path1 This extra restriction on the structure of the tree results in a much harder problem

matching section of transmission line The input impedance is Rin, the output impedance is Rout, and the characteristic impedance of the line is Zo

4 Put the matrix of Prob 3 into echelon form 5 Put the matrix derived in the solution to Prob 4 into diagonal form 6 Reduce the matrix derived in the solution to Prob 5 to a form with the smallest possible absolute values in each row, such that all the numbers in the matrix are integers 7 Reduce the matrix derived in the solution to Prob 6 to unit diagonal form Then state the tentative solution to the three-by-three linear system we derived from the matrix in Prob 3 and stated in solution 3 8 Check the values for x, y, and z derived in the solution to Prob 7 to be sure they re correct To do this, plug the numbers into the equations stated in the solution to Prob 3

In the summer of 1735 Leonhard Euler (pronounced Oiler ), the famous Swiss mathemati cian, was walking the bridges of the East Prussian town of K onigsberg After a while, he noticed in frustration that, no matter where he started his walk, no matter how cleverly he continued, it was impossible to cross each bridge exactly once And from this silly ambition, the eld of graph theory was born Euler identi ed at once the roots of the park s de ciency First, you turn the map of the park into a graph whose vertices are the four land masses (two islands, two banks) and whose edges are the seven bridges:

9 Put the following three-by-three linear system into matrix form Then play the matrix morphing game for awhile, trying to get the unit diagonal form It can t be done! Why x+y+z=1 x+y+z=2 x+y+z=3 10 Put the following three-by-three linear system into matrix form Then play the matrix morphing game for awhile, trying to get the unit diagonal form It can t be done! Why x+y+z=1 2x + 2y + 2z = 2 3x + 3y + 3z = 3

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