sample_tt1_2_ans - Math 136 Sample Term Test 1 2 NOTE Only...

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Math 136 Sample Term Test 1 - 2 NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) List the 3 elementary row operations. Solution: 1. Multiply a row by a non-zero constant 2. Swap two rows 3. Add a multiple of one row to another. b) What can you say about the consistency and the number of parameters (free variables) in the general solution of a system of 5 linear equations in 4 variables. Solution: You can not say anything about the consistency or the number of parameters because the system has more equation than variables and we don’t know the rank of the coefficient matrix. c) What is the area of the parallelogram induced by ~a = (1 , - 2) and ~ b = (4 , - 9). Solution: Area= det 1 - 2 4 - 9 = | 1( - 9) - ( - 2)(4) | = 1. d) Let A = 3 2 1 - 2 1 4 and B = - 2 1 1 1 0 - 1 . Calculate AB . Solution: AB = - 4 4 5 - 5 . e) Let S = { ~v 1 ,~v 2 ,~v 3 } be a set of vectors in R 3 . State the definition of the set S being linearly independent. Solution: S is linearly independent if the only solution to c 1 ~v 1 + c 2 ~v 2 + c 3 ~v 3 = ~ 0 is the trivial solution ( c 1 = c 2 = c 3 = 0). f) Explain why ~a × ( ~ b × ~ c ) must be a vector in the plane with vector equation ~x = s ~ b + t~ c , s, t R . Solution: Suppose that ~n = ~ b × ~ c 6 = ~ 0. Then ~n is orthogonal to both ~ b and ~ c , so it is a normal vector to the plane through the origin that contain ~ b and ~ c . Then ~a × ( ~ b × ~ c ) = ~a × ~n is orthogonal to ~n so it lies in the plane with normal ~n , that is, in the plane containing ~ b and ~ c . Hence, for some real numbers s, t , a × ( ~ b × ~ c ) = s ~ b + t~ c
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