This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.
 Spring '08
 Staff

Click to edit the document details