terms of any basis as Y= span yj, 3=1,., M, meaning that any member of y can be represented as a linear combination of the Orthogonality If our space Y is a Hilbert space with inner product(, Y, we can introduce the notion of orthogonality: two members yi E y and y2 E Y are orthogonal if An orthogonal basis is thus a basis for which the y, are mutually orthogona (yi, 9j)Y ≠J.I, furt hermore,(v;y)Y=‖yi=1, the basis D Exercise 1 Consider the Hilbert space R2 "equipped clidean inner product, ([1, y1],[2, y2])=2132+u132, and hence norm ,yll Is(1, 1),(1,0)a basis for R2? an orthogonal basis? (b)If(1, -1)/v2 is one of our basis vectors, find a second vector such that we have an or thonormal basis Exercise 2 Consider Y= P2([-1, 1]=span (1, r) equipped with the 2 inner product, (3, x)r= y z dz(here y and z are two members of Y (a) Replace x with another basis vector (in fact, polynomial) such that b) Appropriately with what famous French mathematician) are you generating by the above Gram-Schmidt"process Nodal basis for Xh dim(Xn)➠❭➡✟➢❭➤✙➥✤➦✶➧➅➨●➩➭➫✸➯✦➨✏➥✓➲❇➥❲➨✶➥❘➳➸➵❵➥✓➺✦➨✶➩❧➻✟➼✌➽●➾❑➚✾➵⑧➪✶➾✠➶✟➶✠➶②➾✱➹➸➘✏➴❫➤✾➡✔➨●➩❫➲✷➩❫➷✾➠❭➬❽➨✞➠◗➨✶➩r➫▼➤❧➡✟➤➙➯❋➡✠➢ ➦●➧✫➳➱➮✠➨✶➩❈➯❋➡❅➢❭➡✟➺❫➢✼➡✟➥❭➡✠➩r➠❭➡✔✃▼➨✶➥✤➨✾❐✷➲✷➩❫➡✟➨✶➢➣➮✠➦✶➤✁➯❫➲☛➩❽➨✞➠❭➲✷➦✶➩✬➦●➧✯➠❭➬✦➡❅➼✌➽●❒ ❮✧❰❁Ï❀Ð❪Ñ Ò❚Ó➭Ï❭Ô➅❰➭Õ✩❰❉Ö❱×❉Ø❤Ù❭ÏÛÚ Ü➧❢➦✏Ý❫➢❑➥❭➺✦➨✶➮✠➡❲➳❷➲❇➥❑➨❚Þ❘➲☛❐✷➯❋➡✠➢❭➠❝➥❭➺✦➨✶➮✠➡✤ß✤➲❨➠✼➬✸➲☛➩✦➩❫➡✠➢❑➺❫➢❭➦❉✃❉Ý❽➮❀➠❃à✓á☛➾✟á â❭ã❲➴rß❝➡❘➮✠➨✶➩✙➲☛➩r➠❭➢✼➦❉✃❉Ý✦➮✠➡ ➠❭➬✦➡❅➩❫➦●➠✼➲☛➦✏➩❈➦✶➧✯➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐☛➲☛➠❄➫åä❢➠❄ß❝➦❧➤✾➡✟➤➙➯❋➡✠➢✱➥➣➼ræ◗ç❊➳❿➨✶➩✦✃✬➼✶è❆çP➳❿➨✶➢❭➡✸é●ê❀ë✗ì✦é❭íré✞îåï✞ð❱➲❨➧ àÛ➼ræ✞➾✓➼✶è✔â ã ➵↔ñ✬➶ ò➩ó➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐✩➯✦➨✏➥✓➲❇➥❆➲✷➥❃➠✼➬➭Ý✦➥❅➨✬➯✦➨✶➥❭➲✷➥◗➧Û➦✏➢❅ß✤➬❫➲✷➮✱➬✛➠❭➬✦➡❧➼➽ ➨●➢✼➡❧➤➙Ý❉➠✼Ý✦➨●❐✷❐☛➫✛➦✶➢❭➠❭➬❫➦✏➷✶➦✏➩✦➨●❐✣➴ àÛ➼✏ô✓➾❭➼✌➽✟â ã ➵õñ❫➴❘öø÷➵➛➚❽❒ Ü➧❄➴❘➧ÛÝ✦➢✓➠✼➬❫➡✠➢✼➤✾➦✶➢✼➡✶➴➙à❤➼✶ô✓➾❭➼✶ô✎â ã ➵➟ù②➼✶ô❭ù èã ➵ú➪✏➴❲➠❭➬✦➡✛➯✦➨✏➥✓➲❇➥❈➲✷➥ é✞ê❀ëÛì❫é●î❋é●ê❀û✾ï●ðü❒ ý➓þ◗ÿÐ●Ó✁￾✌Ù✄✂✶Ð✆☎✞✝➦✶➩❽➥✓➲❇✃❉➡✠➢❧➠✼➬❫➡➌Þ❲➲✷❐☛➯❋➡✠➢❭➠▼➥❭➺✦➨✶➮✠➡ Ü✟è✡✠ ➡☞☛rÝ❫➲✷➺❫➺❽➡✔✃✍✌✛ß✤➲❨➠✼➬øÝ❽➥✓Ý✦➨✶❐✏✎➅Ý✒✑ ➮②❐✷➲❇✃❉➡✟➨✶➩❃➲✷➩❫➩❫➡✟➢❢➺❫➢✼➦❉✃❉Ý✦➮❀➠✔➴❉à✔✓✕åæ●➾✓➼ræ✗✖ ➾☞✓✕❽è✏➾✓➼✏è✘✖Ûâ➅➵✙✕❁æ✘✕❋è✛✚◗➼ræ❀➼✶è●➴✌➨●➩✦✃❃➬❫➡✟➩✦➮②➡✑➩❫➦✶➢✼➤❿ù✁✓✕⑤➾✓➼✜✖üù✤➵ à✢✕è ✚■➼ è â æ✗✣✼è ❒✥✤❘➦●➠❭➡❆➠✼➬❫➡❅➺✦➨✶➲☛➢✦✓✕✩➾✓➼✜✖✩➢✼➡②➧Û➡✟➢✼➥❝➠✼➦✙➨✙➥✓➲✷➩❫➷✶❐✷➡❆➤✾➡✠➤➙➯❋➡✠➢❚à❤➺❽➦✏➲☛➩r➠❀â➣➲☛➩ Ü✟è ❒ à❤➨râ Ü➥❅à❄➪✶➾✟➪✔â②➾✟à❄➪✏➾❭ñrâ❑➨❧➯✦➨✏➥✓➲❇➥❝➧Û➦✶➢ Ü✟è✛✧ ➨✶➩P➦✶➢❭➠❭➬❫➦✏➷✶➦✶➩❽➨●❐❋➯✦➨✶➥❭➲✷➥ ✧ àÛ➯❋â Ü➧❆à✓➪✶➾✩★❅➪✌â✗✪✬✫✭❈➲✷➥❚➦✶➩❫➡✾➦●➧➣➦✶Ý✦➢❃➯❽➨✶➥❭➲✷➥✯✮✏➡✟➮❀➠✼➦✶➢✱➥✠➴✱✰❽➩✦✃ó➨❊➥✓➡✔➮②➦✶➩❽✃✲✮✏➡✟➮❀➠✼➦✶➢❅➥❭Ý✦➮✱➬❪➠✼➬✦➨✞➠ ß❑➡❆➬✦➨✳✮✏➡❃➨✶➩✬➦✏➢✓➠✼➬❫➦✶➩✦➦✶➢✼➤✾➨✶❐❽➯❽➨✶➥❭➲✷➥✟❒ ý✸þ◗ÿÐ●Ó✬￾✔Ù✄✂✶Ð✵✴✶✝➦✏➩✦➥❭➲✷✃❉➡✟➢❑➳⑧➵ Ü✷ è✶à✗✓✸★❅➪✏➾✠➪✹✖Ûâ✫➵✕➥❭➺✦➨✶➩❆➻✏➪✶➾✗✕✩➾✗✕è ➘❃➡☞☛rÝ❫➲✷➺❫➺❋➡✟✃✸ß✤➲☛➠❭➬▼➠❭➬❫➡ ✺è ➲✷➩❫➩❫➡✟➢❅➺❫➢✼➦➭✃❫Ý✦➮❀➠✔➴❝à❤➼❋➾✼✻râ ã ➵✾✽ æ✿ æ ➼❀✻✦❁✁✕✕à❤➬❫➡✠➢✼➡✙➼❪➨●➩❽✃❂✻❊➨✶➢❭➡❧➠❄ß❝➦P➤❧➡✟➤➙➯❋➡✠➢✱➥❅➦●➧✤➳✙➴ ➠❭➬❽➨✞➠❲➲❇➥✠➴➭➠❄ß❝➦✾➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐❇➥✱â❀❒ à❤➨râ ✟➡✠➺❫❐❇➨✶➮✠➡❃✕è ß✤➲❨➠✼➬✧➨✶➩❫➦●➠✼➬❫➡✠➢❆➯✦➨✏➥✓➲❇➥❄✮✶➡✔➮❀➠❭➦✏➢✙àÛ➲✷➩❪➧❤➨✏➮❀➠✟➴✩➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐✗â❃➥✓Ý❽➮✱➬❪➠✼➬✦➨✞➠❅ß❝➡ ➩❫➦✞ß↔➬✦➨✳✮✏➡❃➨✶➩❈➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐å➯✦➨✶➥❭➲✷➥✟❒ àÛ➯❋â ò➺❫➺❫➢✼➦✶➺✦➢❭➲❇➨✞➠✼➡✠❐✷➫❖➩❫➦✶➢✼➤✙➨●❐✷➲❆❅✟➡✟✃❁➴❑ß✤➬✦➨✞➠✸➺❋➦✶❐✷➫➭➩❫➦✶➤✾➲❇➨●❐❲➥❭➫➭➥✓➠❭➡✟➤➂à✗➠✼➬✦➨✞➠❈➲✷➥✟➴❝➨✏➥❭➥❭➦❉➮②➲❇➨✞➠✼➡✟✃ ß✤➲❨➠✼➬➙ß✤➬✦➨●➠✯➧❤➨✶➤✾➦✶Ý✦➥❈❇❫➢✼➡✠➩❽➮✱➬➙➤✙➨✞➠✼➬❫➡✠➤✙➨✞➠✼➲✷➮✠➲✷➨✶➩❽â⑤➨●➢✼➡✑➫✶➦✏Ý➙➷✏➡✠➩❫➡✟➢✼➨●➠❭➲✷➩❫➷❘➯➭➫❆➠❭➬✦➡➣➨✶➯❽➦❉✮✏➡ ✠✼❊➢✱➨●➤❋✑❍●❫➮✱➬❫➤✾➲✷✃➭➠■✌➙➺✦➢❭➦❉➮②➡✔➥❭➥✟❒ ❏▲❑✜▼❖◆◗P❙❘❉❚ ❯é✛❱✶ï●ð❉➯✦➨✶➥❭➲❇➥❝➧Û➦✶➢❳❲❩❨❉ä ❬ ➽✶➾✑➚✁➵❷➪✶➾✠➶✟➶✠➶②➾✗❭❊➵↔✃❉➲✷➤❊à✢❲❩❨✏â ❪