Exam1Solutions

# Exam1Solutions - Newberger Math 247 Spring 03 Exam 1 name 1...

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Unformatted text preview: Newberger Math 247 Spring 03 Exam 1 name: 1. The augmented matrix of a linear system is 1 3 5 7 3 5 7 9 5 7 9 11 . a. Reduce the augmented matrix to reduced echelon form. 1 3 5 7 3 5 7 9 5 7 9 11 ∼ 1 3 5 7- 4- 8- 12- 8- 16- 24 ∼ 1 3 5 7 0 1 2 3 0 0 0 ∼ 1 0- 1- 2 0 1 2 3 0 0 . b. Write the general solution of the linear system in parametric vector form. The general solution is x 1 =- 2 + x 3 x 2 = 3- 2 x 3 x 3 is free. So in parametric vector form, we have: x 1 x 2 x 3 = - 2 + x 3 3- 2 x 3 x 3 = - 2 3 + 1- 2 1 x 3 , where x 3 is any real number. c. (6 points) Give a geometric description of the solution set. The solution set is a line in R 3 through - 2 3 parallel to 1- 2 1 . Note that since there are three variables, the solution set lies in R 3 . If there had been 2 free variables, the geometric description would be a plane in R 3 . 1 2. The problems on this page refer to the following vectors in R 3 . a 1 = 1 1 , a 2 = 2 1 3 , a 3 = 1 k , b = 4 3 h . 1 2 0 4 0 1 1 3 1 3 k h ∼ 1 2 0 4 0 1 1 3 0 1 k h- 4 ∼ 1 2 4 0 1 1 3 0 0 k- 1 h- 7 (a) (6 points) Find all values of h and k such that b is in Span { a 1 , a 2 , a 3 } ....
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Exam1Solutions - Newberger Math 247 Spring 03 Exam 1 name 1...

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