midterm1_2009_answers

# midterm1_2009_answers - Math 107 First Midterm Examination...

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Math 107 First Midterm Examination February 18, 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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(10 Points) Score Solve 5 x 1 + 3 x 2 + x 3 = - 3 10 x 1 + 10 x 2 + 4 x 3 = - 2 15 x 1 + 17 x 2 + 10 x 3 = 5 We can factor 5 3 1 - 3 10 10 4 - 2 15 17 10 5 5 3 1 - 3 2 4 2 4 3 8 7 14 5 3 1 - 3 2 4 2 4 3 2 3 6 Let us check this factorization: 1 2 1 3 2 1 5 3 1 4 2 3 = 5 3 1 10 10 4 15 17 10 Then we back-solve 5 3 1 4 2 3 x 1 x 2 x 3 = - 3 4 6 = x 1 x 2 x 3 = - 1 0 2 Let us check the back-solve: 5 3 1 4 2 3 - 1 0 2 = - 3 4 6 Let us check the solution in the original linear system: 5 3 1 10 10 4 15 17 10 - 1 0 2 = - 3 - 2 5 70% of the class solved this problem correctly. answer = x 1 = - 1 , x 2 = 0 , x 3 = 2 . 2
(25 Points) Score Suppose that we have factored A = LU where L = 1 4 1 7 3 , U = 2 5 8 - 24 4 16 - 32 1. (2 points) What is the dimension of the range of A (also called the column space of A)? 78% of the class solved this problem correctly. answer = 2 = r . 2. (2 points) What is the dimension of the nullspace of A? 74% of the class solved this problem correctly. answer = 2 = n-r . 3. (2 points) What is the dimension of the nullspace of A ? 17% of the class solved this problem correctly. answer = 1 = m-r . 4. (6 pts) Find a basis for the range of A. 61% of the class solved this problem correctly.

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