Week10 - ~ u 1 and ~ u 2 . This give Proj W ~v = ~v ~ u 1 ~...

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Assignment 10 Applied Linear Algebra Math 232 (Fall 2008) Section 6.3
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Section 6.4 Section 6.5
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Section 6.6
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Assignment 10 Applied Linear Algebra Math 232(Fall 2007) Additional question: Let ~ u 1 = 1 2 1 4 , ~ u 2 = 1 1 - 3 0 . Suppose you know that the vector ~v = 1 1 1 1 is not on the plane W spanned by ~ u 1 and ~ u 2 . Find a nonzero vector ~ y that is orthogonal to the plane spanned by ~ u 1 and ~ u 2 and such that ~ y is in the span of ~v, ~ u 1 , ~ u 2 . Hint: Think of how the orthogonal projection could be useful. A1. We compute the orthogonal projection of ~v onto the plane W spanned by
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Unformatted text preview: ~ u 1 and ~ u 2 . This give Proj W ~v = ~v ~ u 1 ~ u 1 ~ u 1 u 1 + ~v ~ u 2 ~ u 2 ~ u 2 u 2 . This is equal to 4 / 11 ~ u 1-1 / 11 ~ u 2 . Thus the projection vector is b v = 3 / 11 7 / 11 7 / 11 16 / 11 . Let ~ z = ~v-b v . Then ~ z is orthogonal to W and is in the span of ~v, ~ u 1 , ~ u 2 . Computing this, we nd 1 ~ z = 8 / 11 4 / 11 4 / 11-5 / 11 . Note that any nonzero scalar multiple of ~ z will also work. 2...
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Week10 - ~ u 1 and ~ u 2 . This give Proj W ~v = ~v ~ u 1 ~...

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