2008_Spring_midterm_1_solutions

2008_Spring_midterm_1_solutions - Name: PID: TA: Sec. No:...

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Unformatted text preview: Name: PID: TA: Sec. No: Sec. Time: Math 20F. Midterm Exam 1 Solutions April 23, 2008 Turn off and put away your cell phone. You may use one page of notes, but no calculators, books or other assistance. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. # Points Score 1 6 2 14 3 12 4 8 40 1. (6 points) Consider the following vectors in R 3 : v 1 = 1- 2 , v 2 = 2 1 , v 3 = 1 1 , v 4 = - 1 3 . Are { v 1 , v 2 , v 3 , v 4 } linearly independent? If so, explain why, and if not, find a nontrivial relation of linear dependence among these vectors. Solution: Consider the matrix A = 1 2 1- 2 1- 1 1 3 . The solutions to the homogeneous system A x = are precisely the vectors x = x 1 x 2 x 3 x 4 such that x 1 v 1 + x 2 v 2 + x 3 v 3 + x 4 v 4 = . The RREF of A is 1- 5 / 3 1 3 1- 13 / 3 . Setting the free variable x 4 = t , we have the general solution x = (5 / 3) t- 3 t...
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2008_Spring_midterm_1_solutions - Name: PID: TA: Sec. No:...

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