C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes6-3

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes6-3 - W...

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SECTION 6.3 ORTHOGONAL PROJECTIONS AND NEAREST POINTS THE BASIC PROBLEM. Given a subspace W and a vector y , we want to write y as the sum of a vector in W and a vector orthogonal to W . AN EASY CASE. Suppose { u 1 , u 2 , u 3 , u 4 , u 5 , u 6 } is an orthogonal basis for R p and W = Span { u 1 , u 2 } . Write a vector y as the sum of a vector in W and a vector in W . ORTHOGONAL DECOMPOSITION FACT. If we have just an orthogonal basis for W only, then do the same thing to get a vector ˆ y in W , and then z = y - ˆ y will be in W . Given W and y , the ˆ y and z are unique, so they don’t depend on the orthogonal basis we use for W .

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EXAMPLE. Let W be the subspace spanned by the u ’s and write y as the sum of a vector in W and a vector orthogonal to
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Unformatted text preview: W . y = 4 3 3-1 , u 1 = 1 1 1 , u 2 = -1 3 1-2 , u 3 = -1 1 1 BEST APPROXIMATION FACT. The vector ˆ y is the closest vector or point in W to y , that is, || y-ˆ y || < || y-v || for all v ’s in W distinct from ˆ y . THE ORTHONORMAL CASE. If the columns of the matrix U are an orthonormal basis for the subspace W of R n , then proj W y = UU T y . HOMEWORK: SECTION 6.3...
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## This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes6-3 - W...

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