note'in the mi,Mof the sMii e, d oD=0. To see this consi'er that since aD at z =O. an' therefore )ds=kl oD(2")dsy Since _ oD(a)ds=0, as the area uner aDs curve is zero, then d oD=0 The equation at the bottom of the slii e shows that if d has a nulLSpace, an' there exists a soltion, then there exist infinitely many soltions One Mast col It shouM be ma'e. The spike function we generate' is not unique. Simply shifting the nonzero point wouM generate aninfinite number f spike functions which wouM anbe in the nulLSpace of d. That is, d has an incredibly rich nuLLSpaceÛ❣❯❃★●■❏P✾✹❅❦❁◆❃✆❏Pt★✾✬❀❂❁⑤❅★❅✰▼❄✾á●❋ß✮❏Pt❆✾á❣P▼◆❁⑤❅✰✾❥Ü ☎è➼✑ ➳ ✵★ã➐ä✔●❂❣✐✾✹✾✿❏Pt❆❁❄❣✪❱✠●❥❃❆❣P❁❄❅✰✾✹❇ê❏rt❆❈✦❏✪❣✐❁❄❃❆❱✠✾á➼✑ ❁⑤❣✪❃★●■❃▲❑✹✾❲❇r●❹●❥❃★▼❄⑦❘❈✦❏✿➭❘➳ ✵★Ü★❈■❃❆❅❘❏Pt★✾✹❇P✾❲ß➯●■❇r✾ ✒ ➴ ✳ ➴ ✴ ➭➜✏➥➭➻ ✴ ➼✑ ➫✳➭➻ ➲⑥➽◗➾➻ ➳ ✴ ➭ ✴ ✒ ➴ ✳ ➴ ➼✑ ➫✳➭➻ ➲⑥➽◗➾➻ ✓ ●◗❁❄❃❆❱✠✾ ➵ ➴ ✳ ➴ ➼✑ ➫➯➭✤➻✒➲✐➽✏➾➐➻✚➳✭✵❆Ü★❈■❣Þ❏rt★✾✉❈❋❇r✾✹❈❹❍★❃❆❅✰✾✹❇❉➼✑ ❅ ❣✿❱❲❍★❇r♦■✾▲❁❄❣ ❑✹✾❲❇r●❆Ü◗❏rt★✾❲❃ ☎Ú➼✑ ➳✭✵❆ã ä✪t★✾✉✾✜♠✏❍❆❈✦❏r❁◆●❥❃❚❈✦❏❵❏rt★✾➃❊✴●❋❏P❏P●❥❀➙●■ß❭❏Pt★✾❂❣✐▼❄❁❄❅★✾❹❣Pt★●✦Ý❉❣✪❏Pt✤❈✦❏✬❁◆ß ☎ t❆❈❥❣❵❈✇❃◗❍★▼❄▼❭❣✐♣❆❈❥❱✠✾❥Ü✤❈❋❃✤❅ ❏Pt❆✾❲❇r✾á✾❲å✰❁❄❣✐❏r❣✿❈q❣P●■▼❄❍✰❏P❁❄●■❃✻Ü★❏Pt★✾✹❃❘❏rt★✾❲❇r✾▲✾✠å✰❁⑤❣⑥❏✿❁❄❃✚❆❃★❁◆❏P✾✹▼◆⑦❘❀q❈■❃◗⑦✆❣✐●❥▼◆❍★❏P❁❄●■❃❆❣✹ã ✰❃★✾❪▼⑤❈■❣✐❏✇❱❲●■❀q❀q✾❲❃✏❏❦❣✐t❆●■❍★▼⑤❅ ❊✴✾s❀✇❈■❅✰✾❥ã✶ä✪t★✾s❣✐♣★❁❄❖■✾☞ß➯❍★❃✤❱Ø❏P❁❄●■❃➜ÝÞ✾❪❧■✾❲❃❆✾❲❇❤❈✦❏P✾✜❅❿❁⑤❣q❃★●❋❏ ❍★❃★❁⑤♠✏❍★✾■ã ●◗❁◆❀q♣★▼❄⑦s❣✐t❆❁éß✒❏r❁◆❃★❧✆❏Pt★✾❹❃★●■❃❑❲✾❲❇r●✇♣✴●■❁❄❃❥❏áÝ❯●❥❍★▼⑤❅❪❧■✾❲❃❆✾❲❇❤❈✦❏P✾❹❈❋❃❆❅❪❁❄❃✚❆❃★❁◆❏P✾❹❃◗❍★❀➃❊✴✾❲❇ ●❋ß✯❣✐♣★❁❄❖■✾áß➯❍★❃✤❱Ø❏P❁❄●■❃✤❣❯Ý✿t❆❁❄❱❤t☞Ý❯●❥❍★▼⑤❅✆❈❋▼❄▼✮❊✴✾á❁❄❃❘❏Pt★✾▲❃◗❍★▼❄▼✻❣✐♣❆❈❥❱✠✾á●❋ß ☎ã➐ä✪t❆❈✦❏✿❁⑤❣❲Ü ☎ t❆❈■❣✪❈■❃ ❁❄❃❆❱✠❇r✾✹❅✰❁❄❊★▼❄⑦❦❇r❁❄❱❤t☞❃◗❍★▼◆▼✞❣✐♣❆❈❥❱✠✾❥ã ë✵➤