algebra

❇❛s❡s ❖rt♦♥♦r♠❛✐s

✶ ❞❡ ❏✉❧❤♦ ❞❡ ✷✵✶✸

❈❛♣ít✉❧♦ ✶

❇❛s❡s ❖rt♦♥♦r♠❛✐s
❉❡✜♥✐çã♦ ✶✳✶✳ ❉✐③❡♠♦s q✉❡ w ∈ Rn é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s v1 , v2 , . . . , vm q✉❛♥❞♦ ❡①✐st❡♠ α1 , α2 , . . . , αm ∈ R t❛✐s q✉❡ m w = α1 v1 + α2 v2 + . . . + αm vm =

αi vi . i=1 ❉❡✜♥✐çã♦ ✶✳✷✳ ❉✐③❡♠♦s q✉❡

X ⊂ Rn é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ Rn

q✉❛♥❞♦ t♦❞♦ w ∈ Rn ♣♦❞❡ ❡①♣r✐♠✐r✲s❡ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r w = α1 v1 + α2 v2 + . . . + αm vm

❞❡ ✈❡t♦r❡s v1 , v2 , . . . , vm ∈ X.

❉❡✜♥✐çã♦ ✶✳✸✳ ❉✐③❡♠♦s q✉❡

X ⊂ Rn é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥✲
❞❡♥t❡ s❡ ❛ ú♥✐❝❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ♥✉❧❛ ❞❡ ✈❡t♦r❡s ❞❡ X é ❛q✉❡❧❛ ❝✉❥♦s

❝♦❡✜❝✐❡♥t❡s sã♦ t♦❞♦s ✐❣✉❛✐s ❛ ③❡r♦✱ ✐st♦ é✱ s❡

α1 v1 + α2 v2 + . . . + αm vm = 0,

❝♦♠ v1 , v2 , . . . , vm ∈ X ✱ ❡♥tã♦ α1 = α2 = . . . = αm = 0. ❈❛s♦ ❝♦♥trár✐♦✱
❞✐③❡♠♦s q✉❡ X é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ❙❡ v = α1 v1 +. . .+αm vm = β1 v1 +. . .+βm vm ❡ ♦s ✈❡t♦r❡s v1 , . . . , vm sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦ α1 = β1 , . . . , αm = βm ✱
♦✉ s❡❥❛✱ v ❡①♣r✐♠❡✲s❡ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s v1 , v2 , . . . , vm .
❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠❛ ❜❛s❡ ❞❡ Rn é ✉♠ ❝♦♥❥✉♥t♦ B ⊂ Rn ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡✲

♣❡♥❞❡♥t❡ ❞❡ ❣❡r❛❞♦r❡s ❞❡ Rn ✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❯♠ ❝♦♥❥✉♥t♦ X

⊂ Rn ❞✐③✲s❡ ♦rt♦❣♦♥❛❧ q✉❛♥❞♦ ❞♦✐s ✈❡t♦r❡s q✉❛✐sq✉❡r ❡♠ X sã♦ ♦rt♦❣♦♥❛✐s✳ ❙❡ ❛❧é♠ ❞✐ss♦✱ t♦❞♦s ♦s ✈❡t♦r❡s ❞❡ X sã♦
✉♥✐tár✐♦s✱ ❡♥tã♦ X ❝❤❛♠❛✲s❡ ✉♠ ❝♦♥❥✉♥t♦ ♦rt♦♥♦r♠❛❧✳

❉❡✜♥✐çã♦ ✶✳✻✳ ❯♠❛ ❜❛s❡ ❞❡

Rn ❞✐③✲s❡ ♦rt♦❣♦♥❛❧ ✭♦rt♦♥♦r♠❛❧✮ q✉❛♥❞♦ ❢♦r

✉♠ ❝♦♥❥✉♥t♦ ♦rt♦❣♦♥❛❧ ✭♦rt♦♥♦r♠❛❧✮✳

✶

✶✳✶

❊①❡r❝í❝✐♦s

{(1, 2)}

✶✳ ▼♦str❡ q✉❡

♥ã♦ ❣❡r❛

R2 ✳

{(1, 0), (0, 2), (3, 4)} é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡
❣❡r❛❞♦r❡s ❞❡ R2 ✳

✷✳ ▼♦str❡ q✉❡
❞❡

✸✳ ▼♦str❡ q✉❡

{(3, 4), (5, −6)}

✹✳ ▼♦str❡ q✉❡

{(3, −4), (−4, −3)}

✺✳ ▼♦str❡ q✉❡

{( 53 , − 45 ), (− 45 , − 35 )}

✻✳ ▼♦str❡ q✉❡ ♦s ✈❡t♦r❡s

{e1 , e2 }

♦rt♦♥♦r♠❛❧

❞❡

é ✉♠❛ ❜❛s❡ ❞❡

R2 ✳

é ✉♠❛ ❜❛s❡ ♦rt♦❣♦♥❛❧ ❞❡

é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡

R2 ✳

e1 =