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...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online