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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 # 13 U. Mu˘gan July 2, 2008 Homework problems from the 2 nd Edition, SECTION 4.3 3 (3) ) 2 Clearly any three vectors in R 2 are L.D. Since, the L.C. c 1 ~v 1 + c 2 ~v 2 + c 3 ~v 3 = ~ 0 reduces to a homogenous system of two equations in three unknowns c 1 ,c 2 and c 3 with the coefficient matrix A = £ ~v 1 ~v 2 ~v 3 / . Such system has a nontrivial solution. 6 (6) ) The L.C. c 1 ~v 1 + c 2 ~v 2 + c 3 ~v 3 = ~ 0 yields c 1 (1 , , 0) + c 2 (1 , 1 , 0) + c 3 (1 , 1 , 1) = ( c 1 + c 2 + c 3 ,c 2 + c 3 ,c 3 ) = (0 , , 0) . Therefore c 1 = c 2 = c 3 = 0. Hence the given vectors are L.I. 11 (11) ) Set up the linear system to be solved for the linear combination coefficients { c i } , i = 1 , 2 and then show that the reduction of its augmented matrix A to reduced echelon form E . In this problem, the L.C. is c 1 ~v 1 + c 2 ~v 2 = ~w and which is equivalent to the non homogenous system with the following augmented matrix A = 7 3 1 6 3 4 2 5 3 1 and its reduced row echelon form...
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 Spring '10
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