midterm1_columbia+solution

midterm1_columbia+solution - MA 242 LINEAR ALGEBRA C1...

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Unformatted text preview: MA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam Prof. Nikola Popovic, October 5, 2006, 09:30am - 10:50am Problem 1 (15 points). Determine h and k such that the solution set of x 1 + 3 x 2 = k 4 x 1 + hx 2 = 8 (a) is empty, (b) contains a unique solution, and (c) contains infinitely many solutions. (Give separate answers for each part, and justify them.) Solution. Row reducing the augmented matrix of the linear system, we find 1 3 k 4 h 8 ∼ 1 3 k h- 12 8- 4 k . Hence, we have the following possibilities: (a) The system is inconsistent (i.e., the solution set is empty) if the third column of the augmented matrix contains a pivot. This is the case if h- 12 = 0 and 8- 4 k 6 = 0 , that is, if h = 12 and k 6 = 2 . (b) The system has a unique solution if the first two columns contain pivots, that is, if both x 1 and x 2 are basic variables. Hence, h 6 = 12 must hold for the second column to contain a pivot. Note that k is arbitrary. (c) The system has infinitely many solutions if x 2 is a free variable and if the third column does not contain a pivot, which is the case for h = 12 and k = 2 . Problem 2 (15 points). Determine if the following vectors are linearly dependent or linearly independent. ( Justify your answers.) (a) - 4 1 5 , , 4 3 6 , (b) 4 4 ,- 1 3 , 2 5 , 8 1 , (c) - 8 12- 4 , 2- 3- 1 . Solution. (a) The set is linearly dependent, since it contains the zero vector . (Any set containing the zero vector is linearly dependent.) (b) The set is linearly dependent, since it consists of four vectors in R 2 . (Any set containing more vectors than each vector has entries is linearly dependent.)more vectors than each vector has entries is linearly dependent....
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midterm1_columbia+solution - MA 242 LINEAR ALGEBRA C1...

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