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# midterm2_2009_answers - Math 107 Second Midterm Examination...

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Math 107 Second Midterm Examination March 20, 2009 NAME (Please print) Page Score 2 3 4 5 6 7 8 Total Instructions: 1. Do all computations on the examination paper. You may use the backs of the pages if necessary. 2. Put answers inside the boxes (when applicable). 3. Please signify your adherence to the honor code: I, , have neither given nor received aid in completion of this examination. 1

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(20 Points) Score Let b = 0 1 1 , A = 1 2 1 2 4 3 3 6 4 1 2 1 2 0 1 3 0 1 1 3 2 1 1 2 2 1 0 3 1 0 1 3 2 1 1 2 2 1 0 3 1 0 1. (3 points) What is the dimension of the range of A, a.k.a. the column space of A? dim R ( A ) = 2 82 % correct 2. (3 points) What is the dimension of the nullspace of A? dim N ( A ) = 1 82 % correct 3. (10 points) Find the general solution to Ax = b. We factored A P = L U above. To find a particular solution, we forward-solve L y = b for y : 1 2 1 3 1 0 1 = 0 1 1 then we back-solve U x P = y for x with the free variable set to zero, and put the solution back in the original order: 1 1 2 1 0 - 1 1 0 = 0 1 = x P = P x P = - 1 0 1 To find a basis for the homogeneous solutions, we solve U x H = 0 , and put the solution back in the original order: 1 1 2 1 0 - 2 0 1 = 0 0 = x H = P x H = - 2 1 0 The general solution is x P + x H c : x = - 1 0 1 + - 2 1 0 c 18 % correct 4. (4 points) Find a nonzero vector z that is orthogonal to the nullspace of A . z = b = 0 1 1 since b ∈ R ( A ) and R ( A ) ⊥ N ( A ). 41 % correct 2
(15 Points) Score Consider the equation X 1 1 1 1 1 0 0 1 0 1 1 0 = 1 1 1 1 0 2 2 0 0 3 3 0 1. (3 points) How many rows and columns must X have? X has 3 rows and 3 columns 91 % correct 2. (6 points) Find a particular solution X P to this equation. Transpose the equation XA = B above to get A X = B . We can factor [ A , B ]: 1 1 0 1 0 0 1 0 1 1 2 3 1 0 1 1 2 3 1 1 0 1 0 0 1 1 0 1 0 0 1 - 1 1 0 2 3 1 - 1 1 0 2 3 1 0 0 0 0 0 1 1 0 1 0 0 1 - 1 1 0 2 3 1 1 0 0 0 0 1 0 0 0 0 0 We can back-solve U X P = Y

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midterm2_2009_answers - Math 107 Second Midterm Examination...

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