6.3-lay - orthogonal projection of y onto W, and write proj...

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6.3 Generalized Orthogonal projections in R n Homework : 1, 3, 5, 7, 9, 11, 13, 21, 22 Orthogonal Complements If a vector z of orthogonal to every vector in a subspace W of R n , then z is said to be orthogonal to W. The set W = { z R n | z is orthogonal to W} is called the orthogonal complement of W. Ex 1 : Let W = a plane through the origin in R 3 . Let L = the line through the origin and perpendicular to W. Note : a) A vector x in W if and only if x is orthogonal to every vector in a set that spans W. b) W is a vector space. Ex 2 : Let { u 1 , u 2 , u 3 } be an orthogonal basis for R 3 and let W = Span{ u 1 , u 2 }. If y R 3 , then y = c 1 u 1 + c 2 u 2 + c 3 u 3 . Let z 1 = c 1 u 1 + c 2 u 2 and z 2 = c 3 u 3 . Then y = z 2 u 1 = Similarly z 2 u 2 = 0. Hence z 2 is in W . Theorem (Orthogonal Decomposition theorem) Let W be a subspace of R n . If y R n , then y = y + z , where y W and z W and this expression is unique. Furthermore, if { u 1 , u 2 , ……, u p } is an orthogonal basis for W, then y = 1 1 1 1 u u u u y + ……. .+ p p p p u u u u y and z = y – y . In this case, y is called the
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Unformatted text preview: orthogonal projection of y onto W, and write proj w y 1 Ex : Let u 1 = -1 5 2 , u 2 = -1 1 2 , y = 3 2 1 , and W = Span { u 1 , u 2 }. Write y as the sum of a vector in W and a vector orthogonal to W. The Best Approximation Theorem Let W be a subspace of n , y any vector in R n , and y the orthogonal projection of y onto W. Then y is the closest point in W to y , in the sense that ||y y|| < ||y v|| for all v in W distinct from y . y is called the best approximation to y by elements of W 2 Ex : Let u 1 = -1 5 2 , u 2 = -1 1 2 , y = 3 2 1 , and W = Span { u 1 , u 2 }. Find the closest point in W to y and the distance from y to W 3...
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6.3-lay - orthogonal projection of y onto W, and write proj...

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