Álgebra linear no ensino médio

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆❈➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

❆▲❊❳ ❉❊ ❙❖❯❩❆ ▼❆●❆▲❍➹❊❙

➪▲●❊❇❘❆ ▲■◆❊❆❘ ◆❖ ❊◆❙■◆❖ ▼➱❉■❖

❆▲❊❳ ❉❊ ❙❖❯❩❆ ▼❆●❆▲❍➹❊❙

➪▲●❊❇❘❆ ▲■◆❊❆❘ ◆❖ ❊◆❙■◆❖ ▼➱❉■❖

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠✁t✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ➹r❡❛ ❞❡ ❝♦♥❝❡♥trçã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡✲ ♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ▼s✳ P❛✉❧♦ ❈és❛r ❈❛✈❛❧❝❛♥t❡ ❞❡ ❖❧✐✈❡✐r❛✳

❏❯❆❩❊■❘❖ ❉❖ ◆❖❘❚❊ ✲ ❈❊

✷✵✶✹

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s✱ ❝r✐❛❞♦r ❞❡ t✉❞♦ ❡ ❞❡ t♦❞♦s✱ ♣♦r s❡♠♣r❡ ❛t❡♥❞❡r ♠❡✉s ♣❡❞✐❞♦s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❊❧✐③â♥❣❡❧❛ ❚♦rr❡s ❞❡ ❙á ▼❛❣❛❧❤ã❡s✱ q✉❡ ❢♦✐ ♣❡ç❛ ❢✉♥❞❛✲ ♠❡♥t❛❧ ♥❡st❛ ❝♦♥q✉✐st❛✱ ❡ q✉❡✱ s❡♠♣r❡ ♠❡ ❛♣♦✐❛ ♥❛s ♠✐♥❤❛s ❡s❝♦❧❤❛s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ✜❧❤❛ ❆❞rí❝✐❛ ❚♦rr❡s ▼❛❣❛❧❤ã❡s✱ q✉❡ ❤♦❥❡✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ s✉❛ ♠ã❡✱ r❡♣r❡s❡♥t❛♠ ❛ r❛③ã♦ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❏♦ã♦ ❆✐❧t♦♥ ❞❡ ▼❛❣❛❧❤ã❡s ❡ ▼❛r✐❛ ❞❛s ❣r❛ç❛s ❞❡ ❙♦✉③❛ ▲❡ã♦✱ q✉❡✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ♠❡ ♠♦str❛r❛♠ ♦ ❝❛♠✐♥❤♦ ❞❛ ❡❞✉❝❛çã♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s ❆♥❛ P❛✉❧❛ ❞❡ ❙♦✉③❛ ▼❛❣❛❧❤ã❡s ❡ ❋❛❣♥❡r ❞❡ ❙♦✉③❛ ▼❛✲ ❣❛❧❤ã❡s✱ q✉❡ ❛❝✐♠❛ ❞❡ t✉❞♦ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠✐♠✱ ❛té ♠❛✐s ❞♦ q✉❡ ❡✉✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♠❡✉s ❛❧✉♥♦s ❡ ❡①✲❛❧✉♥♦s✱ ♣♦✐s é ♣♦r ✈♦❝ês q✉❡ ❡st♦✉ ♠❡ ❝❛♣❛❝✐✲ t❛♥❞♦✳

❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ■♥❛❧❞♦ ❉✐♦♥ís✐♦ ♥❡t♦✱ q✉❡ ❢♦✐ ♦ ♣r✐♥❝✐♣❛❧ r❡s♣♦♥sá✈❡❧✱ ♣❡❧❛ ♠✐✲ ♥❤❛ ❢♦r♠❛çã♦ ❜ás✐❝❛ ❡♠ ♠❛t❡♠át✐❝❛✱ s❡♠ ❛ q✉❛❧ ❡✉ ❝❡rt❛♠❡♥t❡ ♥ã♦ ❡st❛r✐❛ ❡s❝r❡✈❡♥❞♦ ❡st❛s ♣❛❧❛✈r❛s✳

❆❣r❛❞❡ç♦ ❛♦ Pr♦❢❡ss♦r P❛✉❧♦ ❈és❛r ❈❛✈❛❧❝❛♥t❡ ❞❡ ❖❧✐✈❡✐r❛✱ q✉❡ s❡♠ ❞✉✈✐❞❛s✱ ❡ ♥ã♦ ❞❡s♠❡r❡❝❡♥❞♦ ♦s ❞❡♠❛✐s✱ ❢♦✐ ♦ q✉❡ ♠❛✐s ❝♦♥tr✐❜✉✐✉ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♥❡st❡ ❝✉rs♦✳

❆❣r❛❞❡ç♦ ❛ t♦❞❛ ❡q✉✐♣❡ q✉❡ ❢❛③ ♣❛rt❡ ❞♦ P❘❖❋▼❆❚✱ q✉❡✱ ❝♦♠ t♦❞❛ ❝❡rt❡③❛✱ ♠✉❞❛rá ❛ ❢♦r♠❛çã♦ ❞❡ ♠✐❧❤❛r❡s ❞❡ ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❇r❛s✐❧✳

P♦r ✜♠ ❛❣r❛❞❡ç♦ ❛ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❢❛r❡♠♦s ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r ♣r❡s❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛✳ ◆❡st❛ ❢♦r♠❛✱ s❡rá ♣r♦♣♦st♦ ❛ ✐♥tr♦❞✉çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❡s♣❛➹➓♦ ✈❡t♦r✐❛❧ ❡ ✈❛r✐❡❞❛❞❡ ❛✜♠✱ q✉❡ s❡rã♦ ❡①❡♠♣❧✐✜❝❛❞♦s ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❛s ♠❛tr✐③❡s ❡ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❙❡♥❞♦ ❛ss✐♠ ❛s ♠❛tr✐③❡s ❛♣❛r❡❝❡♠ ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✳ ◆❡st❡ t❡①t♦ ♥➹➾♦ s❡rá ❛❜♦r❞❛❞♦ ❛ ✐❞❡✐❛ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ❛❝r❡❞✐t❛♠♦s q✉❡ ❡st❛ ♣♦❞❡ s❡r✱ s❡♠ ♠✉✐t♦s ♣r❡❥✉í③♦s✱ r❡t✐r❛❞❛ ❞♦ ❝✉rrí❝✉❧♦ ♠❛t❡♠át✐❝♦ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ ✇✐❧❧ ♠❛❦❡ ❛ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ▲✐♥❡❛r ❆❧❣❡❜r❛ ✐♥ ❤✐❣❤ s❝❤♦♦❧ t❤✐s ❛❧t❡r✲ ♥❛t✐✈❡ ❢♦r♠✳ ■♥ t❤✐s ✇❛②✱ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣ts ♦❢ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ ❛✣♥❡ ✈❛r✐❡t②✱ ✇❤✐❝❤ ❛r❡ ✐♥tr♦❞✉❝❡❞ t❤r♦✉❣❤ t❤❡ st✉❞② ♦❢ ♠❛tr✐❝❡s ❛♥❞ ❧✐♥❡❛r s②st❡♠s✱ ✇✐❧❧ ❜❡ ♣r♦♣♦s❡❞✳ ❚❤✉s t❤❡ ❛rr❛②s ❛♣♣❡❛r ❛s ❡❧❡♠❡♥ts ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ❛ ❧✐♥❡❛r s②st❡♠ ❛s ❛♥ ❛✣♥❡ ✈❛r✐❡t②✳ ❚❤✐s t❡①t ✇✐❧❧ ♥♦t ❜❡ ❛❞❞r❡ss❡❞ t❤❡ ✐❞❡❛ ♦❢ ❞❡t❡r♠✐✲ ♥❛♥ts✱ ✇❡ ❜❡❧✐❡✈❡ t❤✐s ❝❛♥ ❜❡ ✇✐t❤♦✉t ♠✉❝❤ ❞❛♠❛❣❡✱ ✇✐t❤❞r❛✇❛❧ ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❝✉rr✐❝✉❧✉♠ ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥✳

❙✉♠ár✐♦

✶ ❊s♣❛ç♦ ❱❡t♦r✐❛❧ ✶✹

✶✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❆❞✐❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ▼❛tr✐③❡s ✶✽

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ▼❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ▼❛tr✐③❡s ❊s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ▼❛tr✐③ ▲✐♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✷ ▼❛tr✐③ ❈♦❧✉♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✸ ▼❛tr✐③ ◗✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✹ ▼❛tr✐③ ◆✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✺ ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ❙✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✻ ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ■♥❢❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✼ ▼❛tr✐③ ❉✐❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✽ ▼❛tr✐③ ■❞❡♥t✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✾ ▼❛tr✐③ ❙✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❖ ❊s♣❛ç♦ ❱❡t♦r✐❛❧ ❞❛s ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ❖✉tr❛s ❖♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✶ ❚r❛♥s♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸ ❱❛r✐❡❞❛❞❡ ❆✜♠ ✸✷ ✸✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✸✹

✹✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✸ ❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s ❡ ▼❛tr✐③❡s ▲✐♥❤❛s ❡q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✹ ❋♦r♠❛ ❊s❝❛❞❛✱ ♣♦st♦ ❡ ♥✉❧✐❞❛❞❡ ❞❡ ✉♠❛ ▼❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺ ❘❡s♦❧✉çã♦ ❡ ❊st✉❞♦ ❞♦s ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✻ ❙✐st❡♠❛ ▲✐♥❡❛r✱ ❱❛r✐❡❞❛❞❡ ❆✜♠ ❡ ❊s♣❛ç♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✼

■♥tr♦❞✉çã♦

❆tr❛✈és ❞❡ ❛❧❣✉♥s ❛♥♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ❝♦♠♦ ♣r♦❢❡ss♦r ❡ ❡st✉❞❛♥t❡ ❞❡ ♠❛t❡♠át✐❝❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦✱ ♦❜s❡r✈❡✐ ✈ár✐♦s ❢❛t♦s ✐♥✲ t❡r❡ss❛♥t❡s ❡ ❝♦♥tr❛❞✐tór✐♦s ♥❛ ❢♦r♠❛ ❞❡ ❝♦♠♦ ❛ ♠❛t❡♠át✐❝❛ é tr❛♥s♠✐t✐❞❛ ♣❛r❛ ♦s ❛❧✉♥♦s✳ ❉❡♥tr❡ ♦s ❢❛t♦s✱ ❡stá ❛ ❢♦r♠❛✱ ♣r❛t✐❝❛♠❡♥t❡ ❤♦♠♦❣ê♥✐❛✱ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ❋❛t♦ ❡st❡✱ ❝♦♥tr❛❞✐tór✐♦✱ ♣♦✐s ♦ ♣❡♥s❛♠❡♥t♦ ❤✉♠❛♥♦ é ✐♥❡✈✐t❛✈❡❧♠❡♥t❡ ❤❡t❡r♦❣ê♥❡♦✳ ❙❡♥❞♦ ❛ss✐♠ ❝♦♠♦ ♣♦❞❡ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s s❡r❡♠ tã♦ s❡♠❡❧❤❛♥t❡s✱ ❛♣❡s❛r ❞❡ s❡r❡♠ ❡s❝r✐t♦s ♣♦r ❞✐✈❡rs♦s ❛✉t♦r❡s❄ ❆♣❡s❛r ❞✐ss♦✱ ♠✉✐t♦ ♣♦✉❝♦ s❡ ❢❛③✱ ♥♦ s❡♥t✐❞♦ ❞❡ ✐♥♦✈❛r ♦s ❝♦♥t❡ú❞♦s ♠❛t❡♠át✐❝♦s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳

P❛r❛ ❝♦♥✜r♠❛r ❡st❡ ❢❛t♦✱ ❛♥t❡s ❞❡ r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✱ ✜③❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❡ ♥♦✈❡ ❞✐❢❡r❡♥t❡s ❝♦❧❡çõ❡s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ◆❡st❛ ❛♥á❧✐s❡ ✜❝♦✉ ❜❡♠ ❝❧❛r❛ ❛ ❢♦r♠❛ ❡stát✐❝❛ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ ♠❛t❡♠át✐❝❛✳ ❆s ❝♦❧❡çõ❡s✱ ♠✉✐t♦ s❡♠❡❧❤❛♥t❡s ❡♥tr❡ s✐✱ ♥ã♦ ❛♣r❡s❡♥t❛✈❛♠ ❞✐❢❡r❡♥ç❛s s✐❣♥✐✜❝❛t✐✈❛s✱ ♥♦ q✉❡ s❡ r❡❢❡r❡ ❛♦ ♠♦❞♦ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❡ ♥♦s ❡①❡r❝í❝✐♦s✳

❆tr❛✈és ❞❡st❛ ❛♥á❧✐s❡ ♣ô❞❡✲s❡ ✐♥❢❡r✐r✱ ❞❡ ❢♦r♠❛ q✉❛❧✐t❛t✐✈❛✱ ❛❧❣✉♥s ❛s♣❡❝t♦s r❡❧❡✈❛♥✲ t❡s ❞❡st❛s ♦❜r❛s✱ ✈❡❥❛♠♦s✳

Pr✐♠❡✐r♦✱ ✜❝♦✉ ❡✈✐❞❡♥t❡ ❛ ❢❛❧t❛ ❞❡ ✐♥♦✈❛çã♦ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s✱ ❛té ♠❡s♠♦ ♦s ❡①❡r❝í❝✐♦s ❛♣r❡s❡♥t❛✈❛♠ s❡♠❡❧❤❛♥ç❛s✱ ❛❧❣✉♥s ❡r❛♠ ♣r❛t✐❝❛♠❡♥t❡ ✐❞ê♥t✐❝♦s✱ ❞✐st✐♥❣✉✐♥❞♦✲s❡ ❛♣❡♥❛s ♣♦r ✈❛❧♦r❡s ♥✉♠ér✐❝♦s✳

◆♦ q✉❡ s❡ r❡❢❡r❡ ❛ á❧❣❡❜r❛ ❧✐♥❡❛r ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ✜❝♦✉ ❡✈✐❞❡♥t❡ q✉❡ ❛ ❣r❛♥❞❡ ♠❛✐♦✲ r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛✱ ❛♣r❡s❡♥t❛ ♦s tó♣✐❝♦s ❞❡st❛ ár❡❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛✳ ❋♦r♠❛ ❡st❛ q✉❡ s❡ r❡str✐♥❣❡ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s ❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❊♠ ♦✐t♦ ❞♦s ♥♦✈❡ ❧✐✈r♦s ❛♥❛❧✐s❛❞♦s ♦s tó♣✐❝♦s ❞❡

á❧❣❡❜r❛ ❧✐♥❡❛r sã♦ ❛♣r❡s❡♥t❛❞♦s ♥❛ s❡❣✉✐♥t❡ ♦r❞❡♠✿ ✐✮ ♠❛tr✐③❡s❀ ✐✐✮ ❞❡t❡r♠✐♥❛♥t❡s❀ ❡ ✐✐✐✮ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❊ ❡♠ ✉♠ ❞♦s ❧✐✈r♦s ♠✉❞❛✈❛ ❡st❛ ♦r❞❡♠ ♥♦ ❛s♣❡❝t♦ ❞❡ ❛♣r❡s❡♥t❛r s✐st❡♠❛s ❧✐♥❡❛r❡s ❛♥t❡s ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✳

❙❡♥❞♦ ❛ss✐♠✱ ✜❝♦✉ ❡✈✐❞❡♥t❡ ❛ s❡♠❡❧❤❛♥ç❛ ❞♦s ❧✐✈r♦s✱ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r✳ ❖✉tr❛ ✐♠♣♦rt❛♥t❡ ❝❛r❛❝t❡ríst✐❝❛✱ é q✉❡ ❡♠ t♦❞❛s ❛s ❝♦❧❡çõ❡s ❛♥❛❧✐s❛❞❛s ❛ ❛♣r❡✲ s❡♥t❛çã♦ ❞♦ ❝♦♥t❡ú❞♦ s❡ ❞❛✈❛ ❛tr❛✈és ❞❡ ❝♦♥❝❡✐t♦s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ✈❛r✐❛❞♦s ❡①❡r❝í❝✐♦s ❞❡ ❝❛rát❡r ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♣r❡✈❛❧❡❝❡♥❞♦ ❛ss✐♠ ✉♠❛ ❣r❛♥❞❡ ê♥❢❛s❡ ♥❛ r❡♣❡t✐çã♦ ❞❡ ❡①❡r✲ ❝í❝✐♦s✳

❘❡❝♦♥❤❡❝❡♠♦s q✉❡ ❡st❛ ❛❜♦r❞❛❣❡♠ t❡♠ s✉❛s ✈❛♥t❛❣❡♥s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ❢♦r♠❛ s✐♠♣❧✐st❛ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s✱ ❢❛❝✐❧✐t❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ ❞♦s ❛❧✉♥♦s ❝♦♠ ❡st❡s t❡♠❛s✳ ▼❛s ❛❝r❡❞✐t❛♠♦s q✉❡ ❡①✐st❡♠ ❛❧❣✉♠❛s ✐♥❝♦❡rê♥❝✐❛s ❧ó❣✐❝❛s ♥❡st❛ ❛♣r❡s❡♥t❛çã♦✳ ❉❡♥tr❡ ❡st❛s q✉❡st✐♦♥❛♠♦s✿ ❆ ♦r❞❡♠ ❞❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s ❡ ❛ ✉t✐❧✐❞❛❞❡ ❡ ❛♣❧✐❝❛çõ❡s ❞❡st❛s ✐♥❢♦r♠❛çõ❡s✱ ❛❧é♠ ❞♦ ♠❛✐s✱ ❞❡✈❡r✐❛♠ ❡①✐st✐r ♥♦✈❛s ❢♦r♠❛s ❞❡ ❛❜♦r❞❛❣❡♠✱ ♣♦✐s ❛ á❧❣❡❜r❛ ❧✐♥❡❛r é ✉♠❛ ♣❛rt❡ ❞❛ ♠❛t❡♠át✐❝❛ ♠✉✐t♦ r✐❝❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡ ❝♦♠♦ t❛❧✱ ♣♦❞❡r✐❛ s❡r ♠❡❧❤♦r ❡①♣❧♦r❛❞❛ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✳

❆ ♦r❞❡♠ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s q✉❡ é ♠❛✐s ❝♦♠✉♠ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✭♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s ❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮ ❛❝r❡❞✐t❛♠♦s ♥ã♦ s❡r ❛ ♠❛✐s ❛❞❡q✉❛❞❛ ♥♦ s❡♥t✐❞♦ t❡ór✐❝♦✳ ❆❧é♠ ❞✐ss♦✱ s❡rá q✉❡ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s✱ s✐st❡♠❛s ❧✐♥❡❛r❡s sã♦ ♦s ♠❡❧❤♦r❡s tó♣✐❝♦s ❞❡ á❧❣❡❜r❛ ❧✐♥❡❛r q✉❡ ❞❡✈❡♠ s❡r ❡♥s✐♥❛❞♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦❄

◆♦ q✉❡ s❡ r❡❢❡r❡ ❛♦ ❡st✉❞♦ ❞❛s ♠❛tr✐③❡s✱ ❛❝r❡❞✐t❛♠♦s q✉❡ s✉❛ ❛♣r❡s❡♥t❛çã♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ é ❛❞❡q✉❛❞❛✳ ▼❛s é ❞❡ ❝❡rt❛ ❢♦r♠❛ ✈❛❣❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❝♦❡rê♥❝✐❛ ❧ó❣✐❝❛ ❞♦ ❛ss✉♥t♦✱ s❡♥❞♦ q✉❡ ✐ss♦ s❡ ❞á✱ ♣♦✐s ♥ã♦ é ❡♥❢❛t✐③❛❞♦ ♣❡❧♦s ❧✐✈r♦s ❛ s✉❛ ♣r✐♥❝✐♣❛❧ ❛♣❧✐❝❛çã♦ ♥❡st❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦✱ q✉❡ é ♥❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳

❖ ❡♥s✐♥♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s é ❡q✉✐✈♦❝❛❞♦ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛ ❢♦r♠❛ q✉❡ é ❛♣r❡s❡♥t❛❞♦✱ ♣♦r ❞♦✐s ♠♦t✐✈♦s✳ Pr✐♠❡✐r♦ ♥ã♦ ❢❛③ ♠✉✐t♦ s❡♥t✐❞♦ ❢❛❧❛r ❞❡ ❞❡t❡r✲ ♠✐♥❛♥t❡s ❛♥t❡s ❞❡ s✐st❡♠❛ ❧✐♥❡❛r✱ ❛✜♥❛❧ ❡st❡ tó♣✐❝♦ é ✉t✐❧✐③❛❞♦✱ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ s✐♠✲ ♣❧❡s♠❡♥t❡ ♣❛r❛ ❡st✉❞❛r ❡ r❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ❛❧é♠ ❞❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s ♠❡♥♦s r❡❧❡✈❛♥t❡s✳ ❙❡❣✉♥❞♦✱ ♣♦❞❡♠♦s ❡st✉❞❛r ❡ r❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s

❡✜❝❛③ ❡ ❡✜❝✐❡♥t❡ s❡♠ ♦ ✉s♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ❡♥tã♦ é r❡❛❧♠❡♥t❡ ♥❡❝❡ssár✐♦ ♦ ❡♥s✐♥♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛❄ ❆❧é♠ ❞♦ ♠❛✐s✱ ❛s ♦✉tr❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❞❡t❡r♠✐♥❛♥✲ t❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♣♦❞❡♠ s❡r ❢❡✐t❛s ❞❡ ♦✉tr❛s ♠❛♥❡✐r❛s✳ ❉❡ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ é ♣♦ssí✈❡❧ ❛ ❡①❝❧✉sã♦ ❞♦ ❡♥s✐♥♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳

❏á ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ❛ss✉♥t♦ q✉❡ é ❝♦♠ ❝❡rt❡③❛ ♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ é ❛♣r❡s❡♥t❛❞♦ ❞❡ ❢♦r♠❛ s❛t✐s❢❛tór✐❛✱ ♠❛s q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛✱ ❛té ♠❛✐s ♣rát✐❝❛ ♣❛r❛ ❡st❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦✳ ❆t✉❛❧♠❡♥t❡✱ ❡①✐st❡ ✉♠ ❣r❛♥❞❡ ê♥❢❛s❡ ❞♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛ ❧✐♥❡❛r ❛tr❛✈és ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ❡ ❛ r❡s♦❧✉çã♦ ❛tr❛✈és ❞❛ r❡❣r❛ ❞❡ ❈r❛♠❡r✳ ▼❛s s❡rá r❡❛❧♠❡♥t❡ ♥❡❝❡ssár✐❛ ❡st❛ ❛❜♦r❞❛❣❡♠❄ ❚❛❧✈❡③✱ ♦ s✐♠♣❧❡s ✉s♦ ❞❛ ❡❧✐♠✐♥❛çã♦✱ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥ s❡❥❛ ♠❛✐s s✐♠♣❧❡s ❡ s✉✜❝✐❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛❧t❡r♥❛t✐✈❛✱ q✉❡ ♥❡st❡ ❝♦♥t❡①t♦ ✜❝❛ ✐s❡♥t❛ ❞❛s ❝rít✐❝❛s ❛♥t❡r✐♦r❡s✱ ♠❛s é ❝❧❛r♦ q✉❡ ♥ã♦ é ✐s❡♥t♦ ❞❡ ♥♦✈❛s ❝rít✐❝❛s✳ ◆❡st❛ ❢♦r♠❛ ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ t❡♠♦s ❝♦♠♦ ❝❛r❛❝t❡ríst✐❝❛s ❛ ✐♥tr♦❞✉çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ✈❛r✐❡❞❛❞❡ ❛✜♠✱ ❛ ❡①❝❧✉sã♦ ❞♦s ❞❡t❡r♠✐✲ ♥❛♥t❡s ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛✱ ❡ ♦ ❡st✉❞♦ ❡ r❡s♦❧✉çã♦ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s ❛tr❛✈és ❞❛ ❡❧✐♠✐♥❛çã♦✱ ❡♥❢❛t✐③❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ♣♦st♦ ❡ ♥✉❧✐❞❛❞❡✱ s❡❣✉♥❞♦ ❧✐♥❤❛s ❞❡ ✉♠❛ ♠❛tr✐③ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳

P♦r ✜♠✱ ❝❛❜❡ ❛❣♦r❛✱ ❞❡st❛❝❛r ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ◗✉❡ é ♠♦str❛r ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ♦✉ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ❛♣r❡s❡♥t❛r ❛ á❧❣❡❜r❛ ❧✐♥❡❛r ♥♦ ❡♥s✐♥♦ ♠é✲ ❞✐♦✳ ❊st❛ ❢♦r♠❛ ♣♦❞❡rá s❡r ✉s❛❞❛ ♣♦r ♣r♦❢❡ss♦r❡s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡ ❢♦r♠❛ r❡❣✉❧❛r✱ s✉❜st✐t✉✐♥❞♦ ♦s t❡①t♦s ❝♦♠✉♥s ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ♦✉ ❝♦♠♦ t❡①t♦ ❝♦♠♣❧❡♠❡♥t❛r ♣❛r❛ ♦ ❛❧✉♥♦✱ ❞❡ ❢♦r♠❛ ❛ ❛♠♣❧✐❛r s❡✉ ❤♦r✐③♦♥t❡ s♦❜r❡ ❛ ♠❛t❡♠át✐❝❛✳

❈❛♣ít✉❧♦ ✶

❊s♣❛ç♦ ❱❡t♦r✐❛❧

✶✳✶ ❉❡✜♥✐çã♦

❆ ♣r✐♠❡✐r❛ ♥♦çã♦ q✉❡ ✈❛♠♦s ❝♦❧♦❝❛r ♥♦ t❡①t♦ s❡rá ❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ q✉❡ é ❛ ✐❞❡✐❛ ❜❛s❡ ❞❡ t♦❞❛ ❛ á❧❣❡❜r❛ ❧✐♥❡❛r✳ ❆tr❛✈és ❞❡st❡ ❝♦♥❝❡✐t♦✱ ✈ár✐♦s ❞♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ❡st✉❞❛❞♦s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳

❆♥t❡s ❞❡ ❞❡✜♥✐r ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝❛❜❡♠ ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s✳ ❚♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ❛♥t❡s ❞❡ t✉❞♦ ✉♠ ❝♦♥❥✉♥t♦✱ ❡ ❝♦♠♦ t❛❧✱ s❡rá ❞❡✜♥✐❞♦ ♣♦r ♣r♦♣r✐❡❞❛❞❡s✱ s❡✉s ❡❧❡♠❡♥t♦s s❡rã♦ t❛✐s q✉❡✱ ❣♦③❛♠ ❞❡st❛s ♣r♦♣r✐❡❞❛❞❡s✳ ◗✉❛♥❞♦ ♠❡♥❝✐♦♥❛r♠♦s ❛ ♣❛❧❛✈r❛ ❡s❝❛❧❛r ❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ ❛ ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ♦✉ s❡❥❛✱ ♥❡st❡ t❡①t♦ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s❡rã♦ r❡❛✐s✳

❯♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ✉♠ ❝♦♥❥✉♥t♦ E✱ q✉❡ s❡✉s ❡❧❡♠❡♥t♦s s❡rã♦ ❝❤❛♠❛❞♦s ✈❡t♦r❡s✱ ♦♥❞❡ ❡stã♦ ❞❡✜♥✐❞❛s ❞✉❛s ♦♣❡r❛çõ❡s✳ ❆ ♣r✐♠❡✐r❛ ♦♣❡r❛çã♦✱ ❝❤❛♠❛❞❛ ❛❞✐çã♦ ✱ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ❞♦✐s ✈❡t♦r❡s u ❡ v ❡♠ E ✉♠ ❡❧❡♠❡♥t♦ w ❡♠ E t❛❧ q✉❡ w =u+v✳ ❆ s❡❣✉♥❞❛ ♦♣❡r❛çã♦✱ ❝❤❛♠❛❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✱ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ❡s❝❛❧❛rα❡ ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ v ❡♠ E✱ ✉♠ ❡❧❡♠❡♥t♦ h ❡♠ E t❛❧ q✉❡ h=α_·v✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r ❡s❝❛❧❛r❡s α ❡β ❡ u✱v ❡w ❡♠ E ❡st❛s ♦♣❡r❛çõ❡s ❞❡✈❡♠ ❣♦③❛r ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ u+v =v+u✱

✷✳ (u+v) +w=u+ (v+w)✱

✸✳ (α_·β)_·u=α_·(β_·u)✱

✹✳ ❊①✐st❡ ✉♠ ✈❡t♦r ❡♠ E✱ ❝❤❛♠❛❞♦ ✈❡t♦r ♥✉❧♦✱ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ 0✱ q✉❡

s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✱ v + 0 =v ♣❛r❛ t♦❞♦v ❡♠ E✱

✺✳ P❛r❛ t♦❞♦ ✈❡t♦r v ❡♠ E✱ ❡①✐st❡ ✉♠ ✈❡t♦r ❞❡♥♦t❛❞♦ ♣♦r ₋v ❡♠ E✱ t❛❧ q✉❡ v + (₋v) = 0✱ ❡st❡ ✈❡t♦r(₋v) s❡rá ❝❤❛♠❛❞♦ ❞❡ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❞♦ ✈❡t♦r v✱

✻✳ (α+β)_·u=α_·u+β_·u✱

✼✳ α_·(u+v) = α_·u+α_·v✱

✽✳ 1_·v =v✱ ♣❛r❛ t♦❞♦ v ❡♠ E✳

❆♥t❡s ❞❡ ♠♦str❛r ❡①❡♠♣❧♦s ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ ❝❛❜❡ ❛q✉✐ ❞❡st❛❝❛r q✉❡✱ ❝❛❞❛ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t❡♠ s✉❛s ♦♣❡r❛çõ❡s ❞❡✜♥✐❞❛s ❞❡ ❛❧❣✉♠❛ ♠❛♥❡✐r❛✱ s❡♥❞♦ q✉❡✱ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♣♦❞❡♠ s❡r ❜❡♠ ❞✐❢❡r❡♥t❡s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ♦✉tr♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ P❛r❛ ❝❛❞❛ ❡s♣❛ç♦✱ ❞❡✈❡♠♦s ❝♦♥❤❡❝❡r s❡✉s ❡❧❡♠❡♥t♦s✱ s✉❛s ♦♣❡r❛çõ❡s ❡ ♣r♦✈❛r♠♦s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❆❣♦r❛ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❞♦✐s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ q✉❡ ❥á sã♦ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ❝♦♥❤❡✲ ❝✐❞♦s✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s r❡❛✐s ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✳

❊①❡♠♣❧♦ ✶✳ ◆♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❢♦✐ ✈✐st♦ q✉❡ ✉♠ ♣❧❛♥♦ ♣♦❞❡ s❡r s✐s✲ t❡♠❛t✐③❛❞♦✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ P ❞♦ ♣❧❛♥♦ ❛ss♦❝✐❛✲s❡ ✉♠ ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s (a, b)✱ ❡ ❛ ❝❛❞❛ ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s (a, b) ❛ss♦❝✐❛✲s❡ ✉♠ ♣♦♥t♦P✳ ❆❧é♠ ❞♦ ♠❛✐s ❡st❛ r❡❧❛çã♦ é ❜✐✉♥í✈♦❝❛✱ ♦✉ s❡❥❛✱ t♦❞♦ ♣♦♥t♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ✉♠ ú♥✐❝♦ ♣❛r ♦r❞❡✲ ♥❛❞♦✱ ❡ ❝❛❞❛ ♣❛r ♦r❞❡♥❛❞♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳ ❊st❛ r❡❧❛çã♦✱ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ▼❛t❡♠át✐❝❛✱ é ❛ ✐❞❡✐❛ ❜ás✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❛❧é♠ ❞♦ ♠❛✐s é ✉♠ ❜❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❙❡❥❛ E ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❉❛❞♦s u ❡ v ❡♠ E ✈❡t♦r❡s t❛✐s u= (a, b) ❡ v = (c, d) ❡ α ✉♠ ♥ú♠❡r♦ r❡❛❧✳ P♦❞❡♠♦s ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

• ❆❞✐çã♦✿ u+v = (a, b) + (c, d) = (a+c, b+d)✱

• ▼✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✿ α_·u=α_·(a, b) = (α_·a, α_·b)✳

▼♦str❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ❡st❛s ♦♣❡r❛çõ❡s✱ ❛ss✐♠ ❞❡✜♥✐❞❛s ♥❡st❡ ❝♦♥❥✉♥t♦ s❛t✐s❢❛③❡♠ ❛s ♦✐t♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡st❡ ♠♦❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

❊①❡♠♣❧♦ ✷✳ ❖✉tr♦ ❡①❡♠♣❧♦✱ ♥ã♦ ♠❡♥♦s r❡❧❡✈❛♥t❡✱ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s r❡❛✐s ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✱ ♦✉ s❡❥❛✱ ❛s ❢✉♥çõ❡s q✉❡ tê♠ ❝♦♠♦ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ❡st❛s ❢✉♥çõ❡s✱ sã♦ ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❉❡st❛ ❢♦r♠❛ s❡❥❛ F ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s r❡❛✐s ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✳ ❚♦♠❛♥❞♦f ❡g ❡♠ F ❞✉❛s ❢✉♥çõ❡s✱ ♦♥❞❡ f(x) ❡ g(x) sã♦ ♦s ✈❛❧♦r❡s ❛ss✉♠✐❞♦s ♣♦r ❡st❛s ❢✉♥çõ❡s ❡♠ ✉♠ ❛r❜✐trár✐♦ ✈❛❧♦r x ❞♦ ❞♦♠í♥✐♦✳ ❉❛❞♦ α ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞❡✜♥✐♠♦s ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

• ❆❞✐çã♦✿ ❆ ❛❞✐çã♦ ❞❡ f ❝♦♠ g✱ r❡s✉❧t❛ ♥❛ ❢✉♥çã♦ f +g t❛♠❜é♠ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✱ q✉❡ é ❞❡✜♥✐❞❛ ♣♦r (f +g)(x) = f(x) +g(x)✳

• ▼✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✿ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❡s❝❛❧❛r α ♣❡❧❛ ❢✉♥çã♦ f✱ ❞❡♥♦t❛❞❛ ♣♦r α_·f✱ é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✱ ❞❡✜♥✐❞❛ ♣♦r (α_·f)(x) = α_·f(x)✳

❚❛♠❜é♠ ♠♦str❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ❡st❛s ♦♣❡r❛çõ❡s✱ ❛ss✐♠ ❞❡✜♥✐❞❛s ♥♦ ❝♦♥❥✉♥t♦ F s❛t✐s❢❛③❡♠ ❛s ♦✐t♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡st❡ ♠♦❞♦ F é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❆❞✐❝✐♦♥❛✐s

❱❡r❡♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❛❞✐❝✐♦♥❛✐s ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s q✉❡ r❡s✉❧t❛♠ ❞❛ ❞❡✜♥✐çã♦✳

Pr♦♣♦s✐çã♦ ✶✳ ❙❡❥❛♠ u✱ v ❡ w ✈❡t♦r❡s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E✳ ❙❡ u+v = u+w ❡♥tã♦v =w✳

❉❡♠♦♥str❛çã♦✳ ❜❛st❛ ♥♦t❛r q✉❡✿

v = 0 +v

= (₋u+u) +v

= ₋u+ (u+v)

= ₋u+ (u+w)

= (₋u+u) +w

= 0 +w

= w .

Pr♦♣♦s✐çã♦ ✷✳ 0_·u= 0 ♣❛r❛ t♦❞♦ u ✈❡t♦r ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡u+ 0_·u= 1_·u+ 0_·u= (1 + 0)_·u= 1_·u=u=u+ 0✳ ▲♦❣♦

u+ 0_·u=u+ 0 ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❡❝♦rr❡ ❞❛ Pr♦♣♦s✐çã♦ ✶ q✉❡0_·u= 0✳

Pr♦♣♦s✐çã♦ ✸✳ ❙❡❥❛ α ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❞❛❞♦ 0 _∈ E✱ ♦♥❞❡ E é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ t❡♠♦s q✉❡ α_·0 = 0✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ α _·0 + α _· 0 = α _· (0 + 0) = α _· 0 = α _· 0 + 0✳ ▲♦❣♦✱

α_·0 +α_·0 = α_·0 + 0 ♣♦rt❛♥t♦✱ s❡❣✉❡✲s❡ ❞❛ Pr♦♣♦s✐çã♦ ✶ q✉❡ α_·0 = 0✳

Pr♦♣♦s✐çã♦ ✹✳ ❙❡❥❛♠ α ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ u ✉♠ ✈❡t♦r ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E✳ ❙❡ α_·u= 0 ❡♥tã♦ α = 0 ♦✉ u= 0✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ α ₆= 0 ❡ u ₆= 0✳ ❈♦♠♦ α ₆= 0 ❞❡❝♦rr❡ ♦

s❡❣✉✐♥t❡✿ α_·u= 0 _⇔ 1_α·α_·u= _α1 _·0_⇔u= 0✱ ❝♦♥tr❛❞✐çã♦✳

Pr♦♣♦s✐çã♦ ✺✳ ❙❡❥❛u ✉♠ ✈❡t♦r ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E✳ ❚❡♠♦s q✉❡ (₋1)_·u=₋u✳ ❉❡♠♦♥str❛çã♦✳ ❜❛st❛ ♥♦t❛r q✉❡u+ (₋1)_·u= 1_·u+ (₋1)_·u= (1 + (₋1))_·u= 0_·u= 0 = u+ (₋u)✳ ❆ss✐♠ u+ (₋1)_·u = u+ (₋u)✳ ❆❣♦r❛ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✶ q✉❡ (₋1)_·u=₋u✳

❈❛♣ít✉❧♦ ✷

▼❛tr✐③❡s

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ▼❛tr✐③

❆♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ ✉♠ ❝♦♥❝❡✐t♦ ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛✱ q✉❡ é ❛ ✐❞❡✐❛ ❞❡ ♠❛tr✐③✳ ❊st❛ ✐❞❡✐❛ s❡rá ♠✉✐t♦ út✐❧✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ q✉❛♥❞♦ ❡st✉❞❛r♠♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ♣♦✐s✱ ❛ ♠❡s♠❛ ❢❛❝✐❧✐t❛ ❜❛st❛♥t❡ ❛ ❢♦r♠❛ ❞❡ r❡s♦❧✉çã♦ ❞♦s s✐st❡♠❛s ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❛ ❡❧✐♠✐♥❛çã♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳

◆❡st❡ ❝♦♥t❡①t♦ ❞✐r❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ é ✉♠❛ t❛❜❡❧❛ ❞❡ ♥ú♠❡r♦s ♦r❣❛♥✐③❛❞❛ ❛tr❛✈és ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✳ ❆ss✐♠✱ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛

 

1 3 0

2 4 ₋2

 

é ✉♠❛ ♠❛tr✐③ ❞❡ ❞✉❛s ❧✐♥❤❛s ❡ três ❝♦❧✉♥❛s✳ ❖❜s❡r✈❡ q✉❡ ✉♠ ❡❧❡♠❡♥t♦ ✜❝❛ ❜❡♠ ❞❡t❡r♠✐♥❛❞♦ s❛❜❡♥❞♦ ❡♠ q✉❡ ❧✐♥❤❛ ❡ ❝♦❧✉♥❛ ♦ ♠❡s♠♦ s❡ ❡♥❝♦♥tr❛✳ ❉❡ ❢❛t♦✱ s❛❜❡♠♦s q✉❡ ♦ ❡❧❡♠❡♥t♦ ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❡ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ é ♦ ♥ú♠❡r♦ ✷✳ ◆❡st❡ t❡①t♦ t♦❞❛ ♠❛tr✐③ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❧❡tr❛ ♠❛✐ús❝✉❧❛ ❞❡ ♥♦ss♦ ❛❧❢❛❜❡t♦✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ A q✉❡ ♣♦ss✉✐ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ é ❞❡ ♦r❞❡♠ m_×n✳ ➱ ❝♦♠✉♠ ❞❡♥♦t❛r♠♦s ✉♠ ❡❧❡♠❡♥t♦ ❣❡♥ér✐❝♦ ❞❡ss❛ ♠❛tr✐③ A ❛tr❛✈és ❞♦ sí♠❜♦❧♦ aij✱ ♦♥❞❡ i ❡ j ❞❡♥♦t❛♠✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❧✐♥❤❛ ❡ ❛ ❝♦❧✉♥❛ ♥❛s q✉❛✐s s❡ ❡♥❝♦♥tr❛ ❡ss❡ ❡❧❡♠❡♥t♦✳ ◆❡ss❡ ❝❛s♦

1_≤i_≤m ❡1_≤j _≤n✳ ❉❡st❛ ❢♦r♠❛✱ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♥ér✐❝❛ ❞❡ss❛ ♠❛tr✐③A ♣♦❞❡

s❡r ❢❡✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

A=

       

a11 a12 . . . a1n

a₂₁ a₂₂ . . . a₂n

. . . .

am1 am2 . . . amn

       

P♦❞❡♠♦s ❛✐♥❞❛ r❡♣r❡s❡♥t❛r ❡st❛ ♠❡s♠❛ ♠❛tr✐③ A ❡s❝r❡✈❡♥❞♦ A = [a_ij]m×_n✱ ♦♥❞❡ 1_≤i_≤m ❡ 1_≤j _≤n✳

❆❧é♠ ❞♦ ♠❛✐s ❝❛❜❡ ❞❡st❛❝❛r q✉❡ ✉♠❛ ♠❛tr✐③A = [aij]m×_n✈❛✐ s❡r ✐❣✉❛❧ ❛ ✉♠❛ ♠❛tr✐③

B = [bij]k×_w q✉❛♥❞♦ m =k✱ n=w ❡a_ij =b_ij ♣❛r❛ t♦❞♦ i ❡ t♦❞♦ j✳

❊①❡♠♣❧♦ ✸✳ ❱❛♠♦s ❛❣♦r❛ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ x ❡ y ♣❛r❛ q✉❡ ❛s ♠❛tr✐③❡s ❛❜❛✐①♦ s❡❥❛♠ ✐❣✉❛✐s✳

A=

 

0 0

32 ₁₀

 

B =

 

logx 0

32 _y2

 

❱ê✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ♣❛r❛ A=B ❞❡✈❡♠♦s t❡r x= 1 ❡ q✉❡ y=_±√10✳

❊①❡♠♣❧♦ ✹✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ❛ ♠❛tr✐③ A = [aij]3×2 t❛❧ q✉❡ aij = −2i+j✳ ❇❛st❛

♥♦t❛r q✉❡ ❛ ♠❛tr✐③ t❡♠ 3 ❧✐♥❤❛s ❡ 2 ❝♦❧✉♥❛s ❡ ❝♦♠♦ aij =−2i+j✱ t❡♠♦s✿

• a₁₁=₋2_·1 + 1 =₋1✱

• a₁₂=₋2_·1 + 2 = 0✱

• a21=−2·2 + 1 =−3✱

• a₂₂=₋2_·2 + 2 =₋2✱

• a31=−2·3 + 1 =−5✱

• a₃₂=₋2_·3 + 2 =₋4✱

❙❡♥❞♦ ❛ss✐♠✱

A=

    

−1 0

−3 ₋2 −5 ₋4

    

✷✳✷ ▼❛tr✐③❡s ❊s♣❡❝✐❛✐s

❊①✐st❡♠ ❝❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ♠❛tr✐③❡s✱ q✉❡ ❣❛♥❤❛♠ ♥♦♠❡s ♣❛rt✐❝✉❧❛r❡s✱ ❞❡✈✐❞♦ ❛ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❱❡❥❛♠♦s✿

✷✳✷✳✶ ▼❛tr✐③ ▲✐♥❤❛

❈❤❛♠❛✲s❡ ♠❛tr✐③ ❧✐♥❤❛✱ t♦❞❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ ❧✐♥❤❛✱ ♦✉ s❡❥❛✱ t♦❞❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ 1_×n✳

❊①❡♠♣❧♦ ✺✳

h

1 3 7 15 31 . . . 2n

−1

i

◆♦t❡ q✉❡ ❡st❛ ♠❛tr✐③ ♣♦ss✉✐ 1 ❧✐♥❤❛ ❡ n ❝♦❧✉♥❛s✱

✷✳✷✳✷ ▼❛tr✐③ ❈♦❧✉♥❛

❈❤❛♠❛✲s❡ ♠❛tr✐③ ❝♦❧✉♥❛✱ t♦❞❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛✱ ♦✉ s❡❥❛✱ t♦❞❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠m_×1✳

❊①❡♠♣❧♦ ✻✳

          

1

3

7

. . .

2m₋₁

          

◆♦t❡ q✉❡ ❡st❛ ♠❛tr✐③ ♣♦ss✉✐ m ❧✐♥❤❛s ❡ 1 ❝♦❧✉♥❛✳

✷✳✷✳✸ ▼❛tr✐③ ◗✉❛❞r❛❞❛

❈❤❛♠❛✲s❡ ♠❛tr✐③ q✉❛❞r❛❞❛✱ t♦❞❛ ♠❛tr✐③ ♥❛ q✉❛❧ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s✱ ♦✉ s❡❥❛✱ s❡♠♣r❡ q✉❡ m=n✳

❊①❡♠♣❧♦ ✼✳

       

1 3 0 ₋7

5 1 4 5

6 9 1 0

7 12 0 1

       

◆♦t❡ q✉❡ ❡st❛ ♠❛tr✐③ ♣♦ss✉✐ 4 ❧✐♥❤❛s ❡ 4 ❝♦❧✉♥❛s✳

✷✳✷✳✹ ▼❛tr✐③ ◆✉❧❛

❯♠❛ ♠❛tr✐③ [aij]m×_n ♦♥❞❡ 1 ≤ i ≤ m ❡ 1 ≤ j ≤ n✱ é ❝❧❛ss✐✜❝❛❞❛ ❝♦♠♦ ♠❛tr✐③ ♥✉❧❛

q✉❛♥❞♦a_ij = 0✱ ♣❛r❛ t♦❞♦ i ❡ t♦❞♦ j✳ ❊①❡♠♣❧♦ ✽✳

       

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

       

✷✳✷✳✺ ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ❙✉♣❡r✐♦r

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ [a_ij]m×_m é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r q✉❛♥❞♦ t❡♠

❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❙❡i > j✱ ❡♥tã♦ aij = 0✳

❊①❡♠♣❧♦ ✾✳

    

1 2 3

0 4 5

0 0 6

    

✷✳✷✳✻ ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ■♥❢❡r✐♦r

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ [aij]m×_m é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ■♥❢❡r✐♦r q✉❛♥❞♦ t❡♠ ❛

s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❙❡i < j✱ ❡♥tã♦ a_ij = 0✳

❊①❡♠♣❧♦ ✶✵✳

    

1 0 0

2 3 0

4 5 6

    

✷✳✷✳✼ ▼❛tr✐③ ❉✐❛❣♦♥❛❧

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛[aij]m×_m é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ q✉❛♥❞♦ t❡♠ ❛ s❡❣✉✐♥t❡

♣r♦♣r✐❡❞❛❞❡✿ ❙❡i₆=j✱ ❡♥tã♦ a_ij = 0✳

❊①❡♠♣❧♦ ✶✶✳

    

1 0 0

0 2 0

0 0 3

    

✷✳✷✳✽ ▼❛tr✐③ ■❞❡♥t✐❞❛❞❡

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛[aij]m×_m é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ q✉❛♥❞♦ t❡♠ ❛s s❡❣✉✐♥✲

t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✐✮ s❡i₆=j ❡♥tã♦ a_ij = 0 ❀ ✭✐✐✮ s❡i=j ❡♥tã♦ a_ij = 1

❊①❡♠♣❧♦ ✶✷✳

       

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

       

✷✳✷✳✾ ▼❛tr✐③ ❙✐♠étr✐❝❛

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛[aij]m×m é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ s✐♠étr✐❝❛ q✉❛♥❞♦ t❡♠ ❛ s❡❣✉✐♥t❡

♣r♦♣r✐❡❞❛❞❡✿ aij =aji ♣❛r❛ t♦❞♦i ❡ j✳

❊①❡♠♣❧♦ ✶✸✳        

1 2 3 4

2 3 4 5

3 4 5 6

4 5 6 7

       

✷✳✸ ❖ ❊s♣❛ç♦ ❱❡t♦r✐❛❧ ❞❛s ▼❛tr✐③❡s

❙❡ t♦♠❛r♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡s q✉❡ ♣♦ss✉❡♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ ❞❡♥♦t❛❞♦ ❛q✉✐ ♣♦r M(m, n)✱ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦

♣♦r ❡s❝❛❧❛r✱ s❛t✐s❢❛③❡♥❞♦ t♦❞❛s ❛s ♦✐t♦ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡st❡ ♠♦❞♦ ❡st❡ ❝♦♥❥✉♥t♦ s❡ ❝❛r❛❝t❡r✐③❛ ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❱❡❥❛♠♦s ❛ ❛❞✐çã♦✿

❉❛❞❛ ❛s ♠❛tr✐③❡s A = [a_ij]m×_n ❡ B = [b_ij]_m×_n ❡♠ M(m, n)✱ ❞❡✜♥✐♠♦s ❛ ❛❞✐çã♦

❞❡st❛s ♠❛tr✐③❡s ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ A+B = C = [cij]m×_n ♦♥❞❡ c_ij = a_ij +b_ij✳ ◆♦t❡

q✉❡ C t❛♠❜é♠ é ✉♠❛ ♠❛tr✐③ ❝♦♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ ♦✉ s❡❥❛✱ C é ✉♠❛ ♠❛tr✐③ ❞❡ M(m, n)✳

❉❡ ♦✉tr♦ ♠♦❞♦✱ ❞❛❞❛s ❛s s❡❣✉✐♥t❡s ♠❛tr✐③❡s

A=        

a₁₁ a₁₂ . . . a₁n

a21 a22 . . . a2n

. . . .

am1 am2 . . . amn

       

, B =

       

b₁₁ b₁₂ . . . b₁n

b21 b22 . . . b2n

. . . .

bm1 bm2 . . . bmn

        t❡♠♦s q✉❡

A+B =

       

a₁₁+b₁₁ a₁₂+b₁₂ . . . a_1n+b_1n a21+b21 a22+b22 . . . a2n+b1n

. . . .

am1+bm1 am2+bm2 . . . amn+bmn

       

❊①❡♠♣❧♦ ✶✹✳ ❉❛❞❛s ❛s ♠❛tr✐③❡s

A=

 

1 ₋1 2 ₋2

3 ₋3 4 ₋4

 , B =

 

1 2 3 4

4 3 2 1

 

t❡♠♦s q✉❡

A+B =

 

1 + 1 ₋1 + 2 2 + 3 ₋2 + 4

3 + 4 ₋3 + 3 4 + 2 ₋4 + 1

 =

 

2 1 5 2

7 0 6 ₋3

 

❊st❛♥❞♦ ❞❡✜♥✐❞❛ ❛ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s✱ ✈❛♠♦s ❞❡✜♥✐r ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✳ ❉❛❞♦s ♦ ❡s❝❛❧❛r α ❡ ❛ ♠❛tr✐③A = [aij]m×_n ❡♠ M(m, n)✱ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦

♣♦r ❡s❝❛❧❛r ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ α_·A = D = [dij]m×n ♦♥❞❡ dij = α·aij✳ ◆♦t❡ q✉❡ D

t❛♠❜é♠ é ✉♠❛ ♠❛tr✐③ ❝♦♠m❧✐♥❤❛s ❡n ❝♦❧✉♥❛s✱ ♦✉ s❡❥❛✱D é ✉♠❛ ♠❛tr✐③ ❞❡M(m, n)✳

❉❡ ♦✉tr♦ ♠♦❞♦✱ ❞❛❞♦ ♦ ❡s❝❛❧❛r α ❡ ❛ ♠❛tr✐③

A=        

a₁₁ a₁₂ . . . a₁n

a₂₁ a₂₂ . . . a_2n . . . .

a_m1 a_m2 . . . a_mn

        t❡♠♦s q✉❡

α_·A=

       

α_·a₁₁ α_·a₁₂ . . . α_·a₁n

α_·a21 α·a22 . . . α·a2n

. . . .

α_·am1 α·am2 . . . α·amn

       

❊①❡♠♣❧♦ ✶✺✳ ❉❛❞❛ ❛ ♠❛tr✐③

A=

 

1 ₋1 2 ₋2

3 ₋3 4 ₋4

 

t❡♠♦s q✉❡

2_·A=

 

2_·1 2_·(₋1) 2_·2 2_·(₋2)

2_·3 2_·(₋3) 2_·4 2_·(₋4)

 =

 

2 ₋2 4 ₋4

6 ₋6 8 ₋8

 

❆❣♦r❛ q✉❡ ❡stã♦ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ♥♦ ❝♦♥❥✉♥t♦M(m, n)✱ ♣❛r❛ s❛❜❡r s❡ ❡st❡ ❝♦♥❥✉♥t♦ é r❡❛❧♠❡♥t❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❞❡✈❡♠♦s

✈❡r✐✜❝❛r t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ q✉❡ t❛♠❜é♠ sã♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❡ ♠❛tr✐③❡s✳

Pr♦♣♦s✐çã♦ ✻✳ ❖ ❝♦♥❥✉♥t♦ M(m, n) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ s❡❥❛ ❝❛r❛❝t❡r✐③❛❞♦ ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❞❡✈❡♠♦s ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✱ ❛❧é♠ ❞❡ ✈❡r✐✜❝❛r s❡ ❡st❛s ♦♣❡r❛çõ❡s ❣♦③❛♠ ❞❛s ♦✐t♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❈♦♠♦ ❥á ❞❡✜♥✐♠♦s ❛s ♦♣❡r❛çõ❡s ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ M(m, n)✱ ❜❛st❛ ✈❡r✐✜❝❛r ❛s ♣r♦♣r✐❡❞❛❞❡s✳

❉❛❞♦s ♦s ♥ú♠❡r♦s r❡❛✐sα ❡ β ❡ ❛s ♠❛tr✐③❡s A✱ B ❡ C ❞❡ M(m, n)✱ t❛✐s q✉❡ A = [aij]✱

B = [bij]❡ C = [cij] t❡♠♦s✿

✶✳ A+B =B+A✳ ❉❡ ❢❛t♦✱ A+B = [aij +bij] = [bij +aij] =B+A.

✷✳ (A +B) + C = A + (B +C)✳ ❉❡ ❢❛t♦✱ (A+ B) +C = [a_ij +b_ij] + [c_ij] = [(aij +bij) +cij] = [aij + (bij+cij)] = [aij] + [(bij+cij)] =A+ (B +C)✳

✸✳ (α_·β)_·A=α_·(β_·A)✳ ❉❡ ❢❛t♦✱(α_·β)_·A= [(α_·β)_·aij] = [α·(β·aij)] =α·(β·A)✳

✹✳ ❊①✐st❡ ✉♠❛ ♠❛tr✐③ ❡♠M(m, n)✱ ❝❤❛♠❛❞❛ ♠❛tr✐③ ♥✉❧❛✱ r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ sí♠❜♦❧♦ 0✱ q✉❡ s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡✱A+0 =A✱ ♣❛r❛ t♦❞♦A❡♠E✳ ❇❛st❛ t♦♠❛r0 = [0ij]

❡♠ M(m, n) t❛❧ q✉❡ 0ij = 0 ♣❛r❛ t♦❞♦i ❡ j✱ ♣♦✐s✱

A+ 0 = [aij+ 0ij] = [aij + 0] = [aij] =A✳

✺✳ P❛r❛ t♦❞❛ ♠❛tr✐③A❡♠M(m, n)✱ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ ❞❡♥♦t❛❞❛ ♣♦r₋A❡♠M(m, n)✱

t❛❧ q✉❡ A+ (₋A) = 0✳ ❊st❛ ♠❛tr✐③ (₋A) s❡rá ❝❤❛♠❛❞❛ ❞❡ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❞❛

♠❛tr✐③ A✱ ♦✉ s❡❥❛✱ ₋A= [₋aij]❡♠ M(m, n)✳ ❉❡ ❢❛t♦✱

A+ (₋A) = [aij + (−aij)] = [0] = [0ij] = 0✳

✻✳ (α+β)_·A=α_·A+β_·A✳ ❉❡ ❢❛t♦

(α+β)_·A= [(α+β)_·aij] = [α·aij +β·aij]

= [α_·a_ij] + [β_·a_ij] =α_·A+β_·A ✼✳ α_·(A+B) =α_·A+α_·B✳ ❉❡ ❢❛t♦✱

α_·(A+B) = [α_·(aij+bij)] = [α·aij +α·bij]

[α_·aij] + [α·bij] =α·A+α·B

✽✳ 1_·A=A✱ ♣❛r❛ t♦❞❛ ♠❛tr✐③ A ❡♠ M(m, n)✳ ❉❡ ❢❛t♦✱

1_·A= [1_·a_ij] = [a_ij] =A

❙❡♥❞♦ ❛ss✐♠✱ ♦ ❝♦♥❥✉♥t♦M(m, n) é r❡❛❧♠❡♥t❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

✷✳✹ ❖✉tr❛s ❖♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s

❱❡r❡♠♦s ❛❣♦r❛ ♦✉tr❛s ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s q✉❡ ✈ã♦ ❛❧é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

✷✳✹✳✶ ❚r❛♥s♣♦s✐çã♦

❆ tr❛♥s♣♦s✐çã♦ ❞❡ ♠❛tr✐③❡s é ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♠❛tr✐③ A = [aij]m×n

✉♠❛ ♠❛tr✐③ B = [bji]n×_m t❛❧ q✉❡ b_ji = a_ij✳ ❊st❛ ♠❛tr✐③ é ❞❡♥♦t❛❞❛ ♣❡❧♦ sí♠❜♦❧♦ At ❡

❝❤❛♠❛❞❛ ❞❡ tr❛♥s♣♦st❛ ❞❡A✱ ♦✉ s❡❥❛At_=B✳

❊①❡♠♣❧♦ ✶✻✳ ❉❛❞❛ ❛ ♠❛tr✐③

A=

 

−1 ₋2 ₋5

1 5 10

 

s✉❛ tr❛♥s♣♦st❛ é ❞❛❞❛ ♣♦r

At=

    

−1 1 −2 5 −5 10

    

❯♠ ❝❛s♦ ✐♥t❡r❡ss❛♥t❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ♠❛tr✐③ s✐♠étr✐❝❛✱ q✉❡ é ✐❣✉❛❧ ❛ s✉❛ tr❛♥s♣♦st❛✳ ❊①❡♠♣❧♦ ✶✼✳ ❉❛❞❛ ❛ ♠❛tr✐③

A=At =

    

1 2 3

2 4 5

3 5 6

    

Pr♦♣♦s✐çã♦ ✼✳ ❙❡❥❛ α ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ A ❡ B ♠❛tr✐③❡s t❛✐s q✉❡ A = [aij]m×n ❡

B = [bij]m×_n✳ ◆❡st❡ ❝♦♥t❡①t♦ sã♦ ✈❡r❞❛❞❡✐r❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

✶✳ (At₎t_=A✱

✷✳ (A+B)t_=At_+Bt_✱

✸✳ (α_·A)t _=α

·At_✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❜❛st❛ ♦❜s❡r✈❛r ❛s s❡❣✉✐♥t❡s ❥✉st✐✜❝❛t✐✈❛s✳ ✶✳ ❙❡❥❛At _{= [b}

ji]n×_mt❛❧ q✉❡b_ji =a_ij ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡A✳ ❙❡❥❛(A

t₎t_{= [c} ij]m×_n

t❛❧ q✉❡ cij = bji ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ At✳ ❈♦♠♦ cij = bji = aij✱ ♣♦❞❡♠♦s

❝♦♥❝❧✉✐r q✉❡ (At₎t_=A✳

✷✳ ❙❡❥❛ (A+B)t _{= [c}

ji]n×_m t❛❧ q✉❡ c_ji = a_ij +b_ij ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ A+B✳

❙❡❥❛♠ At _{= [d}

ji]n×_m t❛❧ q✉❡d_ji =a_ij ❡Bt = [e_ji]_n×_m t❛❧ q✉❡e_ji =b_ij ❛s ♠❛tr✐③❡s

tr❛♥s♣♦st❛s ❞❡ A ❡ B r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉✐ss♦ r❡s✉❧t❛ q✉❡

(A+B)t _{= [c}

ji] = [aij +bij] = [aij] + [bij]

[dji] + [eji] =At+Bt

✸✳ ❙❡❥❛ (α_·A)t _{= [c}

ji]n×_m t❛❧ q✉❡ c_ji = α·a_ij ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ α·A✳ ❙❡❥❛

At _{= [d}

ji]n×_m t❛❧ q✉❡ d_ji = a_ij ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ A✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛

✐❣✉❛❧❞❛❞❡ d_ji=a_ij ♣♦r α ♦❜t❡♠♦s q✉❡ α_·d_ji =α_·a_ij✳ ❉✐ss♦ r❡s✉❧t❛ q✉❡

(α_·A)t _{= [c}

ji] = [α·aij] = [α·dji]

=α_·[dji] =α·At

✷✳✹✳✷ ▼✉❧t✐♣❧✐❝❛çã♦

❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s é ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❞✉❛s ♠❛tr✐③❡s✱ ❝✉❥♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♣r✐♠❡✐r❛ é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛✱ ✉♠❛ ♥♦✈❛ ♠❛tr✐③A_·B q✉❡ ♣♦ss✉✐ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡A❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❡B✳ ❉❡ss❛ ❢♦r♠❛ s❡❥❛♠ A= [a_ik]m×_s ❡ B = [b_kj]_s×_n ❞✉❛s ♠❛tr✐③❡s✳ ❉❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛

♠❛tr✐③A ♣❡❧❛ ♠❛tr✐③ B✱ ❞❡♥♦t❛❞❛ ♣♦r A_·B✱ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ A_·B = [cij]m×_n ♦♥❞❡

c_ij =

n

P

k=1

a_ik·b_kj✳ ❈❛❜❡ ❛q✉✐ ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s✿

✶✳ ❆ ❡①♣r❡ssã♦ cij = n

P

k=1

aik·bkj ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ❧✐♥❤❛

i ❞❛ ♠❛tr✐③ A ♣❡❧❛ ❝♦❧✉♥❛ j ❞❛ ♠❛tr✐③ B✱ ♥♦t❡ q✉❡ ❡st❛✱ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s ❝❛❞❛ ❧✐♥❤❛ ❞❛ ♠❛tr✐③ A t❡♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦ ❞❡ ❝❛❞❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ B✱

✷✳ ◆♦t❡ q✉❡ A_·B é r❡❛❧♠❡♥t❡ ✉♠❛ ♠❛tr✐③✱ ❡ ❝❛❞❛ ❡❧❡♠❡♥t♦cij é ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛

❧✐♥❤❛ i❞❛ ♠❛tr✐③ A♣❡❧❛ ❝♦❧✉♥❛j ❞❛ ♠❛tr✐③ B✱ ❡ ♥❡ss❛ ♦r❞❡♠ ♣♦❞❡♠♦s ♦r❣❛♥✐③❛r ❡st❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ❡♠ ✉♠❛ t❛❜❡❧❛ ❞❡ ♦r❞❡♠m_×n✱ ♥♦t❡ ❛✐♥❞❛✱ q✉❡ ❝♦♠♦ t❡♠♦s m ❧✐♥❤❛s ♥❛ ♠❛tr✐③A ❡n ❝♦❧✉♥❛s ♥❛ ♠❛tr✐③B✱ ♣♦❞❡♠♦s ❢❛③❡r ❡st❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ m_·n ♠❛♥❡✐r❛s✳

❊①❡♠♣❧♦ ✶✽✳ ❉❛❞❛ ❛s ♠❛tr✐③❡s

A=

 

1 2 3

3 2 1

 

❡

B =

    

1 3

2 2

3 1

    

t❡♠♦s q✉❡

A_·B =

 

1_·1 + 2_·2 + 3_·3 1_·3 + 2_·2 + 3_·1

3_·1 + 2_·2 + 1_·3 3_·3 + 2_·2 + 1_·1

 

❙❡♥❞♦ ❛ss✐♠✱

A_·B =

 

14 10

10 14

 

❊st❛ ♦♣❡r❛çã♦ ❣♦③❛ ❞❡ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❡♥✉♥❝✐❛❞❛s ♥❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r✳

Pr♦♣♦s✐çã♦ ✽✳ ❙❡♥❞♦ ♣♦ssí✈❡✐s ❛s r❡❛❧✐③❛çõ❡s ❞❛s ♦♣❡r❛çõ❡s ❛❜❛✐①♦ sã♦ ✈❡r❞❛❞❡✐r❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

✶✳ A_·(B _·C) = (A_·B)_·C✱

✷✳ (A+B)_·C = (A_·C) + (B_·C)✱

✸✳ A_·(B +C) = (A_·B) + (A_·C)✱

✹✳ I1·A=A ❡ A·I2 =A✱ ♦♥❞❡I1 ❡ I2 sã♦ ♠❛tr✐③❡s ✐❞❡♥t✐❞❛❞❡s✱

✺✳ Bt

·At_{= (A}

·B)t_✱

✻✳ 01·A= 0 ❡ A·02 = 0✱ ♦♥❞❡ 01 ❡ 02 sã♦ ♠❛tr✐③❡s ♥✉❧❛s✱

✼✳ (α_·A)_·B =A_·(α_·B) =α_·(A_·B)✳

❉❡♠♦♥str❛çã♦✳ ✶✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡s A = [aik]m×r✱ B = [bkl]r×s ❡ C = [clj]s×n✳

❋❛③❡♥❞♦

A_·B = [dil]m×_s ♦♥❞❡ d_il =

r

P

k=1

aik·bkl

B_·C = [ekj]r×_n ♦♥❞❡ e_kj =

s

P

l=1

bkl·clj

t❡♠♦s q✉❡

(A_·B)_·C = [dil]m×_s·[c_lj]_s×_n = [

s

P

l=1

dilclj]m×_n= [

s P l=1 ( r P k=1

aik·bkl)clj]m×_n= [

r

P

k=1

aik( s

P

l=1

bklclj)]m×_n= [

r

P

k=1

aikekj]m×_n= [a_ik]_m×_r·[e_kj]_r×_n =A·(B ·C)

✷✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡sA = [aik]m×_r✱ B = [b_ik]_m×_r ❡C = [c_kj]_r×_n✳ t❡♠♦s q✉❡

(A+B)_·C = [aik+bik]m×_r·[c_kj]_r×_n = [

r

P

k=1

(aik+bik)ckj]m×_n = [

r

P

k=1

(aikckj +bikckj)]m×_n= [

r

P

k=1

aikckj+ r

P

k=1

bikckj]m×_n = [

r

P

k=1

aikckj]m×_n+ [

r

P

k=1

bikckj]m×_n = (A·C) + (B·C)

✸✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡sA = [aik]m×_r✱ B = [b_kj]_r×_n ❡C = [c_kj]_r×_n✳ t❡♠♦s q✉❡

A_·(B +C) = [aik]m×_r·[b_kj+c_kj]_r×_n= [

r

P

k=1

aik(bkj +ckj)]m×_n = [

r

P

k=1

(aikbkj +aikckj)]m×_n= [

r

P

k=1

aikbkj + r

P

k=1

aikckj]m×_n= [

r

P

k=1

aikbkj]m×_n+ [

r

P

k=1

aikckj]m×_n= (A·B) + (A·C)

✹✳ ❉❛❞❛ A = [aik]m×_n✳ P❛r❛ ❡st❡ ❝❛s♦ I1 s❡rá ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ m ❡ I2

s❡rá ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n✳

◆❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ I1 ♣♦r A ♦ ❡❧❡♠❡♥t♦ ❞❛ i✲és✐♠❛ ❧✐♥❤❛ ❡ j✲és✐♠❛ ❝♦❧✉♥❛ s❡rá

❢♦r♠❛❞♦ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ❧✐♥❤❛ i ❞❛ ♠❛tr✐③ I₁ ♣❡❧❛ ❝♦❧✉♥❛ j ❞❛ ♠❛tr✐③ A✱ só q✉❡ ❛ ❧✐♥❤❛ i ❞❡ I1 t♦❞♦s s❡✉s ❡❧❡♠❡♥t♦s sã♦ ✐❣✉❛✐s ❛ ③❡r♦✱ ❝♦♠ ❡①❝❡çã♦ ❞♦

i✲és✐♠♦ q✉❡ é ✐❣✉❛❧ ❛ ✉♠✱ ❞❡ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥❡st❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❧✐♥❤❛ ♣♦r ❝♦❧✉♥❛ ♦ ú♥✐❝♦ t❡r♠♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é 1_·aij = aij✳ ❙❡♥❞♦ ❛ss✐♠

I_1·A= [aij]m×r =A✳

◆❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ A ♣♦r I2 ♦ ❡❧❡♠❡♥t♦ ❞❛ i✲és✐♠❛ ❧✐♥❤❛ ❡ j✲és✐♠❛ ❝♦❧✉♥❛ s❡rá

❢♦r♠❛❞♦ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ❧✐♥❤❛ i ❞❛ ♠❛tr✐③ A ♣❡❧❛ ❝♦❧✉♥❛ j ❞❛ ♠❛tr✐③ I₂✱ só q✉❡ ❛ ❝♦❧✉♥❛ j ❞❡ I2 t♦❞♦s s❡✉s ❡❧❡♠❡♥t♦s sã♦ ✐❣✉❛✐s ❛ ③❡r♦✱ ❝♦♠ ❡①❝❡çã♦ ❞♦

j✲és✐♠♦ q✉❡ é ✐❣✉❛❧ ❛ ✉♠✱ ❞❡ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥❡st❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❧✐♥❤❛ ♣♦r ❝♦❧✉♥❛ ♦ ú♥✐❝♦ t❡r♠♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é aij ·1 = aij✳ ❙❡♥❞♦ ❛ss✐♠

A_·I₂ = [a_ij]m×_r =A✳

✺✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡sA = [aik]m×_r ❡ B = [b_kj]_r×_n✳ ❋❛③❡♥❞♦

A_·B = [eij]m×_n ♦♥❞❡ e_ij =

r

P

k=1

aikbkj

At _{= [c}

ki]r×m ♦♥❞❡ cki =aik

Bt_{= [d}

jk]n×_r ♦♥❞❡ d_jk =b_kj (A_·B)t_{= [f}

ji]n×m ♦♥❞❡ fji =eij =

r

P

k=1

aikbkj

t❡♠♦s q✉❡

Bt_·At_{= [}Pr

k=1

djkcki]n×m= [

r

P

k=1

bkjaik]n×m = [Pr

k=1

aikbkj]n×m = [eij]n×m = [fji]n×m = (A·B)

t

✻✳ ❉❛❞❛ A = [aik]m×_r✳ ❙❛❜❡♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r ♠❛tr✐③❡s ♥✉❧❛s 0₁ ❡ 0₂✱ s✉❛s

❧✐♥❤❛s ❡ ❝♦❧✉♥❛s tê♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ✐❣✉❛✐s ❛ ③❡r♦✱ ♣♦rt❛♥t♦✱ t♦❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❧✐♥❤❛ ♣♦r ❝♦❧✉♥❛✱ ♥❡st❡s ❝❛s♦s✱ t❡rã♦ ❝♦♠♦ r❡s♣♦st❛ ③❡r♦✳ ❙❡♥❞♦ ❛ss✐♠01·A= 0

❡ A_·02 = 0✳

✼✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡sA = [aik]m×_r ❡ B = [b_kj]_r×_n ❡α ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❋❛③❡♥❞♦

αA= [αaik]m×_r

t❡♠♦s q✉❡

(αA)_·B = [

r

P

K=1

(αaik)bkj]m×_n= [

r

P

K=1

aik(αbkj)]m×_n=A·(α·B)

(αA)_·B = [

r

P

K=1

(αaik)bkj]m×_n = [α

r

P

K=1

aikbkj]m×_n =α·(A·B)

❈❛♣ít✉❧♦ ✸

❱❛r✐❡❞❛❞❡ ❆✜♠

❆ ♥♦çã♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❛✜♠✱ ❢♦✐ ❝r✐❛❞❛ ♣❛r❛ ❝❧❛ss✐✜❝❛r ❝♦♥❥✉♥t♦s✱ q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♥ã♦ s❡ ❡♥q✉❛❞r❛♠ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♠❛s q✉❡ s❡ ❝♦♠♣♦rt❛♠ ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♦ ✈❛r✐❡❞❛❞❡ ❛✜♠ é ♠❡♥♦s ❢♦rt❡ q✉❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ss❡ ❝♦♥❥✉♥t♦ ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ t♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✱ ♠❛s ♥❡♠ t♦❞❛ ✈❛r✐❡❞❛❞❡ ❛✜♠ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❆ss✐♠ ❝♦♠♦ ❛ ✐❞❡✐❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❛ ✐❞❡✐❛ ❞❡ ❝❛r❛❝t❡r✐③❛r ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♦ ✈❛r✐❡❞❛❞❡ ❛✜♠ é r❡❧❡✈❛♥t❡✱ ♣♦✐s ❝♦♠♦ t❛❧✱ ♦ ❝♦♥❥✉♥t♦ ❣♦③❛rá ❞❡ t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✳ ❆♣❡s❛r ❞✐ss♦✱ ♥❡ss❡ t❡①t♦ ♥ã♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❡st❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♣♦✐s ♥♦ss♦ ♦❜❥❡t✐✈♦ é s✐♠♣❧❡s♠❡♥t❡ ❝❛r❛❝t❡r✐③❛r ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✳

✸✳✶ ❉❡✜♥✐çã♦

❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ V ❞❡ E é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✱ q✉❛♥❞♦ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦sx ❡ y ❡♠ V ❡ t ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❛❧q✉❡r✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿

tx+ (1₋t)y é ❡❧❡♠❡♥t♦ ❞❡ V

❊①❡♠♣❧♦ ✶✾✳ ❖ ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠✳ ❙❡❥❛ T ✉♠❛