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Question 1 2 3 4 5 6 7 8 Total Score Name AMS 10 Practice Midterm Exam Solutions Student ID Fall Quarter 2016 1. (3 points) Invert the matrix 1 0 0 0 1 1 0 - 1 1 Answer : 1 0 0 1 0 0 0 1 1 0 1 0 0 - 1 1 0 0 1 = 1 0 0 1 0 0 0 1 1 0 1 0 0 0 2 0 1 1 = 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 2 1 2 So the inverse is 1 0 0 0 1 2 - 1 2 0 1 2 1 2 2. Consider these vectors: (1, 2, 0), (1, 0, 0), (0, 2, 2). (a) (2 points) Are these vectors linearly independent? Answer : Let’s see which values of c 1 , c 2 and c 3 satisfy the following equation: c 1 (1 , 2 , 0) + c 2 (1 , 0 , 0) + c 3 (0 , 2 , 2) = 0 ( c 1 , 2 c 1 , 0) + ( c 2 , 0 , 0) + (0 , 2 c 3 , 2 c 3 ) = 0 c 1 + c 2 = 0 2 c 2 + 2 c 3 = 0 2 c 3 = 0 c 1 = 0 c 2 = 0 c 3 = 0 This means that none of the vectors is a linear combination of the others; in other words they are linearly independent. (b) (1 points) Do these vectors span a plane, volume, or a line in R 3 ? Justify your answer. Answer : Three linearly independent vectors will span a volume. 3. (2 points) Compute b and draw an accurate diagram that shows how b is constructed from the linear combination that defines it. 3 1 - 1 - 2 3 1 + 4 0 0 Answer : 3 - 6 + 0 - 3 - 2 + 0 = - 3 - 5 1 This study resource was shared via CourseHero.com
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1 1 2 3 -6 -1 -2 -3 b 4. Below you have a simple model for car traffic in the Santa Cruz downtown area.
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