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homework set 11-Solutions - BILKENT UNIVERSITY Department...

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B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set 1 # 11 U. Mu˘gan July 2, 2008 Homework problems from the 2 nd Edition, SECTION 4.1 3 (3) ) 2 | ~a - ~ b | = | (2 ~ i - 3 ~ j + 5 ~ k ) - (5 ~ i + 3 ~ j - 7 ~ k ) | = | - 3 ~ i - 6 ~ j + 12 ~ k | = 189 = 3 21 . 2 ~a + ~ b = 2(2 ~ i - 3 ~ j + 5 ~ k ) + (5 ~ i + 3 ~ j - 7 ~ k ) = (4 ~ i - 6 ~ j + 10 ~ k ) + (5 ~ i + 3 ~ j - 7 ~ k ) = 9 ~ i - 3 ~ j + 3 ~ k. Similarly, 3 ~a - 4 ~ b = - 14 ~ i - 21 ~ j + 43 ~ k. 5 (5) ) Since ~v = 1 2 ~u , the vectors ~u and ~v are L.D. 7 (7) ) a~u + b~v = a (0 , 2) + b (3 , 0) = (3 b, 2 a ) = ~ 0 implies that a = b = 0, so the vectors ~u and ~v are L.I. 11 (11) ) Let ~w = a~u + b~v, and the L.C. is equivalent to the following nonhomogenous linear system: 5 2 7 3 ‚ • a b = 1 1 . Therefore, a = 1 , b = - 2, so, ~w = ~u - 2 ~v . 13 (13) ) Similar to the previous problem, 7 3 5 4 ‚ • a b = 5 - 2 . Therefore, a = 2 , b = - 3, so, ~w = 2 ~u - 3 ~v . 15 (15) ) We should calculate the determinant of the matrix A = £ ~u ~v ~w / , whether the three vectors ~u, ~v, and ~w are L.D. (det = 0) or L.I. (det 6 = 0). det 3 5 8 - 1 4 3 2 - 6 - 4 = 0 . 1 I made every effort to avoid the calculation errors and/or typos while I prepared the solution set. You are responsible to check all the solutions and correct the errors if there is any.
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