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term_test_1_w10_soln

term_test_1_w10_soln - Math 136 1 Short Answer Problems...

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Math 136 Term Test 1 Solutions 1. Short Answer Problems [2] a) Calculate proj (1 , 1 , 2) ( - 1 , 2 , 2). Solution: proj (1 , 1 , 2) ( - 1 , 2 , 2) = ( - 1 , 2 , 2) · (1 , 1 , 2) (1 , 1 , 2) 2 (1 , 1 , 2) = 5 6 (1 , 1 , 2). [1] b) If n = a × b , then what is a · n ? Solution: Since a × b gives an vector orthogonal to a , we have that a · n = 0. [1] c) What is the definition of the rank of a matrix? Solution: The number of leading 1s (pivot positions) in the RREF of the matrix. [2] d) Let A = 1 2 - 1 0 3 4 and B = 0 - 6 5 2 1 1 . Calculate AB . Solution: AB = 9 - 3 19 10 [2] e) Find the area of the parallelogram induced by a = (1 , 2 , - 1) and b = (3 , 2 , - 1). Solution: Area= (1 , 2 , - 1) × (3 , 2 , - 1) = (0 , - 2 , - 4) = 20. [3] f) Let S = { v 1 , v 2 , v 3 } be a set of three di ff erent vectors in R 3 . What are the possible geometric configurations of span S ? Solution: If all three are linearly independent we get span S = R 3 . If v 3 = v 1 + v 2 and v 1 , v 2 are linearly independent, then span S is a plane. If v 1 = 2 v 2 = 3 v 3 , then span S is a line. Note that since all three vectors must be di ff erent, we can not have span S = { 0 } . 1
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2 2. Consider the system of linear equations: 2 x 3 + 6 x 4 = 0 - x 1 + 2 x 2 + 3 x 3 + x 4 = 5 x 1 - x 2 - 4 x 3 - x 4 = 1 [1] a) What is the augmented matrix of the system? Solution: 0 0 2 6 0 - 1 2 3 1 5 1 - 1 - 4 - 1 1 . [4] b) Row reduce the augmented matrix of the system into reduced row echelon form, stating the elementary row operations used. Solution: 0 0 2 6 0 - 1 2 3 1 5 1 - 1 - 4 - 1 1 R 1 R 3 1 - 1 - 4 - 1 1 - 1 2 3 1 5 0 0 2 6 0 R 2 + R 1 1 2 R 3 1 - 1 - 4 - 1 1 0 1 - 1 0 6 0 0 1 3 0 R 1 + 4 R 3 R 2 + R 3 1 - 1 0 11 1 0 1 0 3 6 0 0 1 3 0 R 1 + R 2 1 0 0 14 7 0 1 0 3 6 0 0 1 3 0 [2] c) Find the general solution of the system.
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