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Unformatted text preview: Newberger Math 461 Spring 02 Sample Final Exam name: Remark: This exam does not have any questions asking the students to show they know how to show a transformation is not linear, or a subset is not a subspace. You are likely to have one or both of these types of problems. Please look at the quizzes for practice problems of this type. 1. Let A = 1 2 7 2 5 4 5 6 3 , and let b = 3 3 1 . a. (10 points) Reduce the augmented matrix for the system A x = b to Re duced Echelon form. 1 2 7 3 2 5 4 3 5 6 3 1 ∼ 1 2 7 3 9 18 9 0 16 32 16 ∼ 1 2 7 3 0 1 2 1 0 0 0 ∼ 1 0 3 1 0 1 2 1 0 0 0 b. (10 points) Write the solution set for the system A x = b in parametric vector form. The solution set satisfies: x 1 + 3 x 3 = 1 x 2 + 2 x 3 = 1 x 3 is free. So the parametric vector form of the solution is x 1 x 2 x 3 =  3 x 3 + 1 2 x 2 + 1 x 3 =  3 2 1 x 3 + 1 1 . c. (7 points) Give a geometric description of the solution set of the system A x = b This is a line through 1 1 and parallel to 1 1 . Tip: Always give as much information as you can. Saying “a line” is not a complete answer in this case, since you can say precisely which line. 2. Let A = 1 2 7 2 5 4 5 6 3 , and let b = 3 3 1 . *** These are the same as those in problem number 1. *** a. (6 points) Are the columns of A linearly independent? Why or why not? From our work on problem 1, we have that [ A  b ] ∼ 1 2 7 3 2 5 4 3 5 6 3 1 ∼ 1 0 3 1 0 1 2 1 0 0 0 . since A is the coefficient matrix, we can see that A does not have a pivot position in the third column, which means that the homogeneous equation 1 2 A x = has nontrivial solutions, and the columns of A are not linearly independent. b. (6 points) Do the columns of A span R 3 ? Why or why not? Referring to the reduced echelon form of A above, we see that A has a row without a pivot, so the columns of A do not span R 3 ....
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This note was uploaded on 01/12/2010 for the course MATH 247 taught by Professor F,newberger during the Spring '03 term at Stanford.
 Spring '03
 F,newberger
 Math

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