The projection Ily minimizes ly-端,Vz∈Z ly-(lly +u)llx =((y-lly)-v,(y-Ily)-uy y-Iyl-2(y-Iy,v)y+|‖g,Vu∈z Note z=Iy+v∈ Z and Ily∈ Z implies∈z, and hence since(Iy,v)y (y, U)r for allv E Z,(y-lly, u)r=0. The above result states that y-llyl2< ly-all for all z# lly. In words, Ily is the best approximation in Z of y in 1.2.3 Geometry SLIDE 7 Geometry of projection y Ily :(y-Iy,v)y=0,Vv∈Z Not surprisingly, if we wish to find the z Ily on the Z aris closest to y, Ily should be perpendicular to the z in the ( r inner prod usual notion of projection in Rn should be self-evident. In the above picture, z is the orthogonal complement of Z in Y: the space of all members of z a) Show that‖ Iy lr≤ ly llr and y-Iyly≤llly, and interpret this (b) Show that II(IIg)= lly✱✠✲✴✳✵✲✴✳ ✶✸✷✺✹✼✻✾✽✿✷✛❀✪❁ ❂ ❃✿❄❆❅❈❇❊❉ ❋✄●■❍❑❏■▲✪▼❖◆P❍✍◗✜❘✪❙❚▼✔❯❲❱✫❳❲❨❩❙❚❯■❙❬❨❩❙❚❭❪❍❪❫❵❴✏❳✤❛✚❜✠❴❞❝❡❣❢✐❤❵❜❩❥✧❦❑❧ ♠♥●♣♦■q ❴✜❳r❛ts✉❱✫❳✤✈①✇ ② ③✜④ ⑤ ⑥✉⑦✏⑧❵⑨❪⑩✔❶ ❷ ❴❞❝❡①❸❹s✢s❺❳❵❛✚❱✫❳ ❷ ❛❊✇✠❢❪s❺❳❻❛✚❱✫❳ ❷ ❛❼✇ ❷ ❡ ❸✰❴✜❳✤❛✚❱✫❳✵❴✏❝❡❽❛✚❾❣s❿❳❵❛✚❱✫❳➀❢✖✇ ❷ ❡ ② ③✜④ ⑤ ➁✜➂■➃ ⑩✔❶ ✈❩❴✜✇➄❴✏❝❡❣❢✐❤✤✇✦❥✧❦✰➅ ➆❵➇❖➈➊➉✸❜✩❸❹❱✫❳❵✈✞✇➋❥①❦➍➌✺➎✠➏❩❱☛❳➐❥①❦➒➑❆➓✫➔➀→➣➑❿➉✜↔↕✇➐❥①❦✸➙✫➌✺➎➀➏❩➛➜➉✏➎➀➝✉➉❻↔✜➑❆➎✠➝✉➉❩s❿❱☛❳✠❢✢✇ ❷ ❡ ❸ s❺❳➀❢✢✇ ❷ ❡➐➞➇✺➟☛➌❖→❆→➠✇✦❥➐❦❑➙➡s❺❳➢❛❻❱✫❳➀❢✖✇ ❷ ❡ ❸➥➤✼➦❵➧➀➛➜➉↕➌✔➨✉➇✺➩✺➉✫➟✢➉✏↔✜➫➭→➣➈✵↔✜➈➯➌✺➈➯➉✜↔➲➈❆➛➜➌✺➈➳❴✜❳➢❛❻❱✫❳✵❴✏❝❡✙➵ ❴✜❳❻❛①❜➀❴✏❝❡ ➞➇✺➟➸➌✺→❆→➄❜❊➺❸➻❱☛❳✵➦✤➼✜➎❊➽✾➇✺➟✢➏✺↔✉➙➳❱☛❳①➑✴↔r➈❺➛■➉➸➨✉➉✏↔✜➈☛➌✪➔➠➔➀➟✢➇❞➾✔➑❆➓➸➌✺➈➚➑❿➇✺➎①➑❆➎❼❦➪➇➞ ❳①➑❆➎ ➈❺➛■➉❻❴➲➶✔❴ ❡ ➎✠➇✺➟✜➓➸➦ ✱✠✲✴✳✵✲❬➹ ➘➴✽♣✹✠➷①✽✿❀❞✷✛❁ ❂ ❃✿❄❆❅❈❇❽➬ ➮❍❞▼✔❨❩❍✏❘✢▲✪♦✦▼➠➱➡❏■▲✪▼❖◆P❍✍◗✜❘✢❙❬▼➠❯❈✃ ❐➟✜➈❆➛➜➇✢❒✿➇✺➎➀➌✺→➣➑❆➈➚❮➀✃❣s❺❳❻❛❊❱☛❳✠❢✢✇ ❷ ❡ ❸♥➤➜❢ ❤✤✇❲❥➋❦✤❧ ❰✫Ï ➆❵➇❖➈❵↔✜➫➭➟➚➔➀➟✜➑✴↔✉➑❆➎♣❒❖→Ð❮✛➙❻➑➞ ➽✾➉✧➽➳➑✴↔✢➛♥➈➯➇✦Ñ✾➎✠➏✚➈❺➛■➉❲❜Ò❸Ó❱✫❳❹➇❖➎➥➈❺➛■➉✁❦✮➌❞➾✔➑✴↔✧➝❞→❚➇✛↔❞➉✏↔✉➈❻➈➯➇➋❳✠➙ ❳➴❛t❱☛❳Ò↔✢➛■➇❖➫➭→❚➏❽➨✉➉r➔■➉❞➟➚➔➜➉✏➎➀➏✺➑❿➝❞➫➭→❚➌✺➟✁➈➯➇✧➈❺➛■➉❲❦Ó➌❞➾✔➑✴↔➴ÔÕ➑❆➎t➈❺➛■➉➐sP➶❬❢❞➶ ❷ ❡ ➑❆➎➜➎➀➉✏➟❵➔✠➟✪➇❪➏❖➫➜➝✏➈➯➦ ➧➀➛♣➑✴↔❑➌✺➎✠➌❖→❚➇✪❒❖➫➜➉✸➈➯➇❻➇✺➫➭➟☛➫♣↔✜➫➜➌❖→➜➎✠➇❖➈Ö➑❿➇❖➎➐➇➞ ➔✠➟✢➇P×❞➉✪➝❞➈➚➑❿➇✺➎➋➑❆➎❩ØÙ☛Ú➋↔✢➛■➇❖➫➭→❚➏➸➨✉➉Û↔✏➉❞→➞✉Ü ➉❞➩❪➑❿➏✔➉✏➎✼➈➊➦✄➼✏➎ ➈❺➛■➉❵➌✔➨✉➇✺➩✺➉➲➔✠➑❿➝❞➈➚➫➭➟✢➉✜➙➡❦ÛÝÞ➑✴↔↕➈❺➛■➉✄▼✔▲✖❘✪●■▼➠ß✔▼➠❯➜à➠á✼◗❞▼➠❨❩❏■á❬❍❞❨❩❍❞❯✿❘r➇➞ ❦✰➑❆➎❲â❲ã❣➈❺➛■➉❑↔➚➔■➌✔➝✪➉r➇➞ ➌✺→❆→ ➓➸➉✏➓➸➨✉➉✏➟✉↔❵➇➞ âä➇✺➟✜➈❺➛■➇✪❒✔➇❖➎✠➌❖→❈➈➯➇✩➌✺→❆→➡➓❩➉❞➓➸➨✪➉❞➟✉↔✤➇➞ ❦r➦ å✁æÛç★è✺é✔ê✛ë❿ì➠è✧í s❿à❷✁î●■▼✺ïð❘✪●➜à✺❘✧❴❞❱✫❳✵❴ ❡➍ñ ❴✏❳➄❴ ❡ à❖❯➜ò✝❴✜❳➴❛Ò❱✫❳✵❴ ❡óñ ❴✜❳✵❴ ❡✄ô à❖❯✼ò✙❙❬❯✿❘✢❍❞▲✪❏■▲✪❍✏❘❩❘✢●➜❙❬❫ ▲✢❍✍❫✖õ■á❚❘❣ß➠❍❪▼➠❨❩❍✏❘✪▲✢❙✴◗❞à➠á❚á❬♦➠❧ s❺ö ❷➋î●■▼✺ïÒ❘✪●➜à✺❘☛❱rs❿❱☛❳ ❷ ❸➥❱✫❳➀❧ ÷