# final - MATH 51 FINAL EXAM Brumfiel Hutchings Levandosky...

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Unformatted text preview: MATH 51 FINAL EXAM December 11, 2000 Brumfiel Hutchings Levandosky Staffilani White 11:00 01 05 09 13 17 1:15 03 07 11 15 19 Name: Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the honor code. Circle the number of the section for which you are registered on Infopier. During the test, you may not use notes, books, or calculators. Read each question carefully, and show all your work. Put a box around your final answer to each question. You have 3 hours to do all the problems. Question Score Maximum 1 14 2 10 3 16 4 12 5 14 6 10 7 12 8 12 9 10 10 18 11 10 12 10 13 10 14 15 15 15 16 12 Total 200 1 1. Let u 1 = 1- 1 3 u 2 = 1 1- 2 u 3 = 3- 1 4 (a) (6 points) Find the dimension of span( u 1 , u 2 , u 3 ). (b) (8 points) Find all vectors v which are simultaneously orthogonal (i.e. perpen- dicular) to all three vectors u 1 , u 2 and u 3 . 2. (10 points) Suppose B = ( x,y ) is a point on the circle of radius 1 centered at the origin. That is, x and y satisfy x 2 + y 2 = 1. Let A = (- 1 , 0), C = (1 , 0) and assume y 6 = 0 (so that B is not equal to A or C ). A B C Use dot products to show that angle ABC is a right angle. 3. Suppose A is a 5 × 5 matrix with rref( A ) = 1 0- 1 4 0 0 1 2 3 0 0 0 0 1 0 0 0 0 0 0 0 0 For each part below, give the answer when possible. Otherwise answer “not enoughFor each part below, give the answer when possible....
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final - MATH 51 FINAL EXAM Brumfiel Hutchings Levandosky...

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