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Unformatted text preview: Spring 2010 CS530 Analysis of Algorithms Midterm exam 2 M IDTERM EXAM 2 Only a single hand-written crib sheet can be used, no books or notes. Even if I ask for just a yes / no answer, you must always give a proof. You may get some points even if you write I dont know, but if you write something that is wrong, you may get less. (It is not possible to pass just writing I dont know everywhere....) Problem 1. (10pts) Given n n matrices A and B , prove rank ( A + B ) rank ( A ) + rank ( B ) . Solution. The rank of a matrix is the dimension of its column space. Let e 1 ,..., e r be a basis of the column space of A , and f 1 ,..., f s be a basis of the column space of B . Then every column of A is a linear combination of the vectors e i : say, a 1 = 1 e 1 + + r e r . Similarly, every column of B is a linear combination of the vectors f j : say, b 1 = 1 f 1 + + s f s . Then every column of A + B is a linear combination of the vectors e i , f j : say, a 1 + b 1 = 1 e 1 + + r e r + 1 f 1 + + s f s . Problem 2. (10pts) Consider the following linear program. maximize 2 x 1 + 5 x 2 + 6 x 3 + x 4 + 3 x 5 subject to 2 x 1- 3 x 2 + x 3 + 4 x 4 + 5 x 5 10,- 3 x 1 + 4 x 2 + 5 x 3 + x 4 + 9 x 5 5, x...
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This document was uploaded on 10/05/2010.
- Spring '09