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Unformatted text preview: 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. (detare L....
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This note was uploaded on 06/18/2010 for the course COMPUTER S Math 225 taught by Professor Yosumhoca during the Spring '10 term at Bilkent University.
 Spring '10
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