Sectionlet美人2

Math113Homework#4,due2/9/01at5:00PM1.
LetWbeasubspaceofaninnerproductspaceV.
ShowthatW⊥isasubspaceofV.
2.
LetWbeasubspaceofRn.
(a)LetAbeamatrixwhoserowsareabasisforW.
ShowthatW⊥=N(A).
(b)Showthatdim(W⊥)=ndim(W).
(c)Showthat(W⊥)⊥=W.
Hint:rstgiveasimpleargumentthatW(W⊥)⊥,thenusedimensionalconsiderations.
(d)Letn=4andW=span((1,1,2,3),(2,3,5,8)).
FindabasisforW⊥.
3.
Section2.
1,problems11,12.
4.
LetWbeanitedimensionalsubspaceofaninnerproductspaceV.
Recallthatforx∈V,theorthogonalprojectionPW(x)ofxontoWischaracterizedbyPW(x)∈WandxPW(x)∈W⊥.
(a)ShowthatPW:V→Visalineartransformation.
(b)Lete1,embeanorthonormalbasisforW.
ShowthatPW(x)=mi=1x,eiei.
5.
Considerthedatapoints(x1,y1)=(1,1),(x2,y2)=(0,2),(x3,y3)=(1,3),(x4,y4)=(2,5).
Findthequadraticfunctionf(x)=ax2+bx+cthatminimizesthe"error"E(a,b,c)=4i=1(yif(xi))2.
6.
Section6.
2,problem19.
7.
(extracredit)WhendoesequalityholdinthetriangleinequalityProveyouranswer.