Quiz 7 - Solutions

Quiz 7 - Solutions - / 2( u + v ) 1 / 2( u-v ) = 1 2 ( u...

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Math 54 Quiz 7 Mike Hartglass November 6, 2010 1.) Let W be the subspace of R 3 spanned by the vectors 1 0 0 and 1 1 1 . a. ) Find an orthogonal basis for W . Setting x 1 = 1 0 0 and x 2 = 1 1 1 using Gram Schmidt gives v 1 = x 1 and v 2 = x 2 - h x 2 ,v 1 i h v 1 ,v 1 i v 1 = 0 1 1 so an orthogonal basis for W is 1 0 0 , 0 1 1 . b.) Find the orthogonal projection of 0 1 0 onto W . Using the basis in part a. ) (we must use it as it is orthogonal) and calling the above vector v we see that the projection is h v,v 1 i h v 1 ,v 1 i v 1 + h v,v 2 i h v 2 ,v 2 i v 2 = 0 1 / 2 1 / 2 1
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2.) Suppose that u and v are orthonormal vectors in R n . Prove that the vectors 1 2 u + 1 2 v and 1 2 u - 1 2 v are orthonormal. We see that 1
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Unformatted text preview: / 2( u + v ) 1 / 2( u-v ) = 1 2 ( u u-u v + v u-v v ) = 1 2 (1-0+0-1) = 0 so the vectors are orthogonal. k 1 2 u + 1 2 v k 2 = 1 / 2( u + v ) 1 / 2( u + v ) = 1 2 ( u u + u v + v u + v v ) = 1 2 (1+0+0+1) = 1 k 1 2 u-1 2 v k 2 = 1 / 2( u-v ) 1 / 2( u-v ) = 1 2 ( u u-u v-v u + v v ) = 1 2 (1--0 + 1) = 1. This shows the vectors above have norm 1 and are orthogonal so they are orthonormal. 2...
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Quiz 7 - Solutions - / 2( u + v ) 1 / 2( u-v ) = 1 2 ( u...

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