2025lecture_8

2025lecture_8 - Orthogonal Projections Integral Transforms...

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Unformatted text preview: Orthogonal Projections Integral Transforms - p. 39/78 Chapter 8 Orthogonal Projections Orthog. Projections Introduction v- w W Construction of w The main theorem Orthog. projections Summary Examples Orthogonal Projections Integral Transforms - p. 40/78 Introduction We will now come back to our original aim: Given a vector space V , a subspace W , and v V . Find the vector w W which is closest to v . First let us clarify what "closest to" means. The tool to measure distance is the norm , so we want bardbl v w bardbl to be as small as possible. Thus our problem is: Find a vector w W such that bardbl v w bardbl bardbl v u bardbl for all u W . Orthog. Projections Introduction v- w W Construction of w The main theorem Orthog. projections Summary Examples Orthogonal Projections Integral Transforms - p. 40/78 Introduction We will now come back to our original aim: Given a vector space V , a subspace W , and v V . Find the vector w W which is closest to v . First let us clarify what "closest to" means. The tool to measure distance is the norm , so we want bardbl v w bardbl to be as small as possible. Thus our problem is: Find a vector w W such that bardbl v w bardbl bardbl v u bardbl for all u W . Now let us recall that if W = R w 1 is a line, then the vector w on the line W is the one with the property that v w W . We will start by showing that this is always the case. Orthog. Projections Introduction v- w W Construction of w The main theorem Orthog. projections Summary Examples Orthogonal Projections Integral Transforms - p. 41/78 w W closest to v iff v w W Theorem. Let V be a vector space with inner product ( , ) . Let W V be a subspace and v V . If v w W , then bardbl v w bardbl bardbl v u bardbl for all u W and bardbl v w bardbl = bardbl v u bardbl if and only if w = u . Thus w is the member of W closest to v . Orthog. Projections Introduction v- w W Construction of w The main theorem Orthog. projections Summary Examples Orthogonal Projections Integral Transforms - p. 41/78 w W closest to v iff v w W Theorem. Let V be a vector space with inner product ( , ) . Let W V be a subspace and v V . If v w W , then bardbl v w bardbl bardbl v u bardbl for all u W and bardbl v w bardbl = bardbl v u bardbl if and only if w = u . Thus w is the member of W closest to v . Proof. First we remark that bardbl v w bardbl bardbl v u bardbl if and only if bardbl v w bardbl 2 bardbl v u bardbl 2 . Now we simply calculate bardbl v u bardbl 2 = bardbl ( v w ) + ( w u ) bardbl 2 = bardbl v w bardbl 2 + bardbl w u bardbl 2 because v w W and w u W ( ) bardbl v w bardbl 2 because bardbl w u bardbl...
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.

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2025lecture_8 - Orthogonal Projections Integral Transforms...

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