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soln5

# soln5 - Math 152 Spring 2010 Assignment#5 Notes Each...

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Math 152, Spring 2010 Assignment #5 Notes: Each question is worth 5 marks. Due in class: Monday, February 8 for MWF sections; Tuesday, February 9 for TTh sections. Solutions will be posted Tuesday, February 9 in the afternoon. No late assignments will be accepted. 1. (a) Use Gaussian elimination to find the set of solutions to the following homogeneous system. Present your answer in parametric form. x 1 + 4 x 2 + 4 x 3 + x 4 = 0 2 x 1 + x 2 + 3 x 4 = 0 x 1 + x 2 + x 3 + x 4 = 0 4 x 1 x 3 + 5 x 4 = 0 Solution: We first rewrite the system as an augmented matrix, and then use row operations to put it in reduced row echelon form: 1 4 4 1 0 2 1 0 3 0 1 1 1 1 0 4 0 1 5 0 −→ 1 4 4 1 0 0 7 8 1 0 0 3 3 0 0 0 16 17 1 0 R 2 = R 2 2 R 1 R 3 = R 3 R 1 R 4 = R 4 4 R 1 −→ 1 4 4 1 0 0 3 3 0 0 0 7 8 1 0 0 16 17 1 0 R 2 R 3 −→ 1 4 4 1 0 0 1 1 0 0 0 7 8 1 0 0 16 17 1 0 R 2 = 1 3 R 2

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−→ 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 R 1 = R 1 4 R 2 R 3 = R 3 + 7 R 2 R 4 = R 4 + 16 R 2 −→ 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 R 3 = R 3 −→ 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 R 2 = R 2 R 3 R 4 = R 4 + R 3 Then there is a free variable x 4 , and we can write x 1 + x 4 = 0 x 2 + x 4 = 0 x 3 x 4 = 0 0 = 0 −→ x 1 = x 4 x 2 = x 4 x 3 = x 4 x 4 = x 4 −→ vectorx = x 4 [ 1 , 1 , 1 , 1] .
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soln5 - Math 152 Spring 2010 Assignment#5 Notes Each...

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