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MATH 330 Exam 1 Solution

# MATH 330 Exam 1 Solution - Math 330 Exam 1 Feb 11th 2009 NO...

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Unformatted text preview: Math 330 Exam 1, Feb 11th, 2009 NO calculators are allowed on the following. Be sure to show all work. This includes writing down any needed matrices are carefully writing out each and every row oper- ation you perform. Unjustified answers will receive no credit. Problem Possible Earned Unique Solution 15 General Solution 15 Vector Span 15 Linear Independence 15 A Linear Transformation 15 Short/Answer/TRUE/FALSE 15 Application 10 TOTAL 100 Extra Credit 5 Unique Solution - 15 points Convert the following system to an augmented matrix and solve by using row operations to convert it to Reduced Echelon Form . Note that in the solution all the variables are integers!- x- y + z =- 10 x + y = 7 5 x + 4 y- 4 z = 45 We use an augmented matrix which we take to reduced Echelon form: SOLUTION: - 1- 1 1- 10 1 1 7 5 4- 4 45 R 1 ←- R 1 = ⇒ 1 1- 1 10 1 1 7 5 4- 4 45 R 2 ← R 1+ R 2& R 3 ← R 3- 5 R 1 = ⇒ 1 1- 1 10 1- 3- 1 1- 5 R 2 ↔ R 3 = ⇒ 1 1- 1 10- 1 1- 5 1- 3 R 2 ←- R 2 = ⇒ 1 1- 1 10 0 1- 1 5 0 0 1- 3 R 2 ← R 2+ R 3& R 1 ← R 1+ R 3 = ⇒ 1 1 0 7 0 1 0 2 0 0 1- 3 R 1 ← R 1- R 2 = ⇒ 1 0 0 5 0 1 0 2 0 0 1- 3 = ⇒ x y z == ⇒ 5 2- 3 1 General Solution - 15 points Convert the following system to an augmented matrix and solve by using row operations to convert it to Reduced Echelon Form. Then state the general solution in parametric vector form. 2 x 1 + 6 x 2 + 8 x 3 + 12 x 4 = 20 x 1 + 7 x 3 + 9 x 4 = 13 x 3 + 4 x 4 = 5 SOLUTION: 2 6 8 12 20 1 0 7 9 13 0 0 1 4 5 R 1 ← R 1 2 = ⇒ 1 3 4 6 10 1 0 7 9 13 0 0 1 4 5 R 2 ← R 2- R 1 = ⇒ 1 3 4 6 10- 3 3 3 3 1 4 5 R 2 ← R 2- 3 = ⇒ 1 3 4 6 10 0 1- 1- 1- 1 0 0 1 4 5 R 1 ← R 1- 3 R 2 = ⇒ 1 0 7 9 13 0 1- 1- 1- 1 0 0 1 4 5 R 1 ← R 1- 7 R 3& R 2 ← R 2+ R 3 = ⇒ 1 0 0- 19- 22 0 1 0 3 4 0 0 1 4 5 ⇒ x 1 x 2 x 3 x 4 = - 22 + 19 x 4 4- 3 x 4 5- 4 x 4 x 4 = - 22 4 5 + x 4 19- 3- 4 1 2 Vector Span - 15 points Is the fourth vector below in the span of the first three vectors? Completely explain your reasoning but don’t do more calculations than are necessary to answer the question....
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MATH 330 Exam 1 Solution - Math 330 Exam 1 Feb 11th 2009 NO...

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