exam_1_20F_solutions

exam_1_20F_solutions - Name Katie Student ID...

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Name Student ID TA/Section (circle): Katie 12pm-C01 1pm-C02 2pm-C03 Mary 3pm-C04 8pm-C05 9pm-C06 Math 20F, Winter 2010, Midterm Exam 1 Solutions Show all of your work and justify your answers to receive full credit. Write your answers and work clearly and legibly; no credit will be given for illegible solutions. Go back and check your answers if you finish early. # Points Score 1 6 2 3 3 2 4 3 5 2 6 2 7 2 Σ 20
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1. Let A = 1 1 1 2 3 1 1 0 2 and b = 4 7 5 (a) (4 points) Find the solution set of the equation A x = b . Write your answer in parametric vector form . Answer: We row reduce the augmented matrix. 1 1 1 4 2 3 1 7 1 0 2 5 1 1 1 4 0 1 1 1 0 1 1 1 1 1 1 4 0 1 1 1 0 0 0 0 1 0 2 5 0 1 1 1 0 0 0 0 Column three is not a pivot column, so x 3 is a free variable. This gives the equations x 1 + 2 x 3 = 5 x 2 x 3 = 1 or rearranging, x 1 = 5 2 x 3 x 2 = 1 + x 3 In parametric vector form this is x = x 1 x 2 x 3 = 5 2 x 3 1 + x 3 x 3 = 5 1 0 + x 3 2 1 1 (b) (2 points) What is the parametric vector form of the solution set of the system A x = 0 ? Answer: The solution to the system A x = 0 is the same as in part (a) except the vector 5 1 0 is removed.
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