❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦
P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦
†♣♦r
❘ê♥❛❞ ❋❡rr❡✐r❛ ❞❛ ❙✐❧✈❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ▼❡❞❡✐r♦s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦
P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦
♣♦r
❘ê♥❛❞ ❋❡rr❡✐r❛ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✱ ♣❡❧❛ ✈✐❞❛✱ s❛ú❞❡ ❡ ❝♦r❛❣❡♠ q✉❡ ♠❡ ❞❡✉✱ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ❝✉rs♦❀
❆ ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ❞❛s ❉ôr❡s✱ ❛ ♠✐♥❤❛ ❛✈ó ▼❛r✐❛ ❞❛s ◆❡✈❡s ❡ ❛♦s ♠❡✉s ✐r♠ã♦s ❘❡♥ê✱ ▼❛r✐❛ ❘❡♥② ❡ ❘❡♥❛♥✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ q✉❡ ♠❡ ♦❢❡r❡❝❡r❛♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦✱ ❝♦♠♦ t❛♠❜é♠ ❞✉r❛♥t❡ t♦❞❛ ❛ ❥♦r♥❛❞❛ ❛té ❛q✉✐❀
❆ ♠✐♥❤❛ ❡s♣♦s❛ ▼❛r✐❛ ❱✐tór✐❛✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♠♦r✱ ❡st❛♥❞♦ ❛♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ t♦❞♦s ❡ss❡ ❛♥♦s❀
❆ s❡♥❤♦r❛ ▼❛r✐❛③✐♥❤❛ ❡ s❡✉ ❡s♣♦s♦ ❲❛❧❧❛❝❡✱ q✉❡ ♠❡ ❛❝♦❧❤❡r❛♠ ❝♦♠♦ ✉♠ ✜❧❤♦ ❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ♥♦s ♠❡✉s ❡st✉❞♦s❀
❆♦ ♣r♦❢❡ss♦r ❊✈❡r❛❧❞♦✱ ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛✱ ❛♠✐③❛❞❡ ❡ ♦r✐❡♥t❛çã♦✱ ❡st❛♥❞♦ s❡♠♣r❡ ♣r❡s❡♥t❡✱ ❛✉①✐❧✐❛♥❞♦✲♠❡ ♥❛ ❝♦♥❝❧✉sã♦ ❞♦ ♠❡str❛❞♦❀
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ❝♦♠ ♦s q✉❛✐s ❝♦♥✈✐✈✐ ❡ ❝♦♠♣❛rt✐❧❤❡✐ ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛ ❡ ❞❡ tr✐st❡③❛✱ ❛té ❝❤❡❣❛r♠♦s ❛♦ ✜♥❛❧ ❞♦ ❝✉rs♦❀
❆ ❯❋P❇ ❡ ❛ t♦❞♦ ❝♦r♣♦ ❞♦❝❡♥t❡✱ ❝♦♠♦ t❛♠❜é♠ ❛ t♦❞♦s ♦s ❢✉♥❝✐♦♥ár✐♦s❀ ❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❯❊P❇✱ ♦♥❞❡ ❝♦♥❝❧✉✐ ❛ ♠✐♥❤❛ ❣r❛❞✉❛çã♦❀
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ❛❧❣✉♠❛s ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s t✐♣♦s ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❡s♣❡❝✐❛✐s ♥♦ P❧❛♥♦ ❡ ❡♥❝♦♥tr❛♠♦s ❛ ♠❛tr✐③ ❞❡ ❝❛❞❛ ✉♠❛ ❞❡st❛s tr❛♥s❢♦r♠❛çõ❡s✳ ◆❛ s❡❣✉♥❞❛ ♣❛rt❡ ❛❜♦r❞❛♠♦s ❛s tr❛♥s❢♦r♠❛çõ❡s ♥♦ ❊s♣❛ç♦✱ ❞❛♥❞♦ ê♥❢❛s❡ ❛s r♦t❛çõ❡s✳ ❯t✐❧✐③❛♠♦s ♦s â♥❣✉❧♦s ❞❡ ❊✉❧❡r ♣❛r❛ ❞❡t❡r♠✐♥❛r ✉♠❛ r♦t❛çã♦ ♥♦ ❡s♣❛ç♦ ❡♠ t♦r♥♦ ❞♦s ❡✐①♦s ❝❛rt❡✲ s✐❛♥♦s ❡ ❞❡✜♥✐♠♦s ✉♠❛ ❡q✉❛çã♦ q✉❡ ♣❡r♠✐t❡ r♦t❛❝✐♦♥❛r ✉♠ ✈❡t♦r❡s ❡♠ t♦r♥♦ ❞❡ ✉♠ ❡✐①♦ q✉❛❧q✉❡r✳ ❚❛♠❜é♠ ❛❜♦r❞❛♠♦s ♦s ❡s♣❛ç♦s ❤♦♠♦❣ê♥❡♦s ♦❜❥❡t✐✈❛♥❞♦ ❛ r❡♣r❡s❡♥✲ t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ tr❛♥s❧❛çã♦✳ P♦r ú❧t✐♠♦✱ ✉s❛♠♦s ❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ ❞♦s ◗✉❛tér♥✐♦s ♣❛r❛ ❛♣r❡s❡♥t❛r ✉♠❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❞❡ ❢❛③❡r r♦t❛çõ❡s ❞❡ ✈❡t♦✲ r❡s ❡ ❝♦♠♣♦s✐çã♦ ❞❡ r♦t❛çõ❡s ♥♦ ❡s♣❛ç♦✳ ❘❡ss❛❧t❛♠♦s q✉❡ ❡st❡ ❡st✉❞♦ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♦❜❥❡t♦s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦✳
P❛❧❛✈r❛s ❈❤❛✈❡✿ ❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s✱ ➶♥❣✉❧♦s ❞❡ ❊✉❧❡r✱ ●r✉♣♦ ❞♦s ◗✉❛tér♥✐♦s✱ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦s✳
❆❜str❛❝t
❆❜str❛❝t✿ ■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② s♦♠❡ ❣❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ t❤❡ s♣❛❝❡✳ ■♥✐t✐❛❧❧②✱ ✇❡ ♣r❡s❡♥t s♦♠❡ s♣❡❝✐❛❧ t②♣❡s ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ✜♥❞ t❤❡ ♠❛tr✐① ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ✇❡ ❞✐s❝♦✉rs❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ s♣❛❝❡✱ ❡♠♣❤❛s✐③✐♥❣ t❤❡ r♦t❛t✐♦♥s✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛♥❣❧❡s ♦❢ ❊✉❧❡r t♦ ❞❡t❡r♠✐♥❡ ❛ r♦t❛t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ ❛r♦✉♥❞ t❤❡ ❈❛rt❡s✐❛♥ ❛①❡s ❛♥❞ ❞❡✜♥❡ ❛♥ ❡q✉❛t✐♦♥ ✇❤✐❝❤ ❛❧❧♦✇s t♦ r♦t❛t❡ ❛ ✈❡❝t♦r ❛r♦✉♥❞ ❛♥② ❛①✐s✳ ❲❡ ❛❧s♦ ❞✐s❝✉ss t❤❡ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ❛✐♠✐♥❣ t❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ tr❛♥s❧❛t✐♦♥✳ ❋✐♥❛❧❧②✱ ✇❡ ✉s❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ q✉❛t❡r♥✐♦♥s ❣r♦✉♣ t♦ ♣r❡s❡♥t ❛ s❡❝♦♥❞ ❢♦r♠ t♦ r♦t❛t✐♦♥ ✈❡❝t♦rs ❛♥❞ ❝♦♠♣♦s✐t✐♦♥ ♦❢ r♦t❛t✐♦♥s ✐♥ t❤❡ s♣❛❝❡✳ ❲❡ ❡♠♣❤❛s✐③❡ t❤❛t t❤✐s st✉❞② ✐s ❡ss❡♥t✐❛❧ t♦ ❞❡s❝r✐❜❡ t❤❡ ♠♦t✐♦♥ ♦❢ ♦❜❥❡❝ts ✐♥ t❤❡ ♣❧❛♥❡ ❛♥❞ ✐♥ t❤❡ s♣❛❝❡✳
❑❡② ✇♦r❞s✿ ●❡♦♠❡tr✐❝ ❚r❛♥s❢♦r♠❛t✐♦♥s✱ ❆♥❣❧❡s ♦❢ ❊✉❧❡r✱ ◗✉❛t❡r♥✐♦♥s ●r♦✉♣s✱ ❍♦♠♦❣❡♥❡♦✉s ❙♣❛❝❡
❙✉♠ár✐♦
✶ ❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ P❧❛♥♦ ✶ ✶✳✶ ❚r❛♥s❢♦r♠❛çã♦ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ▼❛tr✐③ ❆ss♦❝✐❛❞❛ ❛s ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✷ ❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦ P❧❛♥♦ ✳ ✳ ✼ ✶✳✶✳✸ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s❝❛❧❛ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✹ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✺ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✶✳✻ Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✶✳✼ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❈✐s❛❧❤❛♠❡♥t♦ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✶✳✽ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❚r❛♥s❧❛çã♦ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✶✳✾ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷ ❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ ❊s♣❛ç♦ ✸✵ ✷✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✳✶ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❚r❛♥s❧❛çã♦ ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s❝❛❧❛ ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❘❡✢❡①õ❡s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✳✹ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✳✺ ❘♦t❛çã♦ ♥♦ ❊s♣❛ç♦ ✉s❛♥❞♦ ♦s ➶♥❣✉❧♦s ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✶✳✻ Pr♦❞✉t♦ ❱❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✶✳✼ ❘♦t❛çã♦ ♥♦ ❊s♣❛ç♦ ❡♠ t♦r♥♦ ❞❡ ✉♠ ❊✐①♦ q✉❛❧q✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✶✳✽ ■♥st❛♥❝✐❛çã♦ ❞❡ ❖❜❥❡t♦s ❡ ❍✐❡r❛rq✉✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✺✵
✸ ◗✉❛tér♥✐♦s ✺✷
✸✳✶ ❋✉♥❞❛♠❡♥t♦s ❍✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✶✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ◗✉❛tér♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❆❧❣é❜r✐❝❛s ❞♦s ◗✉❛tér♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✶✳✸ ◗✉❛tér♥✐♦s ❯♥✐tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✶✳✹ ◗✉❛tér♥✐♦s ❡ ❘♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✶✳✺ ❈♦♠♣♦s✐çõ❡s ❞❡ ❘♦t❛çõ❡s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ♣ ❡♠ ♣′ ♥♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✶✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❞✐❧❛t❛çã♦ ♥♦ ♣❧❛♥♦ R2
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ ① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ♣❧❛♥♦ s♦❜r❡ ♦ ❡✐①♦ ① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✻ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✼ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✶✳✽ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ y ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✶✳✾ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✵ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛ y=−x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✶✳✶✶ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛ y=x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✶✳✶✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✶✳✶✸ ❉❡❞✉çã♦ ❞✐r❡t❛ ❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✶✹ ❚r❛♥s❢♦r♠❛çã♦ ✈❡rs✉s ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✶✺ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✶✳✶✻ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❚r❛♥s❧❛çã♦ ♥♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✶✼ ■♠❡rsã♦ ❞♦ R2
♥♦ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ xh, yh ❡ w ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷✳✶ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ♣ ❡♠ ♣′ ♥♦ ❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦ xOy ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✷✳✸ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦ xOz ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✹ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦ yOz ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✺ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✷✳✻ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦ ♥♦ ❡s♣❛ç♦ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✼ ❘♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦s ❡✐①♦s ❝❛rt❡s✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✽ ❘♦t❛çã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x ♥♦ ❡s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✷✳✾ ❘♦t❛çã♦ ❡♠ t♦r♥♦ ❞❡ ✉♠ ❡✐①♦ q✉❛❧q✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✶✵ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✶✶ ■♥st❛♥❝✐❛çã♦ ❞❡ ♦❜❥❡t♦s ♥✉♠ ❜r❛ç♦ ♠❡❝â♥✐❝♦ s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✶✷ ❊✐①♦s ❧♦❝❛✐s ❞♦ ❜r❛ç♦ ♠❡❝â♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
■♥tr♦❞✉çã♦
◆♦ ❡♥s✐♥♦ ♠é❞✐♦✱ q✉❛♥❞♦ ♦s ❛❧✉♥♦s t❡♠ ♦s ♣r✐♠❡✐r♦s ❝♦♥t❛t♦s ❝♦♠ ♦s ❝♦♥t❡ú❞♦s ❞❡ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s✱ tr✐❣♦♥♦♠❡tr✐❛ ❡ ❢✉♥çõ❡s ✭❡st❡ ❥á ✐♥tr♦❞✉③✐❞♦ ♥♦ ✾♦ ❛♥♦
❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✮✱ ❡❧❡s tê♠ ❛ ✐❞❡✐❛ ❡q✉✐✈♦❝❛❞❛ q✉❡ ❡ss❡s ❝♦♥t❡ú❞♦s sã♦ ❛♣❡♥❛s ♠❡r♦s ♣r♦❝❡❞✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ❝á❧❝✉❧♦s s❡♠ ♥❡♥❤✉♠❛ ✉t✐❧✐❞❛❞❡✱ ❛❧é♠ ❞♦ ❝♦❧é❣✐♦✳ ❱✐s❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦✱ ❞❡s♠✐t✐✜❝❛r ❡st❛ ✐❞❡✐❛✳ ❈♦♠♦ ♦❜s❡r✈❛❞♦ ❡♠ ❬✻❪ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s t❡♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥❛ ár❡❛ ❞❛ ❝♦♠♣✉t❛çã♦ ❣rá✲ ✜❝❛✱ ♣♦✐s ♣❡r♠✐t❡ ❛❧t❡r❛r✱ ♠♦❞❡❧❛r ❡ ♠❛♥✐♣✉❧❛r ♦s ♦❜❥❡t♦s q✉❡ ❡stã♦ ❝♦♥t✐❞♦s ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❡♥❛✳ P♦r ❡①❡♠♣❧♦✱ ♠✉❞❛♥ç❛s ❡♠ ♦r✐❡♥t❛çõ❡s✱ t❛♠❛♥❤♦ ❡ ❢♦r♠❛ ❞♦s ♦❜❥❡t♦s✳ ❊❧❛s t❡♠ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❞❡s❝r✐çã♦ ❞❛ ❢♦r♠❛ ❡ ❞♦s ♠♦✈✐♠❡♥t♦s ❡♠ ❝❡♥ár✐♦s ✈✐rt✉❛✐s✳ ❯♠❛ ❝❡♥❛ ♣♦r ♠❛✐s ❝♦♠♣❧❡①❛ q✉❡ ♣❛r❡ç❛✱ ♣♦❞❡ s❡r r❡❞✉③✐❞❛ ❛ ✉♠❛ ♠❛✐s s✐♠♣❧❡s✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ❝❡♥❛ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✉❜❝♦♠♣♦♥❡♥t❡s✱ ❛ss✐♠ ❛❧❣✉♠❛s ♣♦❞❡♠ s❡r r❡❞✉③✐❞❛s ❛ ❢♦r✲ ♠❛s ❣❡♦♠étr✐❝❛s ♣❧❛♥❛s ♠❛✐s s✐♠♣❧❡s ❝♦♠♦ tr✐â♥❣✉❧♦s✱ q✉❛❞r❛❞♦s✱ ♣❡♥tá❣♦♥♦s ❡♥tr❡ ♦✉tr♦s ♦✉ ✜❣✉r❛s ❡s♣❛❝✐❛✐s ♠❛✐s ✉s✉❛✐s ❝♦♠♦ ❝✉❜♦✱ ❝✐❧✐♥❞r♦✱ ❝♦♥❡ ✱ ❡s❢❡r❛✱ ❡t❝✳ ❆ ♣❛r✲ t✐r ❞❡ss❛s ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ✉t✐❧✐③❛♥❞♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❢♦r♠❛ ❡ ♠♦✈✐♠❡♥t♦✱ ♣♦❞❡♠♦s ❣❡r❛r ♠♦❞❡❧♦s ❞❡ ❝❡♥❛s ♠❛✐s ❝♦♠♣❧❡①♦s✳ Pr❡t❡♥❞❡♠♦s ♠♦str❛r q✉❡ ♦s ❝♦♥✲ ❝❡✐t♦s ❞❡ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s✱ tr✐❣♦♥♦♠❡tr✐❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❢✉♥çõ❡s sã♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s✳ ❊st❡ tr❛❜❛❧❤♦ s❡rá ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦✱ ❛ss✐♠ ❝♦♠♦ ♦ ❡st✉❞♦ ❞♦s q✉❛tér♥✐♦s✱ ❡str✉t✉r❛ q✉❡ ❢❛❝✐❧✐t❛ ♦ ❡st✉❞♦ ❞❛s r♦t❛çõ❡s ♥♦ ❡s♣❛ç♦✳ ❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❡stá ❡s❝r✐t♦ ❝♦♠♦ s❡❣✉❡✿
◆♦ ❈❛♣ít✉❧♦ ✶✱ ✐♥tr♦❞✉③✐♠♦s ♦s ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♥♦ P❧❛♥♦ t❡♥❞♦ ❝♦♠♦ ❢♦❝♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s✿ ❞✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦✱ ❡s✲ ❝❛❧❛✱ ❡s♣❡❧❤❛♠❡♥t♦ ♦✉ r❡✢❡①ã♦✱ r♦t❛çã♦✱ ❝✐s❛❧❤❛♠❡♥t♦ ❡ tr❛♥s❧❛çã♦✳ ❆✐♥❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ ♦❜t✐✈❡♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡ ❝❛❞❛ tr❛♥s❢♦r♠❛çã♦✳
❖ ❈❛♣ít✉❧♦ ✷ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♠❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ ❊s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❡st✉❞❛♠♦s ❛s r♦t❛çõ❡s ❡♠ t♦r♥♦ ❞♦s ❡✐①♦s ♦r❞❡♥❛❞♦s✳ ❚❛♠❜é♠ ✐♥tr♦❞✉③✐♠♦s ♦s â♥❣✉❧♦s ❞❡ ❊✉❧❡r ♦❜❥❡t✐✈❛♥❞♦ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ❞❡ r❡♣r❡s❡♥t❛r ✉♠❛ r♦t❛çã♦ ♥♦ ❊s♣❛ç♦✳
❋✐♥❛❧♠❡♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ●r✉♣♦ ❞♦s ◗✉❛tér♥✐♦s ❡ ✉s❛♠♦s ❡st❛ ❡str✉t✉r❛ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ✉♠❛ r♦t❛çã♦ ♥♦ ❊s♣❛ç♦✳
❈❛♣ít✉❧♦ ✶
❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦
P❧❛♥♦
✶✳✶ ❚r❛♥s❢♦r♠❛çã♦ ♥♦ P❧❛♥♦
◆❡st❛ s❡çã♦ ✈❛♠♦s ♥♦s ❞❡❞✐❝❛r ❛♦ ❡st✉❞♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♥♦ ♣❧❛♥♦✱ t❡♥❞♦ ❝♦♠♦ ❢♦❝♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✿ ❡s❝❛❧❛✱ r♦t❛çã♦✱ r❡✢❡①ã♦ ❡ ❝✐s❛❧❤❛♠❡♥t♦✳ Pr✐✲ ♠❡✐r♦ ✈❛♠♦s ❞❡✜♥✐r ♦ q✉❡ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥♦R2
✳ ❚❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ♦✉ ❛♣❧✐❝❛çã♦ ♥♦ ♣❧❛♥♦✳
❉❡✜♥✐çã♦ ✶ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ♥♦R2 é ✉♠❛ ❛♣❧✐❝❛çã♦ ❚
:R2
−→R2✱ q✉❡ ❛ss♦❝✐❛
❛ ❝❛❞❛ ✈❡t♦r ♣ ❞♦ ♣❧❛♥♦ ✉♠ ♥♦✈♦ ✈❡t♦r ♣′ t❛❧ q✉❡✿
❚(♣) =♣′
♦✉
❚
x y
=
x′
y′
,
♦♥❞❡ ♣ =
x y
✱ ❝♦rr❡s♣♦♥❞❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r −♦♣✱ s❡♥❞♦ ♦ ❛ ♦r✐❣❡♠ ❞♦→ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦✳
❖❜s❡r✈❡ q✉❡ ♦ ✈❡t♦r ♣ é tr❛♥s❢♦r♠❛❞♦ ♥♦ ✈❡t♦r ♣′ ♣♦r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❚ :
R2
−→R2
❝♦♠♦ ✐♥❞✐❝❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✿
❚r❛♥s❢♦r♠❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✶✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ♣ ❡♠ ♣′ ♥♦ ♣❧❛♥♦
❖❜s❡r✈❛çã♦ ✶ ◆♦ q✉❡ s❡❣✉❡✱ ❞❡♥♦t❛r❡♠♦s ♦ ✈❡t♦r −♦♣ ♣♦r ♣→
❉❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥♦ R2
♣❛r❛ ♠♦str❛r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ✈❡t♦r ♣ ❛♣ós s❡r ❛♣❧✐❝❛❞❛ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❚✳
❊①❡♠♣❧♦ ✶ ❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ ❚:R2
−→R2
❞❡✜♥✐❞❛ ♣♦r
❚
x y
=
x+y x−y
.
◆♦t❡ q✉❡ ♦ ✈❡t♦r ♣ =
3 4
é tr❛♥s❢♦r♠❛❞♦ ♣♦r ❚ ♥♦ ✈❡t♦r
♣′ =
3 + 4 3−4
=
7
−1
.
❊①✐st❡♠ tr❛♥s❢♦r♠❛çõ❡s q✉❡ t❡♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛♠ ❞❛s ♦✉tr❛s✱ ❡❧❛s sã♦ ❝❤❛♠❛❞❛s ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✱ ♠❛✐s ♣♦❞❡♠♦s ♥♦s ❢❛③❡r ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✳ ◗✉❛♥❞♦ é q✉❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ é ❧✐♥❡❛r❄ P❛r❛ r❡s♣♦♥❞❡r ❡st❛ ♣❡r❣✉♥t❛ ♣r❡❝✐s❛r❡♠♦s ♣r✐♠❡✐r♦ ❞❡✜♥✐r ♦ q✉❡ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳
❉❡✜♥✐çã♦ ✷ ❙❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ❊✱ ♥ã♦✲✈❛③✐♦✱ s♦❜r❡ ♦ q✉❛❧ ❡stã♦ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛✲ çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✱ ✐st♦ é✿
• ∀✉,✈∈❊,✉+✈ ∈❊.
• ∀λ∈R,∀ ∈❊, λ✉∈❊.
❚r❛♥s❢♦r♠❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❖ ❝♦♥❥✉♥t♦ ❊ ♠✉♥✐❞♦ ❞❡ss❛s ❞✉❛s ♦♣❡r❛çõ❡s é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ s❡ ❢♦r❡♠ ✈❡r✐✜❝❛❞♦s ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿
✶✳ P❛r❛ q✉❛✐sq✉❡r ✈❡t♦r❡s ✉,✈,✇∈❊✱
(✉+✈) +✇ =✉+ (✈+✇). (associatividade)
✷✳ ❊①✐st❡ ✉♠ ✈❡t♦r ❡♠ ❊✱ ❞❡♥♦t❛❞♦ ♣♦r 0 ❡ ❝❤❛♠❛❞♦ ✈❡t♦r ♥✉❧♦✱ ♣❛r❛ ♦ q✉❛❧
✉+ 0 = 0 +✉=✉. (elemento neutro)
♣❛r❛ q✉❛❧q✉❡r ✈❡t♦r ✉∈❊.
✸✳ P❛r❛ ❝❛❞❛ ✈❡t♦r ✉∈❊✱ ❡①✐st❡ ✉♠ ✈❡t♦r ❡♠ ❊✱ ❞❡♥♦t❛❞♦ ♣♦r −✉✱ ♣❛r❛ ♦ q✉❛❧
✉+ (−✉) = (−✉) +✉= 0. (inverso aditivo)
✹✳ P❛r❛ q✉❛✐sq✉❡r ✈❡t♦r❡s ✉,✈ ∈❊✱
✉+✈ =✈+✉. (comutatividade)
✺✳ P❛r❛ q✉❛❧q✉❡r ❡s❝❛❧❛r λ∈R ❡ q✉❛✐sq✉❡r ✈❡t♦r❡s ✉,✈∈❊✱
λ(✉+✈) =λ✉+λ✈.
✻✳ P❛r❛ q✉❛✐sq✉❡r ❡s❝❛❧❛r❡s λ1, λ2 ∈R ❡ q✉❛❧q✉❡r ✈❡t♦r ✉∈❊✱
(λ1+λ2)✉=λ1✉+λ2✉.
✼✳ P❛r❛ q✉❛✐sq✉❡r ❡s❝❛❧❛r❡s λ1, λ2 ∈R ❡ q✉❛❧q✉❡r ✈❡t♦r ✉∈❊✱
(λ1λ2)✉=λ1(λ2✉).
✽✳ 1✉=✉✱ ♣❛r❛ q✉❛❧q✉❡r ✈❡t♦r ✉∈❊✳
❖❜s❡r✈❛çã♦ ✷ ❈❤❛♠❛r❡♠♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❊ ❞❡ ✈❡t♦r❡s✱ ✐♥❞❡✲ ♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ s✉❛ ♥❛t✉r❡③❛✳ ◆♦ q✉❡ s❡❣✉❡✱ ❡①❝❡t♦ r❡❢❡rê♥❝✐❛ ❝♦♥trár✐❛✱ q✉❛♥❞♦ ❢❛❧❛r♠♦s q✉❡ ❊ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ✜❝❛ s✉❜t❡♥❞✐❞♦ q✉❡ ❊ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ R✱ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
❉❡✜♥✐çã♦ ✸ ❙❡❥❛♠ ❊ ❡ ❲ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❚:❊−→❲✱ é
❞✐t❛ ❧✐♥❡❛r s❡✿
▼❛tr✐③ ❆ss♦❝✐❛❞❛ ❛ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✶
✶✳ ❚(✉+✈) =❚(✉) +❚(✈)✱ ♣❛r❛ t♦❞♦ ✉,✈∈❊;
✷✳ ❚(λ.✈) = λ.❚(✈)✱ ♣❛r❛ t♦❞♦ ✈∈❊ ❡ ♣❛r❛ t♦❞♦ ❡s❝❛❧❛r λ∈R✳
❆ ❉❡✜♥✐çã♦3❛✜r♠❛ q✉❡ s❡✱ ❚ :❊−→❲ é ❧✐♥❡❛r✱ ❡❧❛ ♣r❡s❡r✈❛ ❛s ❞✉❛s ♦♣❡r❛çõ❡s
❜ás✐❝❛s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ✐st♦ é✱ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✳ ❖❜s❡r✈❛çã♦ ✸ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❚ : ❊ −→ ❊ ✭❝❛s♦ ❡♠ q✉❡ ❊ = ❲ ✮ é
❝❤❛♠❛❞♦ ❞❡ ♦♣❡r❛❞♦r ❧✐♥❡❛r s♦❜r❡ ❊✳ ◆❡st❡ ❈❛♣ít✉❧♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❊=❲=R2
.
P♦❞❡♠♦s t❛♠❜é♠ ✈❡r✐✜❝❛r s❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❚ : ❊ −→ ❲ é ❧✐♥❡❛r ♣❡❧♦
s❡❣✉✐♥t❡ ❧❡♠❛✿
▲❡♠❛ ✸✳✶ ❯♠❛ ❛♣❧✐❝❛çã♦ ❚:❊−→❲ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r s❡✱ ❡ s♦♠❡♥t❡
s❡✱ ∀λ1, λ2 ∈R✱ ❡ ∀✉,✈∈❊ t✐✈❡r♠♦s q✉❡
❚(λ1♣1 + λ2♣2) = λ1❚(♣1) + λ2❚(♣2).
❉❡♠♦♥str❛çã♦✿ ❙❡ ❚ é ❧✐♥❡❛r✱ ✉t✐❧✐③❛♥❞♦ 1 ❡ 2❞❛ ❉❡✜♥✐çã♦ 3 t❡♠♦s q✉❡
❚(λ1✉+λ2✈) = ❚(λ1✉) +❚(λ2✈)
= λ1❚(✉) +λ2❚(✈).
❆ r❡❝í♣r♦❝❛ s❡❣✉❡ ❞❡ q✉❡
❚(λ1✉) = ❚(λ1✉+ 0✈)
= λ1❚(✉) + 0❚(✈)
= λ1❚(✉)
❡
❚(1✉+ 1✈) = ❚(✉+✈) = 1❚(✉) + 1❚(✈) = ❚(✉) +❚(✈),
❧♦❣♦ ❚ é ❧✐♥❡❛r✳
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s t❡♠♦s q✉❡ s❡ ❛♣❧✐❝❛r♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❚ ❛ ✉♠❛ ❝♦♠✲ ❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ✈❡t♦r❡s ❡ ♦❜t❡r♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s tr❛♥s❢♦r♠❛❞♦s ♣♦r ❚✳ ❊ss❛ tr❛♥s❢♦r♠❛çã♦ s❡rá ❧✐♥❡❛r✳ ❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s é q✉❡ ❚(0) = 0✱ ✐st♦ é✱ ❛ tr❛♥s❢♦r♠❛çã♦ ❧❡✈❛ ❡❧❡♠❡♥t♦
♥❡✉tr♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❡♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ♦✉tr♦ ❡s♣❛ç♦✳ ❊ss❛ ♣r♦♣r✐❡❞❛❞❡ ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱
❚(0) =❚(0✉) = 0❚(✉) = 0.
▼❛tr✐③ ❆ss♦❝✐❛❞❛ ❛ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✶
✶✳✶✳✶ ▼❛tr✐③ ❆ss♦❝✐❛❞❛ ❛s ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s
◆❡st❛ s❡çã♦ ✈❛♠♦s tr❛t❛r ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❚ ❛ss♦❝✐❛❞❛s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✳ P❛r❛ ❝❛❞❛ ♣ ∈ Rn,❚(♣) é ❞❛❞♦ ♣♦r ▼♣, ♦♥❞❡ ▼ é ✉♠❛ ♠❛tr✐③ m×n✳
P❛r❛ s✐♠♣❧✐✜❝❛r✱ ♠✉✐t❛s ✈❡③❡s ❞❡♥♦t❛♠♦s ❡ss❛ tr❛♥s❢♦r♠❛❞❛ ♠❛tr✐❝✐❛❧ ♣♦rx−→▼♣✳
❖❜s❡r✈❡ q✉❡ ♦ ❞♦♠í♥✐♦ ❞❡ ❚ é ♦Rnq✉❛♥❞♦ ▼ t❡♠n❝♦❧✉♥❛s✱ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ ❚
é ♦Rm q✉❛♥❞♦ ❝❛❞❛ ❝♦❧✉♥❛ ❞❡ ▼ t❡♠ m❡❧❡♠❡♥t♦s✳ ❙❡♠♣r❡ q✉❡ ✉♠❛ tr❛♥s❢♦r♠❛❞❛
❧✐♥❡❛r ❛♣❛r❡❝❡ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦✉ é ❞❡s❝r✐t❛ ❡♠ ♣❛❧❛✈r❛s✱ ❣❡r❛❧♠❡♥t❡ q✉❡r❡♠♦s ✉♠❛ ✧ ❢ór♠✉❧❛ ✧ ♣❛r❛ ❚(♣)✳ ❆ ❞✐s❝✉ssã♦ q✉❡ s❡❣✉❡ ♠♦str❛ q✉❡ t♦❞❛ tr❛♥s❢♦r♠❛❞❛
❧✐♥❡❛r ❚ : Rn −→ Rm é ♥❛ ✈❡r❞❛❞❡✱ ✉♠❛ tr❛♥s❢♦r♠❛❞❛ ♠❛tr✐❝✐❛❧ x −→ ▼♣ ❡ q✉❡
♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❛ tr❛♥s❢♦r♠❛çã♦ ❚ ❡stã♦ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♥❤❡❝✐❞❛s ❞❛ ♠❛tr✐③ ▼✳ ❆ ❝❤❛✈❡ ♣❛r❛ s❡ ❞❡t❡r♠✐♥❛r ▼ é ♦❜s❡r✈❛r q✉❡ ❚ ✜❝❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ s✉❛ ❛çã♦ ♥❛s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡
n×n,■n.
❆♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ ❞✐③ q✉❡ t♦❞❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ▼♣ ♦♥❞❡ ▼ é ❞❡♥♦♠✐♥❛❞❛ ❞❡ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r✲ ♠❛çã♦✳
Pr♦♣♦s✐çã♦ ✶ ✿ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❚ é ❧✐♥❡❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❚(♣) =▼♣✳
❉❡♠♦♥str❛çã♦✿ ❙❡ ❚(♣) = ▼♣✱ ❡♥tã♦ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s
s❡❣✉❡
▼(λ1♣1+λ2♣2) = ▼(λ1♣1) +▼(λ2♣2)
= λ1▼♣1+λ2▼♣2
= λ1❚(♣1) +λ2❚(♣2)
= ❚(λ1♣1+λ2♣2),
♣♦rt❛♥t♦ ❚(♣) = ▼♣ é ❧✐♥❡❛r✳
❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ s❡ ❚ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✱ ❡❧❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ ❢♦r♠❛ ▼ ♣ ✳ ❈♦♠♦ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ tr❛♥s❢♦r♠❛çõ❡s ♥♦ ♣❧❛♥♦✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ✐r❡♠♦s ♣❛rt✐❝✉❧❛r✐③❛r ❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ R2✳
❙❡❥❛ ♣∈R2✱ ✐st♦ é✱ ♣
=
x y
t❡♠♦s q✉❡
❚
x y
= ❚
x
0
+
0
y
= ❚
x
1 0
+y
0 1
= x❚
1 0
+y❚
0 1
.
❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶ ❋❛③❡♥❞♦✲s❡ ❚ 1 0 = a b e ❚ 0 1 = c d , t❡♠♦s q✉❡✿ ❚ x y = x a b +y. c d = ax bx + cy dy =
ax+cy bx+dy
= a c b d x y = ▼♣,
♦♥❞❡ ▼ é ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
❖❜s❡r✈❛çã♦ ✹ ◆♦t❡ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ♣∈R2
♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ ❢♦r♠❛
x y =x 1 0 +y 0 1 ,
♦✉ s❡❥❛✱ t♦❞♦ ✈❡t♦r ❞♦ ♣❧❛♥♦ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❡1 =
1 0
❡ ❡2 =
0 1
. ❆ss✐♠✱
▼= ❚(❡1) ❚(❡2)
.
❆♥❛❧♦❣❛♠❡♥t❡ s❡ ❚:R3
−→R3
❡♥tã♦
▼= ❚(❡1) ❚(❡2) ❚(❡3)
,
♦♥❞❡ ❚(❡1) = ❚
1 0 0
,❚(❡2) =❚
0 1 0
e ❚(❡3) =❚
0 0 1 ✳
❚r❛t❛r❡♠♦s ❛❣♦r❛ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ❣❡♦♠étr✐❝❛s ❞♦ R2✳ ❱❡r❡♠♦s ❛❧✲
❣✉♠❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ s✉❛s ✐♥t❡r♣r❡t❛çõ❡s ❣❡♦♠étr✐❝❛s✳
❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
✶✳✶✳✷ ❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦
P❧❛♥♦
❆♥t❡s ❞❡ ❢❛❧❛r♠♦s ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ ♥♦ ♣❧❛♥♦✱ ✈❛♠♦s ♦❜s❡r✈❛r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ✉♠ ✈❡t♦r ✈ ∈ ❊, onde ❊ = R2 q✉❛♥❞♦ é ❛♣❧✐❝❛❞♦ ✉♠❛ tr❛♥s❢♦r♠❛✲
çã♦ ❞❡ ❞✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦✳ ❱❡❥❛♠♦s ❛❧❣✉♠❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❞✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ❛ s❡❣✉✐r✱ ❛ss✐♠ ❝♦♠♦ s✉❛s ♠❛tr✐③❡s ❞❡ tr❛♥s❢♦r♠❛çã♦✿
❛✮ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r✿ ❙❡❥❛ ❚ :R2
−→R2
,t❛❧ q✉❡
❚
x y
=λ
x y
=
λx λy
=
λ 0 0 λ
x y
,
❝♦♠ λ∈R✳
❖❜s❡r✈❛♠♦s q✉❡✿
• s❡ |λ|>1✱ ❚ ❞✐❧❛t❛ ♦ ✈❡t♦r❀
• s❡ |λ|<1✱ ❚ ❝♦♥tr❛✐ ♦ ✈❡t♦r❀
• s❡ λ= 1✱ ❚ ♥ã♦ ❛❧t❡r❛ ♦ ✈❡t♦r✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❚ é ❛ ✐❞❡♥t✐❞❛❞❡ ■ ❀
• s❡ λ <0✱ ❚ tr♦❝❛ ♦ s❡♥t✐❞♦ ❞♦ ✈❡t♦r✳
❋✐❣✉r❛ ✶✳✷✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❞✐❧❛t❛çã♦ ♥♦ ♣❧❛♥♦ R2
❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❊①❡♠♣❧♦ ✷ ❆ tr❛♥s❢♦r♠❛çã♦ ❚:R2
−→R2
✱ ❚
x y
= 3
x y
r❡♣r❡s❡♥t❛ ✉♠❛ ❞✐❧❛t❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r✳
❜✮ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ ①✿ ❙❡❥❛ ❚ :R2
−→R2
,t❛❧ q✉❡
❚
x y
=
λx y
=
λ 0 0 1
x y
,
❝♦♠ λ >0✳
❖❜s❡r✈❡ q✉❡✿
• s❡ λ >1✱ ❚ ❞✐❧❛t❛ ♦ ✈❡t♦r❀
• s❡ 0< λ <1✱ ❚ ❝♦♥tr❛✐ ♦ ✈❡t♦r✳
❊ss❛ tr❛♥s❢♦r♠❛çã♦ t❛♠❜é♠ é ❝❤❛♠❛❞❛ ❞✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ❤♦r✐③♦♥t❛❧ ❞❡ ✉♠ ❢❛t♦rλ✳
❋✐❣✉r❛ ✶✳✸✿ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ ①
◆❛s ✜❣✉r❛s 1.3 ❡ 1.4 ♦s ❢❛t♦r❡s ❞❡ ❞✐❧❛t❛çã♦ ❡ ❝♦♥tr❛çã♦ λ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s λ= 2 ❡λ= 1
2. r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❝✮ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ ②✿
❚r❛♥s❢♦r♠❛çã♦ ▲✐♥❡❛r ❞❡ ❉✐❧❛t❛çã♦ ♦✉ ❈♦♥tr❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❙❡❥❛ ❚ :R2
−→R2
,t❛❧ q✉❡
❚
x y
=
x λy
=
1 0 0 λ
x y
,
❝♦♠ λ >0✳
❋✐❣✉r❛ ✶✳✹✿ ❉✐❧❛t❛çã♦ ♦✉ ❝♦♥tr❛çã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ②
❖❜s❡r✈❛çã♦ ✺ ◆♦ ❝❛s♦ ❞❡ λ= 0✱ t❡rí❛♠♦s✿
❚
x y
=
x
0y
=
x
0
❡ ❚ s❡r✐❛ ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ♣❧❛♥♦ s♦❜r❡ ♦ ❡✐①♦ ①✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♣❡❧❛ ✜❣✉r❛✳
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s❝❛❧❛ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✺✿ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ♣❧❛♥♦ s♦❜r❡ ♦ ❡✐①♦ ①
❙❡ λ= 0 ♥♦ ❝❛s♦ ❜✮✱ ❚ s❡r✐❛ ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ♣❧❛♥♦ s♦❜r❡ ♦ ❡✐①♦ ②✳
✶✳✶✳✸ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s❝❛❧❛ ♥♦ P❧❛♥♦
❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❚ : R2
−→ R2
♦♥❞❡ ❛ ❛❜s❝✐ss❛ ① é ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✉♠ ❢❛t♦r s1 ❡ ❛ ♦r❞❡♥❛❞❛ ② ♣♦r ✉♠ ❢❛t♦r s2 ❡ ❛ ♠❛tr✐③ ❞❡
tr❛♥s❢♦r♠❛çã♦ é ❞❛❞❛ ♣♦r✿
▼=
s1 0
0 s2
,
♣♦✐s
❚(♣) =
s1x
s2y
=
s1 0
0 s2
x y
.
❊①❡♠♣❧♦ ✸ ❱❛♠♦s ❛♣❧✐❝❛r ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ ♥♦ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s ❆(1,1)✱
❇(5,2) ❡ ❈(3,3); ❝♦♠ s1 =
1
2 ❡ s2 = 2✳ ❆ss✐♠ s❡ ❆
′,❇′ e ❈′ sã♦ ♦s ♥♦✈♦s ✈ért✐❝❡s
❞♦ tr✐â♥❣✉❧♦ ❛♣ós ❛ ❛♣❧✐❝❛çã♦ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛✱ t❡♠♦s q✉❡✿ ❆′
1 2,2
✱
❇′(5
2,2)❡ ❈ ′
3 2,6
✳ ❖❜s❡r✈❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦ ♣❛r❛ ❡♥t❡♥❞❡r ♦ q✉❡ ❛❝♦♥t❡❝❡✉ ❝♦♠ ♦ tr✐â♥❣✉❧♦ ❆❇❈.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✻✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ ♥♦ R2
P♦ré♠✱ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r✱ ❛ ❡s❝❛❧❛ ♠♦❞✐✜❝❛ ❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧ ❞♦ ♥♦ss♦ ❡❧❡♠❡♥t♦✳ P❛r❛ ❝♦♥t♦r♥❛r ❡ss❡ ♣r♦❜❧❡♠❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ♣♦♥t♦ ❞❡ss❡ ❡❧❡♠❡♥t♦ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♦r✐❣❡♠ ✭t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ♣✐✈♦t✮✳ ❊♥tã♦ ♠♦✈❡rí❛♠♦s ❡ss❡ ♣♦♥t♦ ♣❛r❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ tr❛♥s❧❛çã♦ ✭q✉❡ s❡rá ✈✐st❛ ♠❛✐s ❛ ❞✐❛♥t❡✮ ❡ só ❞❡♣♦✐s ❛♣❧✐❝❛❞❛ ❛ ❡s❝❛❧❛✳ ❆♣ós ❛ ❛♣❧✐❝❛çã♦ ❞❛ ❡s❝❛❧❛ ♥♦ ♥♦ss♦ ❡❧❡♠❡♥t♦ ❧❡✈❛rí❛♠♦s ❞❡ ✈♦❧t❛ ♦ ♣✐✈♦t ♣❛r❛ ❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧✳
◆❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s❝❛❧❛ t❡♠♦s q✉❡ ♦s ❡s❝❛❧❛r❡s s1 ❡ s2 sã♦ ♥ú♠❡r♦s r❡❛✐s
♣♦s✐t✐✈♦s✳ ◗✉❛♥❞♦ ❡❧❡s ✈❛r✐❛♠ ♥♦ ✐♥t❡r✈❛❧♦ ✭✵✱✶✮ t❡♠♦s ✉♠❛ r❡❞✉çã♦ ❞❛ ❞✐♠❡♥sã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡ s❡ s1 ❡ s2❢♦r❡♠ ♠❛✐♦r❡s q✉❡ ✶ t❡r❡♠♦s ✉♠ ❛✉♠❡♥t♦✱ ♣♦ré♠ s❡ ❢♦r❡♠
♥❡❣❛t✐✈♦s ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ s❡r✐❛♠ ❡s♣❡❧❤❛❞♦s ❡♠ t♦r♥♦ ❞❡ ✉♠ ❡✐①♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳
✶✳✶✳✹ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦
❆ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ♦✉ r❡✢❡①ã♦ t❡♠ ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r✲ ♠❛çã♦ ✐❣✉❛❧ ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ ❛❧❣✉♥s ❞❡ s❡✉s t❡r♠♦s ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❝♦♠ s✐♥❛❧ ♥❡❣❛t✐✈♦✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s t✐♣♦s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❡s♣❡❧❤❛♠❡♥t♦✳
✶✳ ❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ①✿ ❙❡❥❛ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
x
−y
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✼✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x
◆♦t❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ①✱ é ❞❛❞❛ ♣♦r✿
▼=
1 0 0 −1
.
❉❡ ❢❛t♦✱
x
−y
=
1 0 0 −1
x y
.
✷✳ ❊s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ②✿ ❙❡❥❛ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
−x
y
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✽✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ y
◆♦t❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ② é ❞❛❞❛ ♣♦r✿
▼=
−1 0 0 1
.
❉❡ ❢❛t♦✱
−x
y
=
−1 0 0 1
x y
.
✸✳ ❊s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠✿ ❙❡❥❛ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
−x
−y
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✾✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠
◆♦t❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ é ❞❛❞❛ ♣♦r✿
▼=
−1 0 0 −1
.
❉❡ ❢❛t♦✱
−x
−y
=
−1 0 0 −1
x y
.
✹✳ ❊s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛ y=−x✿
❙❡❥❛ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
−y
−x
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❊s♣❡❧❤❛♠❡♥t♦ ♦✉ ❘❡✢❡①ã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✶✵✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛y =−x
◆♦t❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ r❡t❛
y=−x é ❞❛❞❛ ♣♦r✿
▼=
0 −1
−1 0
.
❉❡ ❢❛t♦✱
−y
−x
=
0 −1
−1 0
x y
.
✺✳ ❊s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛ y=x✿
❙❡❥❛ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
y x
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✶✶✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❛ r❡t❛y =x
◆♦t❡ q✉❡ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❡s♣❡❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛ r❡t❛
y=x é ❞❛❞❛ ♣♦r✿
▼=
0 1 1 0
.
❉❡ ❢❛t♦✱
y x
=
0 1 1 0
x y
.
✶✳✶✳✺ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ P❧❛♥♦
❆♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♥♦ ♣❧❛♥♦✳ ❊❧❛ ♣❡r♠✐t❡ r♦t❛❝✐♦✲ ♥❛r ✉♠ ✈❡t♦r ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ ❞❡ ✉♠ â♥❣✉❧♦θ❛tr❛✈és ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ♠❛tr✐③
❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♣❡❧❛ ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r q✉❡ q✉❡r❡♠♦s r♦t❛❝✐♦♥❛r✳ ❉❡✜♥✐çã♦ ✹ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♥♦ ♣❧❛♥♦ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ ❞❡ ✉♠ â♥❣✉❧♦ θ é ✉♠❛ ❛♣❧✐❝❛çã♦
❘:R2 −→R2,
t❛❧ q✉❡
❘
x y
=
cos θ −sen θ
sen θ cos θ
x y
.
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
Pr♦♣♦s✐çã♦ ✷ ❆ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♥♦ R2
é ❞❛❞❛ ♣♦r✿
▼=
cos θ −sen θ
sen θ cos θ
.
❉❡♠♦♥str❛çã♦✿ P❛r❛ r❡❛❧✐③❛r ❡st❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s ♣r❡❝✐s❛r ❞❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s tr✐❣♦♥♦♠étr✐❝❛s✿
sen (α+θ) = sen α cos θ + sen θ cos α ✭✶✳✶✮
cos (α+θ) = cos α cosθ − sen α sen θ ✭✶✳✷✮
❋✐❣✉r❛ ✶✳✶✷✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ♥♦ R2
P❡❧❛ ✜❣✉r❛ ✈❡♠♦s q✉❡
sen α = y
r →y=r sen α
cos α= x
r →x=r cos α
❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ✈❡t♦r ♣ ❡♠ ❢✉♥çã♦ ❞❡ cos ❡sen✱ ✐st♦ é✱
♣ =
r cos α
r sen α
.
❋❛③❡♥❞♦ ❛ r♦t❛çã♦ ❞♦ ✈❡t♦r ♣ ❞❡ ✉♠ â♥❣✉❧♦ θ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ t❡♠♦s✿
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘♦t❛çã♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶ ❘(♣) = ❘ x y =
r cos (α+θ)
r sen (α+θ)
, ❝♦♠♦ ❘(♣) =♣′ ❡ ♣′ = x′ y′
✱ ❝♦♥❝❧✉✐r♠♦s q✉❡
x′ y′ =
r cos (α+θ)
r sen (α+θ)
.
❯t✐❧✐③❛♥❞♦ ❛s ❢ór♠✉❧❛s tr✐❣♦♥♦♠étr✐❝❛s (1.1)❡ (1.2)♦❜t❡♠♦s
x′ y′ =
r cos α cos θ − r sen α sen θ
r cos α sen θ + r sen α cos θ
.
❉❡s❞❡ q✉❡ x=r cos α ❡ y=r sen α t❡♠♦s q✉❡
x′ y′ =
x cos θ − y sen θ x senθ + y cos θ
=
cos θ −sen θ
sen θ cos θ
x y
,
♦✉ s❡❥❛✱ ❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦θ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ é
▼=
cos θ −sen θ
sen θ cos θ
,
♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳
P♦❞❡rí❛♠♦s t❡r ❡♥❝♦♥tr❛❞♦ ❛ ♠❛tr✐③ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ r♦t❛çã♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s s❡ ♦❜s❡r✈❛r♠♦s q✉❡ ❛s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈❡t♦r❡s {❡1,❡2} tr❛♥s❢♦r♠❛❞♦s ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ ❘✱ ✐st♦ é✿
❘(❡1) =❘
1 0 = cos θ sen θ ❡
❘(❡2) =❘
0 1 =
−sen θ
cos θ
.
Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✶✸✿ ❉❡❞✉çã♦ ❞✐r❡t❛ ❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦
❈♦♠♦ ▼ = ❘(❡)1 ❘(❡)2
✱ s❡❣✉❡ q✉❡
▼=
cos θ −sen θ
sen θ cos θ
.
✶✳✶✳✻ Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦
❙❡ ✉ ❡ ✈ sã♦ ✈❡t♦r❡s ❡♠ Rn, ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉ ❡ ✈ ❝♦♠♦ ♠❛tr✐③❡s n
×1✳ ❆
tr❛♥s♣♦st❛ ✉T é ✉♠❛ ♠❛tr✐③1×n✱ ❡ ♦ ♣r♦❞✉t♦ ♠❛tr✐❝✐❛❧ ✉T ·✈ é ✉♠❛ ♠❛tr✐③1×1✱
q✉❡ ✈❛♠♦s ❡s❝r❡✈❡r ❝♦♠♦ ✉♠ ♥ú♠❡r♦ r❡❛❧✭❡s❝❛❧❛r✮✳ ❖ ♥ú♠❡r♦ ✉T ·✈ é ❝❤❛♠❛❞♦ ♦
♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ✉ ♣♦r ✈ ❡ é ❡s❝r✐t♦ ♠✉✐t❛s ✈❡③❡s ❝♦♠♦ ✉·✈✳ P♦rt❛♥t♦ ❙❡
✉=
u1
u2
✳✳✳
un
, ✈=
v1
v2
✳✳✳
vn
❡♥tã♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ✉ ❡ ✈ é✿
u1 u2 · · · un
v1
v2
✳✳✳
vn
=u1v1+u2v2+· · ·+unvn.
Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ❈❛♣ít✉❧♦ ✶
❆s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ sã♦ ❛s s❡❣✉✐♥t❡s✿ ❙❡❥❛♠ ✉,✈ ❡ ✇ ✈❡t♦r❡s ❡♠Rn
❡ s❡❥❛λ ✉♠ ❡s❝❛❧❛r✳ ❊♥tã♦
• ✉·✈=✈·✈;
• (✉+✈)·✇ =✉·✇+✈·✇;
• (λ✈)·✇ =λ(✈·✇) =✈·(λ✇);
• ✉·✉>0e ✉·✉= 0 s❡ ❡ s♦♠❡♥t❡ s❡ ✉= 0.
❉❡✜♥✐çã♦ ✺ ❙❡❥❛ ♦s ✈❡t♦r❡s ✉,✈∈R2
✱ t❛❧ q✉❡ ✉= (x1, y1) e ✈= (x2, y2)♦ ♣r♦❞✉t♦
✐♥t❡r♥♦ ❞❡ ✉ ♣♦r ✈ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦r ✉·✈✱ s❡rá ❞❡t❡r♠✐♥❛❞♦ ♣♦r x1x2+y1y2✱
✐st♦ é✿
✉·✈ =x1x2+y1y2.
❉❡✜♥✐çã♦ ✻ ❉❛❞♦ ✉♠ ✈❡t♦r ✈ ∈ R2
✱ ❛ ♥♦r♠❛✱ ♠ó❞✉❧♦ ♦✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✈ é ♦ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥❡❣❛t✐✈♦✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r k✈k✱ ❞❡✜♥✐❞♦ ♣♦r✿
k✈k=√✈·✈.
Pr♦♣♦s✐çã♦ ✸ ❙❡❥❛ ✉♠❛ ♠❛tr✐③ ▼ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♥♦ ♣❧❛♥♦ ✱ ❡♥tã♦✿
▼T▼
=
k✉k2 ✉
·✈
✈·✉ k✈k2
,
♦♥❞❡ ▼T é ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ ▼.
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ✉,✈ ∈ R2
✱ t❛✐s q✉❡ ❚(❡1) = ✉ ❡ ❚(❡2) = ✈✱ ♦♥❞❡ ✉ =
a b
❡ ✈=
c d
.❊♥tã♦ ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ s❡rá
▼=
a c b d
,
♣♦rt❛♥t♦ ❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ▼ s❡rá✿
▼T =
a b c d
.
▲♦❣♦
Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ❈❛♣ít✉❧♦ ✶
▼T▼ =
a b c d
a c b d
=
a2
+b2
ac+bd ca+db c2
+d2
=
✉·✉ ✉·✈
✈·✉ ✈·✈
=
k✉k2
✉·✈
✈·✉ k✈k2
,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦str❛r✳
❖❜s❡r✈❛çã♦ ✻ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❞♦✐s ✈❡t♦r❡s ✉ ❡ ✈ ♥♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❊ t❛♠✲ ❜é♠ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
✉·✈=k✉kk✈kcos β
♦♥❞❡ β é ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s ✉ ❡ ✈✳
Pr♦♣♦s✐çã♦ ✹ ❖s ✈❡t♦r❡s ✉ ❡ ✈ sã♦ ♦rt♦❣♦♥❛✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱
✉·✈= 0.
❉❡♠♦♥str❛çã♦✿ ❙❡ ✉ é ♦rt♦❣♦♥❛❧ ❛ ✈✱ ♦✉ s❡❥❛✱ ♣❡r♣❡♥❞✐❝✉❧❛r✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❡❧❡s é 90◦ ♣❡❧❛ ❖❜s❡r✈❛çã♦ 6 t❡♠♦s
✉·✈ = k✉kk✈kcos 90◦ = k✉kk✈k0 = 0,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦str❛r✳
❉❡✜♥✐çã♦ ✼ ❉✐③❡♠♦s q✉❡ ✉♠ ✈❡t♦ru é ✉♥✐tár✐♦ s❡ s✉❛ ♥♦r♠❛ ❢♦r ✐❣✉❛❧ ❛ 1✱ ✐st♦ é✿
k✉k= 1.
◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ ✉ ❡stá ♥♦r♠❛❧✐③❛❞♦✳
Pr♦♣♦s✐çã♦ ✺ ❚♦❞♦ ✈❡t♦r ✉ ♥ã♦✲♥✉❧♦ ♣♦❞❡ s❡r ♥♦r♠❛❧✐③❛❞♦✱ ❢❛③❡♥❞♦✲s❡✿
✉= ✉
k✉k.
Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ❈❛♣ít✉❧♦ ✶
❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡ q✉❡ ✉
k✉k ·
✉
k✉k =
✉·✉ k✉k2 =
k✉k2
k✉k2 = 1
❡✱ ♣♦rt❛♥t♦ ✉
k✉k é ✉♥✐tár✐♦✳
❉❡✜♥✐çã♦ ✽ ❉♦✐s ✈❡t♦r❡s ✉ ❡ ✈ sã♦ ♦rt♦♥♦r♠❛✐s s❡✱ s♦♠❡♥t❡ s❡✱ ❛♠❜♦s ❢♦r❡♠ ✉♥✐✲ tár✐♦ ❡ ♦rt♦❣♦♥❛✐s ❡♥tr❡ s✐✱ ✐st♦ é✿
✉·✈= 0
❡
k✉k=k✈k= 1.
❉❡✜♥✐çã♦ ✾ ❙❡ ♦s ✈❡t♦r❡s ❝♦❧✉♥❛ ❞❡ ✉♠❛ ♠❛tr✐③ ▼ sã♦ ✉♥✐tár✐♦s ❡ ♦rt♦❣♦♥❛✐s ❞♦✐s ❛ ❞♦✐s✱ ❞✐③❡♠♦s q✉❡ ▼ é ✉♠❛ ♠❛tr✐③ ♦rt♦♥♦r♠❛❧✳
❊①❡♠♣❧♦ ✹ ❆ ♠❛tr✐③ ▼ ❞❡ r♦t❛çã♦ ♥♦ ♣❧❛♥♦ é ♦rt♦♥♦r♠❛❧✱ ♣♦✐s t❡♠♦s ✉=
cos θ
sen θ
❡ ✈=
−sen θ
cos θ
✱ ❧♦❣♦
✉·✈= cos θ (−sen θ) + sen θ cos θ = 0
❡
k✉k=k✈k=√cos2 θ + sen2 θ=√1 = 1.
❖❜s❡r✈❛çã♦ ✼ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❝♦♠✉t❛t✐✈♦✱ ♦✉ s❡❥❛✱
✉·✈ =✈·✉.
Pr♦♣♦s✐çã♦ ✻ ❙❡❥❛ ▼ ✉♠❛ ♠❛tr✐③ ♦rt♦♥♦r♠❛❧✱ ❡♥tã♦
▼T
=▼−1
.
❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr❡♣♦s✐çã♦ 3✱ s❡❣✉❡ q✉❡
▼❚▼= k✉k2 ✉·✈
✈·✉ k✈k2
=
1 0 0 1
=■,
♦♥❞❡ ■ é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♣♦rt❛♥t♦ ▼T =▼−1
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦str❛r✳
Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❘♦t❛çã♦ ❈❛♣ít✉❧♦ ✶
❆✜r♠❛çã♦ ✶ ◆✉♠❛ r♦t❛çã♦✱ ❡♠ t❡r♠♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s t❛♥t♦ ❢❛③ r♦t❛❝✐♦♥❛r♠♦s ♦ ✈❡t♦r ♣ ❡♠ ✉♠ â♥❣✉❧♦ θ ♦✉ ❡s❝r❡✈❡r♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡st❡ ✈❡t♦r ♥✉♠ s✐st❡♠❛ ❞❡
❝♦♦r❞❡♥❛❞❛s r♦t❛❝✐♦♥❛❞♦ ❞❡ −θ✱ ♦✉ s❡❥❛ ✿
u v
=
x′ y′
=
cos θ −sen θ
sen θ cos θ
.
❋✐❣✉r❛ ✶✳✶✹✿ ❚r❛♥s❢♦r♠❛çã♦ ✈❡rs✉s ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈❡t♦r❡s ✉ ❡ ✈ ❡♠ r❡❧❛çã♦ ❛♦s ❡✐①♦s ❝❛rt❡s✐❛♥♦s x ❡y sã♦✿ ✉=
ux
uy
❡ ✈=
vx
vy
,♦♥❞❡ ux = cos (−θ);
uy = −sen (−θ);
vx = sen (−θ);
vy = cos (−θ);
❛ss✐♠✿
✉=
cos (−θ)
−sen (−θ)
e ✈=
sen (−θ) cos (−θ)
,
❝♦♠♦ ❛s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈❡t♦r❡s tr❛♥s✲ ❢♦r♠❛❞♦s✱ ❡♥tã♦ ❛ ♠❛tr✐③ ▼ ❞❡ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ −θ s❡rá✿
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❈✐s❛❧❤❛♠❡♥t♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
▼=
cos (−θ) sen (−θ)
−sen (−θ) cos (−θ)
,
♠ás cos (−θ) = cos θ ❡ sen (−θ) =−sen θ ✱ ❧♦❣♦✿
▼=
cos θ −sen θ
sen θ cos θ
, ♣♦rt❛♥t♦ u v = x′ y′ =
cos θ −sen θ
sen θ cos θ
x y
,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
❚❡♠♦s ❛✐♥❞❛ q✉❡ s❡ ❡s❝r❡✈❡♠♦s ❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ −θ t❡rí❛♠♦s
❛ ♠❛tr✐③
❆=
cos (−θ) −sen (−θ) sen (−θ) cos (−θ)
=
ux uy
vx vy
=
cos θ sen θ
−sen θ cos θ
.
❖❜s❡r✈❡ q✉❡ ❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❆ é ✐❣✉❛❧ ❛ ▼ ✱ ❛ss✐♠
❆T =
cos θ −sen θ
sen θ cos θ
=▼.
❆tr❛✈és ❞❡ss❛ ✐❣✉❛❧❞❛❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ r♦t❛❝✐♦♥❛r ✉♠ ✈❡t♦r ❡♠ ✉♠ â♥❣✉❧♦−θ ❜❛st❛ t♦♠❛r ❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ▼✱ ✐st♦ é✱ ❞❛❞♦ ✉♠ ✈❡t♦r
♣ ❞♦ R2
t❡♠♦s q✉❡✿ ♣′ = x′ y′ =
cos θ −sen θ
sen θ cos θ
x y =▼♣ ❡ ♣′′ = x′′ y′′ =
cos θ sen θ
−sen θ cos θ
x y
=▼T♣,
♦♥❞❡ ♣′ é ♦ ✈❡t♦r r♦t❛❝✐♦♥❛❞♦ ❞❡ ♣ ❡♠ ✉♠ â♥❣✉❧♦θ ❡ ♣′′ é ♦❜t✐❞♦ ❞♦ ♠❡s♠♦ ✈❡t♦r ♣
♥✉♠❛ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ −θ✳
✶✳✶✳✼ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❈✐s❛❧❤❛♠❡♥t♦ ♥♦ P❧❛♥♦
❆❣♦r❛ ✈❛♠♦s ❢❛❧❛r ❞❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ♣r❡s❡r✈❛r ✉♠❛ ❝♦♦r❞❡♥❛❞❛ ❡ ♠♦✈❡r ❛ ♦✉tr❛✱ ♠❛✐s ❡ss❡ ♠♦✈✐♠❡♥t♦ ❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❞❛ ❝♦♦r❞❡♥❛❞❛ ✐♥❛❧t❡r❛❞❛✳
❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❈✐s❛❧❤❛♠❡♥t♦ ♥♦ P❧❛♥♦ ❈❛♣ít✉❧♦ ✶
Pr♦♣♦s✐çã♦ ✼ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x é ✉♠❛
❛♣❧✐❝❛çã♦ ❚:R2
−→R2
, t❛❧ q✉❡
❚
x y
=
x+y tg γ y
=
1 tg γ
0 1
x y
,
♦♥❞❡γ é ♦ â♥❣✉❧♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ y✳ ❆ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦
❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ x é✿
▼=
1 tg γ
0 1
,
❋✐❣✉r❛ ✶✳✶✺✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♠ x
❉❡♠♦♥str❛çã♦✿ P❛r❛ ♦❜t❡r ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❛❝✐♠❛✱ ♦❜s❡r✈❡ q✉❡ s❡ ❚ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦ ❡♥tã♦ ♣❛r❛ t♦❞♦ ♣∈R2 t❡♠♦s
❚(♣) =
x′ y′
=
x+y tg γ y
=
1 tg γ
0 1
x y
.
❆ ♦r❞❡♥❛❞❛ ♣❡r♠❛♥❡❝❡ ✐♥❛❧t❡r❛❞❛ ❡ ❛ ❛❜s❝✐ss❛ s♦❢r❡ ✉♠ ❛❝rés❝✐♠♦ k ∈R✱ ✐st♦ é✿
y′ =y e x′ =x+k
❝♦♠♦ tg γ = k
y →k=y tg γ✱ t❡♠♦s
x′ =x+y tg γ,
♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳
❖❜s❡r✈❛çã♦ ✽ ❈❛s♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ s❡❥❛ ❡♠ ❞✐r❡çã♦ ❛♦ ❡✐①♦ y✱ ❞✐❣❛♠♦s ❝♦♠ ♦
â♥❣✉❧♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ β ❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ s❡rá✿
