Solutions7 - Math 601 Solutions to Homework 7 $ 1 Find a...

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Math 601 Solutions to Homework 7 1. Find a vector in the plane that is orthogonal to the vector . B #B $B œ! # $ % " # $ Ô × Õ Ø Answer: The dot product of the vector with must be zero. This gives the equation: a b a b B ßB ßB #ß$ß% " # $ #B $B %B œ! " # $ We must therefore solve the following system of linear equations: B  #B  $B œ ! #B  $B  %B œ ! Ê B œ B B œ B B " # $ " # $ " $ "( ( # $ # ( $ is free Any vector satisfying these equations is a correct solution to this problem. The simplest solution is the vector . a b "(ß#ß( Note: The vector is also a correct solution to this problem as stated. The statement of a b !ß!ß! the problem should have included the word “nonzero” to exclude this possibility. 2. Consider the following subspace of : & W œ ß ß # & # # $ & " # $ ! " # ! " $ Span Ú Þ Ý á Ý á Ý á Ý á Û ß Ý á Ý á Ý á Ý á Ü à Ô × Ô × Ô × Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Ö Ù Õ Ø Õ Ø Õ Ø (a) Find an orthonormal basis for . W (b) Find an orthonormal basis for . W ¼ Answers: (a) We use the Gram-Schmidt process: 1. , so is a unit vector. l l a b a b È #ß#ß"ß!ß! œ # # " œ$ œ #ß#ß"ß!ß! # # # " " $ u 2. , so a b a b a b &ß$ß#ß"ß" † #ß#ß"ß!ß! œ "!'# œ' " " $ $ is orthogonal to . a b a b a b a b &ß$ß#ß"ß"  ' #ß#ß"ß!ß! œ "ß"ß!ß"ß" " $ " u
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, so is a unit vector orthogonal to . l l a b a b "ß"ß!ß"ß" œ# œ "ß"ß!ß"ß" u u # " " # 3. , and a b a b a b #ß&ß$ß#ß$ † #ß#ß"ß!ß! œ %"!$ œ$ " " $ $ , so a b a b a b #ß&ß$ß#ß$ † "ß"ß!ß"ß" œ #&#$ œ' " " # # a b a b a b a b a b a b #ß&ß$ß#ß$  $ #ß#ß"ß!ß!  ' "ß"ß!ß"ß" œ "ß!ß#ß"ß!
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