Transformações Lineares
1656 palavras
7 páginas
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞♦ ❙❡♠✐ár✐❞♦✲❯❋❊❘❙❆❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ◆❛t✉r❛✐s
❈✉rs♦✿ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ❡ ❈♦♠♣✉t❛çã♦
❉✐s❝✐♣❧✐♥❛✿ ➪❧❣❡❜r❛ ▲✐♥❡❛r
❆❧✉♥♦✭❛✮✿
❚✉r♥♦✿
▼❛♥❤ã
❚❛r❞❡
◆♦✐t❡
❋♦✐ ♣❛r❛ ❡♥tr❡❣❛r ❞✐❛ ✶✾✴✶✵
✶✳ ❙❡ T : V → W é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✱ ♠♦str❡ q✉❡✿
✭❛✮ Ker(T ) é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ V ✳
✭❜✮ Im(T ) é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ W ✳
❙♦❧✉çã♦✿
✭❛✮ ❉❡ ❢❛t♦✱ s❡ T é ❧✐♥❡❛r✱ T (0) = 0✱ ♦✉ s❡❥❛ 0 ∈ Ket(T )✳ ❆❣♦r❛ s❡ u, v ∈ Ker(T )✱ ❡♥tã♦
T (u) = 0 = T (v)✳ ❆❣♦r❛✱
T (u + v)
=
T (u) + T (v) (♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛ T )
=
0 + 0 (♣♦✐s T (u) = 0 = T (v))
=
0.
❊✱ ♣♦rt❛♥t♦✱ T (u + v) = 0 ⇒ (u + v) ∈ Ker(T )✳ ❙❡❥❛♠ λ ∈ R ❡ u ∈ Ker(T )✳ ❊♥tã♦✱ T (u) = 0✳
❖r❛✱
T (λu)
=
λT (u) (♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛ T )
=
λ0 (♣♦✐s T (u) = 0)
=
0.
❊✱ ♣♦rt❛♥t♦✱ T (λu) = 0 ⇒ λu ∈ Ker(T )✳ ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ Ker(T ) é ✉♠ s✉❜❡s♣❛ç♦
✈❡t♦r✐❛❧ ❞❡ V ✳
✭❜✮ ❉❡ ❢❛t♦✱ s❡ T é ❧✐♥❡❛r✱ T (0) = 0✱ ♦✉ s❡❥❛ ❡①✐st❡ 0 ∈ V ♦♥❞❡ T (0) = 0✱ ❛ss✐♠ 0 ∈ Im(T )✳ ❙❡❥❛♠ w1 , w2 ∈ Im(T )✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ (w1 + w2 ) ∈ Im(T )✳ ■st♦ é✱ ❡①✐st❡ v ∈ V t❛❧ q✉❡ w1 + w2 = T (v)✳ ❖r❛✱ s❡ w1 , w2 ∈ Im(T )✱ ❡♥tã♦ ❡①✐st❡♠ ✈❡t♦r❡s v1 , v2 ∈ V t❛✐s q✉❡ w1 = T (v1 ) ❡ w2 = T (v2 ).
❆❣♦r❛✱ s♦♠❛♥❞♦✲s❡ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❡st❛s ❞✉❛s ❡q✉❛çõ❡s ✈❡t♦r✐❛✐s✱ ✈❡♠ w1 + w2
=
T (v1 ) + T (v2 )
=
T (v1 + v2 ) (♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛ T )
=
T (v) (❢❛③❡♥❞♦ v = (v1 + v2 ) ∈ V ).
■st♦ é✱ ❡①✐st❡ v ∈ V t❛❧ q✉❡ w1 + w2 = T (v)✱ ❜❛st❛ t♦♠❛r♠♦s v = v1 + v2 ∈ V ❡✱ ♣♦rt❛♥t♦✱
(w1 + w2 ) ∈ Im(T )✳ ❆❣♦r❛✱ s❡ λ ∈ R ❡ w ∈ Im(T )✱ ❡♥tã♦ q✉❡r❡♠♦s ♠♦str❛r q✉❡ λw ∈ Im(T )✳
■st♦ é✱ ❡①✐st❡ v ∈ V t❛❧ q✉❡ λw = T (v)✳ ❖r❛✱ s❡ w ∈ Im(T )✱ ❡♥tã♦ ❡①✐st❡ u ∈ V t❛❧ q✉❡ w = T (u)✳ ❆❣♦r❛✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡st❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ ♣♦r λ ❡ ✉s❛♥❞♦ ❛
❧✐♥❡❛r✐❞❛❞❡ ❞❡ T ✱ ✈❡♠ λw = λT (u) = T (λu) = T (v),
❢❛③❡♥❞♦ v = λu ∈ V ✳ ■st♦ é✱ ❡①✐st❡ v ∈ V t❛❧ q✉❡ λw = T (v)✱ ❜❛st❛ t♦♠❛r♠♦s v = λu ∈ V ❡✱
♣♦rt❛♥t♦✱ λw ∈ Im(T )✳ ❉❛í✱ ❝♦♥❝❧✉í♠♦s q✉❡ Im(T ) é ✉♠ s✉❜❡s♣❛ç♦