homework set 15-A-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 # 15-A U. Mu˘gan July 2, 2008 Homework problems from the 2 nd Edition, SECTION 4.6 4) Yes, the three vectors are mutually orthogonal, since ~v 1 .~v 2 = 3 + 4 + 9 - 12 - 4 = 0 , ~v 1 .~v 3 = 6 + 4 + - 12 - 2 + 4 = 0 , ~v 2 .~v 3 = 18 + 4 - 12 + 6 - 16 = 0 . 7) Let ~u = CB, ~v = CA and ~w = AB , then a = | ~u | , b = | ~v | and c = | ~w | so as to verify that a 2 + b 2 = c 2 . For given vectors a 2 = 19 , b 2 = 25 and c 2 = 44 . 17) Denote by A the matrix having the given vectors as its row vectors, and by E the reduced echelon form of A . From E we find the general solution of the homogeneous system A˜x = ˜ 0 in terms of parameters s, t, .... We then get basis vectors ~u 1 , ~u 2 , ... for the orthogonal complement V by setting each parameter in turn equal to 1 (and the others then equal to 0). The echelon form E of the given matrix A is E = 1 0 - 7 19 0 1 3 5 . x 1 = s, x 4 = t, x 2 = - 3 s + 5 t, x 1 = 7 s - 19 t. Therefore ~u 1 = (7 , - 3 , 1 , 10) , ~u 2 = ( - 19 , 5 , 0 , 1) .
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