MTexam - Name(print ID number Section(circle 1 2 3...

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Unformatted text preview: Name (print): ID number: Section (circle): 1 2 3 University of Waterloo MATH 136 Linear Algebra I MIDTERM EXAM Term: 1089 Date: October 6, 2008 Time: 7:00 p.m. to 9:00 p.m. Number of pages: 10 (including cover page) Exam type: Closed book Aids: No calculators or other electronic aids are allowed Instructions: Write your name and ID number, and circle your section number, at the top of this page. Answer the questions in the spaces provided, using the backs of pages for overflow or rough work. For examiners’ use only Question Mark Possible marks 1 8 2 8 3 8 4 10 5 8 6 8 7 10 8 3 (bonus) Total 60 1. Consider the system of equations x 1 + 2 x 2- x 3 = a 2 x 1 + x 2 + 3 x 3 = b x 1- 4 x 2 + 9 x 3 = c. (a) [4 marks] Show that the system is inconsistent unless c = 2 b- 3 a . The system has associated augmented matrix 1 2- 1 a 2 1 3 b 1- 4 9 c , which reduces to 1 2- 1 a- 3 5 b- 2 a 3 a- 2 b + c . Thus the system is inconsistent unless 3 a- 2 b + c = 0 ; that is, unless c = 2 b- 3 a . (b) [4 marks] Let a =- 3, b = 3 and c = 15. Write the solution set to the system of equations in parametric vector form. After substituting the given values for a , b and c , the augmented matrix associated with the system reduces to 1 7 / 3 3 1- 5 / 3- 3 . The variable x 3 is free; we assign to it a parameter t , getting x 1 = 3- 7 3 t, x 2 =- 3 + 5 3 t and x 3 = t. Thus the solution set to the system of equations is given by x 1 x 2 x 3 = 3- 3 + t - 7 / 3 5 / 3 1 , t ∈ R . 2 2. Let x 1 = - 1 1 2 , x 2 = 3 1 1 and x 3 = 1 1 1- 4 . (a) [4 marks] Write 2- 6- 1- 8 as a linear combination of x 1 , x 2 and x 3 . We wish to find constants a , b and c such that a x 1 + b x 2 + c x 3 = 2- 6- 1- 8 . This is equivalent to solving the matrix equation associated with the augmented matrix - 1 1 2 1 3 1- 6 1 1- 1 2 1- 4- 8 ....
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MTexam - Name(print ID number Section(circle 1 2 3...

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