lecture35 - Review for Midterm 3 Bring a number 2 pencil to...

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Review for Midterm 3Bring anumber 2 pencilto the exam!Extra help session: today and tomorrow,4–7pm, in AH 441Room assignments for Thursday, Nov 20, 7-8:15pm:if your last name starts with A-E:213 Greg Hallif your last name starts with F-L:100 Greg Hallif your last name starts with M-Sh:66 Libraryif your last name starts with Si-Z:103 Mumford HallBig topics:Orthogonal projectionsLeast squaresGram–SchmidtDeterminantsEigenvalues and eigenvectorsOrthogonal projectionsIfv1,,vnis anorthogonal basisofV, andxis inV, thenx=c1v1+ +cnvnwithcj=(x,vj)(vj,vj).Suppose thatVis a subspace ofW, andxis inW, then theorthogonal projectionofxontoVis given byxˆ=c1v1+ +cnvnwithcj=(x,vj)(vj,vj).The basisv1,,vnhas to be orthogonal for this formula!!This decomposesx=xˆinV+xinV, where the errorxis orthogonal toV.(thisdecomposition is unique)v1v2xˆxxArmin Straub[email protected]1
The correspondingprojection matrixrepresentsxxˆwith respect to thestandard basis.Example 1.(a)What is the orthogonal projection of110ontospanbraceleftBigg100,010bracerightBigg?Solution: The projection is110.(b)What is the orthogonal projection of110ontospan110,111?000.braceleftBiggbracerightBigg
(c)What is the orthogonal projection of11,1

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