notes6-3

notes6-3 - Let W be the subspace spanned by the following u...

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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 ˆ y in W and a vector z orthogonal to W . THINK ABOUT THIS IN R 3 AND DRAW PICTURES. Suppose we have an orthogonal basis for W , say for instance u 1 , u 2 , u 3 . How can we write ˆ y as a linear combination of u 1 , u 2 , u 3 and how would this be related to y itself?

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ORTHOGONAL DECOMPOSITION FACT. Suppose u 1 , u 2 ,..., u p is an orthogonal basis for a subspace W of R q and y is any vector in R q . Calculate ˆ y = y · u 1 u 1 · u 1 u 1 + y · u 2 u 2 · u 2 u 2 + ··· + y · u p u p · u p u p and then z = y - ˆ y . The vector ˆ y is in W and is sometimes denoted by proj W y . The vector z is in W , and y = ˆ y + z . Finally, given W and y , the vectors ˆ y and z are unique, so in particular they don’t depend on the orthogonal basis we use for W . Let’s check this uniqueness thing. EXAMPLE.
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Unformatted text preview: Let W be the subspace spanned by the following u ’s and write y as the sum of a vector in W and a vector orthogonal to 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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notes6-3 - Let W be the subspace spanned by the following u...

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