espacio vectorial ejercicios (2017)

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Universidad Universidad Complutense de Madrid (UCM)
Grado Matemáticas y Estadística - 1º curso
Asignatura Algebra Lineal
Año del apunte 2017
Páginas 26
Fecha de subida 08/07/2017
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ejercicios resueltos de espacios vectoriales

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Pr♦❜❧❡♠❛s r❡s✉❡❧t♦s ❞❡ ❊s♣❛❝✐♦s ❱❡❝t♦r✐❛❧❡s✿ ✶✳✲ P❛r❛ ❝❛❞❛ ✉♥♦ ❞❡ ❧♦s ❝♦♥❥✉♥t♦s ❞❡ ✈❡❝t♦r❡s q✉❡ s❡ ❞❛♥ ❛ ❝♦♥t✐♥✉❛❝✐ó♥ ❡st✉❞✐❛ s✐ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ♦ ❜❛s❡✿ ❛✮ {(2, 1, 1, 1) , (1, 1, 1, 1) , (3, 1, 1, 2) , (0, 1, 2, 1) , (2, −1, 1, −1)} ❡♥ R4 ❙♦❧✉❝✐ó♥✿ ❈♦♠♦ s♦♥ ✺ ✈❡❝t♦r❡s ❞❡ R4 ❝♦♥ t♦❞❛ s❡❣✉r✐❞❛❞ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡s✱ ♣✉❡s ❤❛② ♠ás ✈❡❝t♦r❡s q✉❡ ❝♦♠♣♦♥❡♥t❡s t✐❡♥❡ ❝❛❞❛ ✈❡❝t♦r✶ ✳ ❆❤♦r❛✱ ❡s❝r✐❜✐♠♦s ❧♦s ✈❡❝t♦r❡s ❝♦♠♦ ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② ❝❛❧❝✉❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③ A✿    A=         2 1 3 0 2 1 1 1 1 1 1 1 1 2 1 2 1 −1 1 −1     E1 ↔ E2   ∼       1 0 0 0 0  1 1 1 −1 −1 −1   −E2 −2 −2 −1   ∼ 1 2 1  −3 −1 −1  1 0 0 0 0             1 0 0 0 0      1 1 0 0 0 1 1 0 1 2 1 1 1 0 2     E3 ↔ E4   ∼       1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 2 1 2 3 0 2 1 1 1 1 1 1 1 1 2 1 2 1 −1 1 −1   E2 − 2E1  E3 − 3E1   ∼  E4 − 2E1  1 1 1 E3 + 2E2 1 1 1   E4 − E2 −2 −2 −1   ∼ 1 2 1  E5 + 3E2 −3 −1 −1  1 0 0 0 0    E5 − 2E4   ∼       1 1 0 0 0 1 1 1 0 2 1 1 0 1 2  1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0   E5 − 2E3   ∼     =U   ❈♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s U ❤❛② ✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s s❡ ❞❡❞✉❝❡ q✉❡ ❧♦s ✈❡❝t♦r❡s ♦r✐❣✐♥❛❧❡s s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡s✱ ❝♦♠♦ ②❛ ❞✐❥✐♠♦s✳ P♦r ♦tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡ ❡❧ s✉❜❡s♣❛❝✐♦ ❞❡ R4 ❣❡♥❡r❛❞♦ ♣♦r ❧❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ A ❝♦✐♥❝✐❞❡ ❝♦♥ ❡❧ s✉❜❡s♣❛❝✐♦ ❞❡ R4 ❣❡♥❡r❛❞♦ ♣♦r ❧❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ② ❛❞❡♠ás s❛❜❡♠♦s q✉❡ ❧❛s ✜❧❛s ❞✐st✐♥t❛s ❞❡ ❝❡r♦ ❞❡ ✉♥❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s s✐❡♠♣r❡ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ❉❡ ♠❛♥❡r❛ q✉❡✿ (2, 1, 1, 1) , (1, 1, 1, 1) , (3, 1, 1, 2) , (0, 1, 2, 1) , (2, −1, 1, −1) = ✶ ❖s r❡❝✉❡r❞♦ q✉❡ ❡❧ ❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ ❢❛♠✐❧✐❛ ❞❡ ✈❡❝t♦r❡s ❞❡ Kn Kn t✐❡♥❡ ❞✐♠❡♥s✐ó♥ ♥✱ ❞❡ ♠❛♥❡r❛ q✉❡ ❝✉❛❧q✉✐❡r ❢♦r♠❛❞❛ ♣♦r ♠ás ❞❡ ♥ ✈❡❝t♦r❡s ❡s ❧✐♥❡❛❧♠❛♥t❡ ❞❡♣❡♥❞✐❡♥t❡✳ ✶ = (1, 1, 1, 1) , (0, 1, 1, 1) , (0, 0, 1, 0) , (0, 0, 0, 1) ❈♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ❤❛② ❝✉❛tr♦ ✜❧❛s ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ ❧❛s ✜❧❛s ❞✐st✐♥t❛s ❞❡ ❝❡r♦✱ ❡st❛s ✜❧❛s t♦♠❛❞❛s ❝♦♠♦ ✈❡❝t♦r❡s ❞❡ R4 s♦♥ ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r✱ ❞❡ ❤❡❝❤♦ s♦♥ ✉♥❛ ❜❛s❡ ❞❡ R4 ✱ ❧✉❡❣♦ ❧♦s ✈❡❝t♦r❡s ♦r✐❣✐♥❛❧❡s s♦♥ ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ❞❡ R4 ✳ ❊♥ r❡s✉♠❡♥✱ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s ❞❡ R4 {(2, 1, 1, 1) , (1, 1, 1, 1) , (3, 1, 1, 2) , (0, 1, 2, 1) , (2, −1, 1, −1)} ❡s ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡ ② ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ❞❡ R4 ✳ ❈♦♠♦ ❡s ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡ ♥♦ ❡s ✉♥❛ ❜❛s❡ ❞❡ R4 ✱ ♣❡r♦ ❛❧ s❡r ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ❞❡ R4 ♣♦❞❡♠♦s ❡①tr❛❡r ❞❡ é❧ ❛❧❣✉♥❛ ❜❛s❡ ❡❧✐♠✐♥❛♥❞♦ ❛❧❣ú♥ ✈❡❝t♦r q✉❡ s❡❛ ❝♦♠❜✐♥❛❝✐ó♥ ❧✐♥❡❛❧ ❞❡ ❧♦s ❞❡♠❛s✳ P❛r❛ ❤❛❝❡r ❡st♦ ♦❜s❡r✈❡♠♦s ❧♦ s✐❣✉✐❡♥t❡✱ ②❛ ❤❡♠♦s ❞✐❝❤♦ q✉❡ ❧❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ A ② ❧❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ U ❣❡♥❡r❛♥ ❡❧ ♠✐s♠♦ s✉❜❡s♣❛❝✐♦ ❞❡ R4 ✳ ❆ ❧❛ ✈✐st❛ ❞❡ ❧❛ ♠❛tr✐③ U ✈❡♠♦s q✉❡ s✉s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s ❢♦r♠❛♥ ✉♥❛ ❜❛s❡ ❞❡ R4 ✱ ❡st♦ ♥♦s ♣♦❞rí❛ ❤❛❝❡r ♣❡♥s❛r q✉❡ ❧❛s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s ❞❡ A s♦♥ ✉♥❛ ❜❛s❡ ❞❡ R4 ✳ ❊♥ ❣❡♥❡r❛❧ ❡st♦ ♥♦ ❡s ❝✐❡rt♦ ♣✉❡s ❞✉r❛♥t❡ ❡❧ ♣r♦❝❡s♦ ❞❡ ❡s❝❛❧♦♥❛♠✐❡♥t♦ ❞❡ ❧❛ ♠❛tr✐③ A ♣♦❞rí❛♠♦s ❤❛❜❡r ♣❡r♠✉t❛❞♦ ❧❛ q✉✐♥t❛ ✜❧❛ ❞❡ A ❝♦♥ ❛❧❣✉♥❛ ❞❡ ❧❛s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s✱ ❞❡ ♠❛♥❡r❛ q✉❡ ❧❛s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s ❞❡ U ♥♦ s❡ ❝♦rr❡s♣♦♥❞❡rí❛♥ ❝♦♥ ❧❛s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s ❞❡ A✳ ❙✐♥ ❡♠❜❛r❣♦✱ ❡♥ ❡st❡ ❡❥❡♠♣❧♦ ❝♦♥❝r❡t♦✱ ❝♦♠♦ ❡♥ ❡❧ ♣r♦❝❡s♦ ❞❡ ❡s❝❛❧♦♥❛♠✐❡♥t♦ ❞❡ ❧❛ ♠❛tr✐③ A✱ ❧❛s ❞♦s ú♥✐❝❛s ♣❡r♠✉t❛❝✐♦♥❡s ❞❡ ✜❧❛s q✉❡ ❤❛ ❤❛❜✐❞♦ s♦♥✿ F ila1 ↔ F ila2 ② F ila3 ↔ F ila4 ✱ ❧❛s ❝✉❛tr♦ ♣r✐♠❡r❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ A ❢♦r♠❛♥ ✉♥❛ ❜❛s❡ ❞❡ R4 ✳ ▲✉❡❣♦✱ {(2, 1, 1, 1) , (1, 1, 1, 1) , (3, 1, 1, 2) , (0, 1, 2, 1)}✱ ❡s ✉♥❛ ❜❛s❡ ❞❡ R4 ✳ ❜✮ {(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1)} ❡♥ Z57 ❈♦♠♦ s♦♥ ❝✉❛tr♦ ✈❡❝t♦r❡s ❞❡ K 5 ✱ ❞♦♥❞❡ K = Z7 ✱ ♥♦ ♣✉❡❞❡♥ s❡r ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ❞❡ Z57 ✱ ❧✉❡❣♦ ♥♦ s♦♥ ✉♥❛ ❜❛s❡ ❞❡ Z57 ✳ P❛r❛ ✈❡r s✐ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s ❤❛❝❡♠♦s ✐❣✉❛❧ q✉❡ ❡♥ ❡❧ ❡❥❡r❝✐❝✐♦ ❛♥t❡r✐♦r✱ ❡s ❞❡❝✐r✱ ❡s❝r✐❜✐♠♦s ❧♦s ✈❡❝t♦r❡s ❝♦♠♦ ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② ❝❛❧❝✉❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③ A✿  1  2 A=  1 0  1  0   0 0 1 1 2 0 1 1 6 0 1 2 1 0 1 0 0 0 1 1 2 0 1 1 6 0  1 E2 + 5E1 2   ∼ 1  E3 − E1 1  1 0   E3 + E2 0  ∼ 1  1 1 1  0 1 0   0 0 0 0 0 0   1  0   0 0 1  0   0 0 1 1 0 0 ✷ 1 1 0 0   1 0   E2 ↔ E3 0  ∼ 1 1 6 1 0 1 0 0 0 1 6 1 0 1 0 0 0 1 1 0 0  1 0   E3 ↔ E4 0  ∼ 1 1 0  =U 1  0 ❈♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ❛♣❛r❡❝❡ ✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❧♦s ✈❡❝t♦r❡s ✜❧❛ ❞❡ ❧❛ ♠❛tr✐③ A ♥♦ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ▲✉❡❣♦✱ ❧♦s ✈❡❝✲ t♦r❡s {(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1)}✱ ♥♦ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s ♥✐ s♦♥ ✉♥ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ❞❡ Z57 ② ❡✈✐❞❡♥t❡♠❡♥t❡ t❛♠♣♦❝♦ s♦♥ ✉♥❛ ❜❛s❡ ❞❡ Z57 ✳ P♦r ♦tr♦ ❧❛❞♦✱ ❧♦ q✉❡ s✐ ♣♦❞❡♠♦s ❛s❡❣✉r❛r ❡s q✉❡ ❡❧ s✉❜❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ ❞❡ Z57 ❣❡♥❡r❛❞♦ ♣♦r ❧♦s ✈❡❝t♦r❡s✱ {(1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1)} ❡s ✐❣✉❛❧ ❛❧ s✉❜❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ ❞❡ Z57 ❣❡♥❡r❛❞♦ ♣♦r ❧♦s ✈❡❝t♦r❡s✱ {(1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1)} ❡s ❞❡❝✐r✱ (1, 1, 1, 1, 1) , (2, 1, 2, 1, 2) , (1, 2, 1, 2, 1) , (0, 0, 0, 0, 1) = (1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1) s✐❡♥❞♦ {(1, 1, 1, 1, 1) , (0, 1, 0, 1, 0) , (0, 0, 0, 0, 1)} ✉♥❛ ❜❛s❡ ❞❡ ❞✐❝❤♦ s✉❜❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧✳ ✷✳✲ P❛r❛ ❡❧ ❝♦♥❥✉♥t♦ S ❞❡ ✈❡❝t♦r❡s s✐❣✉✐❡♥t❡ s❡ ♣✐❞❡ ✈❡r q✉❡ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ ❛ñ❛❞✐r❧❡s ✈❡❝t♦r❡s ❤❛st❛ ❝♦♥✈❡rt✐r❧♦s ❡♥ ❜❛s❡ ② ❝❛❧❝✉❧❛r ❧❛s ❝♦✲ ♦r❞❡♥❛❞❛s ❞❡❧ ✈❡❝t♦r v ❡♥ ❞✐❝❤❛ ❜❛s❡✳ S = {(1, 2, 1, 1) , (1, 1, 0, 2)}✱ v = (1, 1, 3, 1)✱ ❡♥ Z45 ✳ ❙♦❧✉❝✐ó♥✿ ❱❡❛♠♦s✱ ❡♥ ♣r✐♠❡r ❧✉❣❛r q✉❡ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ P❛r❛ ❡❧❧♦✱ ❝♦♠♦ s✐❡♠♣r❡✱ ❡s❝r✐❜✐♠♦s ❧♦s ✈❡❝t♦r❡s ❝♦♠♦ ❧❛s ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② ❛ ❝♦♥t✐♥✉❛❝✐ó♥ ❤❛❧❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③✳ 1 1 2 1 1 0 1 2 E2 + 4E1 ∼ 1 0 2 4 1 4 1 1 ❝♦♠♦ ❛❧ ❡s❝❛❧♦♥❛r ❧❛ ♠❛tr✐③ ♥♦ ❛♣❛r❡❝❡ ♥✐❣✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s✱ ❧♦s ✈❡❝t♦r❡s s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ❆❤♦r❛ ❝♦♠♣❧❡t❛♠♦s ❧♦s ✈❡❝t♦r❡s ❤❛st❛ ❢♦r♠❛r ✉♥❛ ❜❛s❡ ❞❡ Z45 ✳ ❙❡❛ ♣✉❡s✱ B = {(1, 2, 1, 1) , (1, 1, 0, 2) , (0, 0, 1, 0) , (0, 0, 0, 1)} ✉♥❛ ❜❛s❡ ❞❡ Z45 ✳ P❛r❛ ❝❛❧❝✉❧❛r ❧❛s ❝♦♦r❞❡♥❛❞❛s ❞❡❧ ✈❡❝t♦r v ❡♥ ❧❛ ❜❛s❡ B ❡st❛❜❧❡❝❡♠♦s ❧❛ ❝♦♠❜✐♥❛❝✐ó♥ ❧✐♥❡❛❧ ✈❡❝t♦r✐❛❧✱ (1, 1, 3, 1) = x1 · (1, 2, 1, 1) + x2 · (1, 1, 0, 2) + x3 · (0, 0, 1, 0) + x4 · (0, 0, 0, 1) q✉❡ ❞❛ ❧✉❣❛r ❛❧ s✐❣✉✐❡♥t❡ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s✿ x1 2x1 x1 x1 + + x2 x2 + 2x2 + x3 + ✸ + x4 = = = = 1 1 3 1        ❈♦♠♦ B ❡s ✉♥❛ ❜❛s❡ ❞❡ Z45 ② ❞❛❞♦ q✉❡ ❧❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♥ ✈❡❝t♦r r❡s♣❡❝t♦ ❞❡ ✉♥❛ ❜❛s❡ s♦♥ ú♥✐❝❛s✱ ❡❧ s✐st❡♠❛ ❧✐♥❡❛❧ ❛♥t❡r✐♦r ❡s ❝♦♠♣❛t✐❜❧❡ ❞❡t❡r♠✐♥❛❞♦✱ ❡s ❞❡❝✐r✱ t✐❡♥❡ s♦❧✉❝✐ó♥ ú♥✐❝❛✳ P❛r❛ ❤❛❧❧❛r ❡st❛ s♦❧✉❝✐ó♥ ❡s❝r✐❜✐♠♦s ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛ ② ❝❛❧❝✉❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳   1   2    1  1 0 1 0 0 1 1 2 0 0   1   E2 + 3E1 0 1   E3 + 4E1  ∼ 0 3   E4 + 4E1 1 1   1 1 0 E3 − E2   0 4 0  ∼  E4 + E2   0 0 1  1   0    0  1 0 0 4 0 0 4 1 0 0 1 0 1  ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳  1   4    2   0 ✳✳ ✳ 1   ✳ 0 ✳✳ 4    ✳✳ 0 ✳ 3   ✳✳ 0 0 0 1 ✳ 4 s✐ s❡❣✉✐♠♦s ❤❛st❛ ❡♥❝♦♥tr❛r ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ r❡❞✉❝✐❞❛ ❞✐s♣♦♥❞r❡♠♦s ❞❡ ❧♦s ✈❛❧♦r❡s ❞❡ ❧❛s ✐♥❝ó❣♥✐t❛s ❞✐r❡❝t❛♠❡♥t❡✱ ❞❡ ♠❛♥❡r❛ q✉❡ ❛sí ❧♦ ❤❛❝❡♠♦s✱  E1 + E2 ∼ ✳✳ ✳ ✳ 4 0 0 ✳✳ ✳ 0 1 0 ✳✳ ✳ 0 0 0 1 ✳✳  1   0    0  0 0 0  0   4   4E2  ∼ 3   0 4  ✳✳ ✳ ✳ 1 0 0 ✳✳ ✳ 0 1 0 ✳✳ ✳ 0 0 0 1 ✳✳  1   0    0  0 0 0  0   1    3   4 ❧✉❡❣♦ ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s t✐❡♥❡ ♣♦r s♦❧✉❝✐ó♥✿ x1 x2 x3 x4 =0 =1 =3 =4        ❞❡ ♠❛♥❡r❛ q✉❡ ♣♦❞❡♠♦s ❞❡❝✐r q✉❡ ❧❛s ❝♦♦r❞❡♥❛❞❛s ❞❡❧ ✈❡❝t♦r v = (1, 1, 3, 1) r❡s♣❡❝t♦ ❞❡ ❧❛ ❜❛s❡ B s♦♥ v = (0, 1, 3, 4)B ✳ ❖❜s❡r✈❛❝✐ó♥ ✿ ❊❧ ♣r♦❜❧❡♠❛ ②❛ ❤❛ ❝♦♥❝❧✉✐❞♦✱ ♣❡r♦ ❛♥t❡s ❞❡ ♣❛s❛r ❛ ♦tr♦ ✈❛♠♦s ❛ ❤❛❝❡r ❧❛ s✐❣✉✐❡♥t❡ ♦❜s❡r✈❛❝✐ó♥ ❣❡♥❡r❛❧✿ ▲❧❛♠❡♠♦s Mm×n (K) ❛❧ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❧❛s ♠❛tr✐❝❡s ❞❡ ♠ ✜❧❛s ② ♥ ❝♦❧✉♠♥❛s ❝♦♥ t♦❞❛s s✉s ❡♥tr❛❞❛s ❡♥ ❡❧ ❝✉❡r♣♦ K ✳ ❙❡❛♥ A ∈ Mm×n (K) ② ¯b ∈ Mn×1 (K) ❧❛s s✐❣✉✐❡♥t❡s ♠❛tr✐❝❡s✿  a11  a21  A= ✳  ✳✳ am1 a12 a22 ··· ··· am2 ··· ··· ✳✳ ✳ ✹   a1n b1  b2 a2n    ✳✳  ; ¯b =  ✳✳   ✳ ✳ amn bn      ✈❡❛♠♦s q✉❡ ✈❛❧❡ ❡❧ ♣r♦❞✉❝t♦ ♠❛tr✐❝✐❛❧ A · ¯b✳ ❙❡❣ú♥ ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡❧ ♣r♦✲ ❞✉❝t♦ ♠❛tr✐❝✐❛❧ s❛❜❡♠♦s q✉❡ A · ¯b ∈ Mm×1 (K)✱ s❡❛ ♣✉❡s✱    b1 a11 a12 · · · a1n  a21 a22 · · · a2n   b2      A · ¯b =  ✳ ✳✳ ✳✳  ·  ✳✳  =   ✳✳  ✳  ✳ ··· ✳ bn am1 am2 · · · amn   a11 b1 + a12 b12 + . . . + a1n bn  a21 b1 + a22 b2 + . . . + a2n bn    =  ✳✳   ✳ am1 b1 + am2 b2 + . . . + amn bn       a1n a12 a11  a2n   a22   a21        =  ✳  b1 +  ✳  b2 + . . . +  ✳  bn ✳ ✳ ✳  ✳   ✳   ✳  amn am2 am1  =a ¯ 1 b1 + a ¯ 2 b2 + . . . + a ¯n bn ❞♦♥❞❡   a1j  a2j    a ¯j =  ✳  ; 1 ≤ j ≤ n  ✳✳  amj s♦♥ ❧❛s ❝♦❧✉♠♥❛s ❞❡ ❧❛ ♠❛tr③ A✳ ❉❡ ♠❛♥❡r❛ q✉❡✱ ♠✉❧t✐♣❧✐❝❛r A · ¯b ✭❡♥ ❡❧ ♦r❞❡♥ ✐♥❞✐❝❛❞♦✮✱ ❞♦♥❞❡ A ❡s ✉♥❛ ♠❛tr✐③ ❞❡ ♦r❞❡♥ m × n ② ¯b ❡s ✉♥❛ ♠❛tr✐③ ❝♦❧✉♠♥❛ ❞❡ n ❡♥tr❛❞❛s ✭❡s ❞❡❝✐r✱ ❞❡ ♦r❞❡♥ n × 1✮ ❞❛ ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♥❛ ❝♦♠❜✐♥❛❝✐ó♥ ❧✐♥❡❛❧ ❞❡ ❧❛s ❝♦❧✉♠♥❛s ❞❡ ❆ ❝♦♥ ❝♦❡✜❝✐❡♥t❡s ✭♦ ♣❡s♦s✮ ❧❛s ❡♥tr❛❞❛s ❞❡ ❧❛ ❝♦❧✉♠♥❛ ¯b✳ ❈♦♠♦ ❝♦♥s❡❝✉❡♥❝✐❛ ❞❡ ❧♦ ❞✐❝❤♦✱ ✈❡♠♦s q✉❡ r❡s♦❧✈❡r ✉♥ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s s❡ ♣✉❡❞❡ ✐♥t❡r♣r❡t❛r ❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦✿ ❉❛❞♦ ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s✱ ❝♦♥ ♠ ❡❝✉❛❝✐♦♥❡s ② ♥ ✐♥❝ó❣♥✐t❛s✿ a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2      am1 x1 + am2 x2 + . . . + amn xn = bm     ✳✳ ✳ s✐ ❧❧❛♠❛♠♦s  a11  a21  A= ✳  ✳✳ am1 a12 a22 ··· ··· am2 ··· ··· ✳✳ ✳   a1n x1  x2 a2n    ✳✳  ; X =  ✳✳   ✳ ✳ amn xn ✺    ;    b1  b2   ¯b =   ✳✳   ✳  bm ❡❧ s✐st❡♠❛ s❡ tr❛♥s❢♦r♠❛ ❡♥ ❧❛ ❡❝✉❛❝✐ó♥ ♠❛tr✐❝✐❛❧ A · X = ¯b ② s✐ ❛❤♦r❛ ✐♥t❡r♣r❡t❛♠♦s ❡❧ ♣r♦❞✉❝t♦ A · X ❝♦♠♦ ❝♦♠❜✐♥❛❝✐ó♥ ❞❡ ❧❛s ❝♦❧✉♠♥❛s ❞❡ A✱ ② ❞❡♥♦t❛♠♦s ❛ ❧❛s ❝♦❧✉♠♥❛s ❞❡ A ❝♦♠♦ a¯1 , a¯2 , . . . , a¯n ✱ ❡❧ s✐st❡♠❛ s❡ tr❛♥s❢♦r♠❛ ❡♥ ❧❛ s✐❣✉✐❡♥t❡ ❡❝✉❛❝✐ó♥ ✈❡❝t♦r✐❛❧ a ¯ 1 x1 + a ¯ 2 x2 + . . . + a ¯n xn = ¯b P♦r t♦❞♦ ❧♦ ❞✐❝❤♦✱ ❝✉❛♥❞♦ s❡ ♥♦s ♣✐❞❛ q✉❡ ❛✈❡r✐❣ü❡♠♦s ❝ó♠♦ s❡ ❡s❝r✐❜❡ ✉♥ ✈❡❝t♦r u = (b1 , b2 , . . . , bm ) ∈ K m ✱ ❝♦♠♦ ❝♦♠❜✐♥❛❝✐ó♥ ❧✐♥❡❛❧ ❞❡ ❧♦s ✈❡❝t♦r❡s v1 , v2 , . . . , vh ∈ K m ✱ ❧♦ q✉❡ ❤❛r❡♠♦s ❡s ❡s❝r✐❜✐r ✉♥❛ ♠❛tr✐③ A ♣♦♥✐❡♥❞♦ ❛ ❧♦s ✈❡❝t♦r❡s vj ❝♦♠♦ s✉s ❝♦❧✉♠♥❛s ② ❛ ❝♦♥t✐♥✉❛❝✐ó♥ ♣❧❛♥t❡❛r❡♠♦s ❡❧ s✐❣✉✐❡♥t❡ s✐s✲ t❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s✱ ❡s❝r✐t♦ ❡♥ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✿ ✳✳  ✳   v1  ✳✳ ✳ ✳✳ ✳ v2 ✳✳ ✳  x1 ✳✳ ✳   x2      vh  ✳✳  =    ✳   ✳✳ ✳ xh  ··· ··· ···    b1 b2   ✳✳  ✳  bm ❙✐ ❡st❡ s✐st❡♠❛ ❡s ✐♥❝♦♠♣❛t✐❜❧❡ ✐♥❞✐❝❛rí❛ q✉❡ u ∈ / v1 , v2 , . . . , vh ✳ ❙✐ ♣♦r ❡❧ ❝♦♥tr❛r✐♦ ❡s ❝♦♠♣❛t✐❜❧❡✱ ❧♦s ✈❛❧♦r❡s ❞❡ ❧❛s ✐♥❝ó❣♥✐t❛s ♥♦s ❞❛rí❛♥ ❧❛ ❝♦♠❜✐♥❛❝✐ó♥ ❧✐♥❡❛❧ ❜✉s❝❛❞❛✳ ✸✳✲ ❊♥❝✉❡♥tr❛ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡❧ s✐❣✉✐❡♥t❡ s✉❜❡s♣❛❝✐♦ ② ❝♦♠♣r✉❡❜❛ s✐ ❡❧ ✈❡❝t♦r ❞❛❞♦ ❡stá ♦ ♥♦ ❡♥ ❡❧ s✉❜❡s♣❛❝✐♦✳ (1, 2, 1, 1) , (1, 1, 1, 2) ❡♥ R4 ✱ v = (1, 0, 1, 0)✳ ❙♦❧✉❝✐ó♥✿ ❙❡❛ (x, y, z, t) ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ R4 ✱ ❡♥t♦♥❝❡s s❛❜❡♠♦s q✉❡ (x, y, z, t) ∈ (1, 2, 1, 1) , (1, 1, 1, 2) ⇐⇒ ❡①✐st❡♥ ❡s❝❛❧❛r❡s x1 , x2 ∈ R t❛❧❡s q✉❡ (x, y, z, t) = x1 · (1, 2, 1, 1) + x2 · (1, 1, 1, 2) ⇐⇒  x1 + x2 = x    2x1 + x2 = y ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s ❡s ❝♦♠♣❛t✐❜❧❡ ⇐⇒ x1 + x2 = z    x1 + 2x2 = t   ✳✳ 1 1 ✳ x     ✳  2 1 ✳✳ y   ❛ ✉♥❛ ❝✉❛♥❞♦ tr❛♥s❢♦r♠❡♠♦s ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛     1 1 ✳✳✳ z    1 ✻ 2 ✳✳ ✳ t ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛✱ ♦❝✉rr❛ q✉❡ ❡♥ ❞✐❝❤❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♥♦ ♥♦s q✉❡❞❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❝♦♠♦ ❝♦❧✉♠♥❛ ♣✐✈♦t❡✳   1   2    1  ✳ 1 ✳✳ ✳ 1 ✳✳ ✳ 1 ✳✳ ✳ 1 2 ✳✳  x   E2 − 2E1 y   E3 − E1  ∼ z   E4 − E1 t ✳ 1 ✳✳   1   0    0   1   0    0   x ✳ −1 ✳✳ y − 2x ✳ 0 ✳✳ z − x ✳ 0 1 ✳✳ t − x      E3 + E2   ∼    1   0    0  x ✳ −1 ✳✳ y − 2x ✳ 1 ✳✳ t − x ✳ 0 0 ✳✳ z − x ✳ 1 ✳✳  ✳ 1 ✳✳    E3 ↔ E4   ∼    x ✳ −1 ✳✳ y − 2x ✳✳ 0 ✳ −3x + y + t ✳ 0 0 ✳✳ z−x        ② ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ♣❛r❛ q✉❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ ♥♦ s❡❛ ❝♦❧✉♠♥❛ ♣✐✈♦t❡ ❤❛ ❞❡ ♦❝✉rr✐r q✉❡ −3x + y + t = 0 ② t❛♠❜✐é♥ q✉❡ z − x = 0 (⇔ x−z = 0)✳ ▲✉❡❣♦ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧✐❝✐t❛s ❞❡❧ s✉❜❡s♣❛❝✐♦ (1, 2, 1, 1) , (1, 1, 1, 2) −3x + y + t = 0 x−z =0 s♦♥ (x, y, z, t) ∈ R4 / (1, 2, 1, 1) , (1, 1, 1, 2) = ✳ ❊s ❞❡❝✐r✱ (x, y, z, t) ∈ R4 / −3x + y + t = 0 x−z =0 ❙❡❛ ❛❤♦r❛ v = (1, 0, 1, 0) ∈ R4 ✱ ♣❛r❛ ✈❡r s✐ v ∈ (1, 2, 1, 1) , (1, 1, 1, 2) só❧♦ ❤❡♠♦s ❞❡ ❝♦♠♣r♦❜❛r s✐ ❝✉♠♣❧❡ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧✐❝✐t❛s✱ ❡♥t♦♥❝❡s ❝♦♠♦ −3 + 0 + 0 = 0 1−1=0 ⇒v∈ / (1, 2, 1, 1) , (1, 1, 1, 2) ✹✳✲ ❈❛❧❝✉❧❛ ✉♥❛ ❜❛s❡ ② ❧❛ ❞✐♠❡♥s✐ó♥ ❞❡ ❧♦s s✐❣✉✐❡♥t❡s s✉❜❡s♣❛❝✐♦s✿ ❛✮ (x, y, z, t) / x−y+z−t=0 2x + z + t = 0 ❡♥ R4 ✳ s♦❧✉❝✐ó♥✿ ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r ❡❧ s✉❜❡s♣❛❝✐♦ ❡s ❡❧ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉❝✐♦♥❡s ❞❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❤♦♠♦❣é♥❡♦✿ x 2x − y + z + z − + t = t = 0 0 ❝✉②❛s s♦❧✉❝✐♦♥❡s s♦♥✿   1 2 ✳✳ ✳ ✳ 0  E2 − 2E1  1 −1 1 −1 ✳✳ 0  ✳ ✳ ∼ 1 ✳✳ 0 0 2 −1 3 ✳✳ 0  −1 1 0 1  −1 ✼  ❛ ❧❛ ✈✐st❛ ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛✱ ✈❡♠♦s q✉❡ ❡❧ s✐st❡♠❛ ❡s ❝♦♠♣❛t✐❜❧❡ ✐♥❞❡t❡r✲ ♠✐♥❛❞♦✱ s✐❡♥❞♦ ❡q✉✐✈❛❧❡♥t❡ ❛❧ s✐❣✉✐❡♥t❡ s✐st❡♠❛ ❤♦♠♦❣é♥❡♦✿ x − y 2y + z − z − + t = 3t = 0 0 ↔ x − y 2y = −z + t = z − 3t ❞♦♥❞❡ ♣♦❞❡♠♦s ✈❡r q✉❡ ❧❛s ✐♥❝ó❣♥✐t❛s x ❡ y s♦♥ ❧❛s ✐♥❝ó❣♥✐t❛s ♣r✐♥❝✐♣❛❧❡s ✭t❛♠✲ ❜✐é♥ ❧❧❛♠❛❞❛s ❜ás✐❝❛s✮ ② ❧❛s ✐♥❝ó❣♥✐t❛s z ② t s♦♥ ✐♥❝ó❣♥✐t❛s ❧✐❜r❡s✳ ❉❡s♣❡❥❛♥❞♦ y ❞❡ ❧❛ s❡❣✉♥❞❛ ❡❝✉❛❝✐ó♥ ② s✉st✐t✉②❡♥❞♦ s✉ ✈❛❧♦r ❡♥ ❧❛ ♣r✐♠❡r❛ ❡❝✉❛❝✐ó♥ ♦❜t❡♥❡♠♦s ❧❛s s♦❧✉❝✐♦♥❡s ✿  1 1 x = y = z ∈ t ∈ −2z − 2t 3 1 2z − 2t R ✭❧✐❜r❡✮ R ✭❧✐❜r❡✮       ▲✉❡❣♦ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉❝✐♦♥❡s ❡s✿ (x, y, z, t) ∈ R4 / (x, y, z, t) = 1 1 3 1 − z − t, z − t, z, t / z, t ∈ R 2 2 2 2 = = 1 1 1 3 z − , , 1, 0 + t − , − , 0, 1 / z, t ∈ R 2 2 2 2 = ❆sí✱ (x, y, z, t) / 1 1 3 1 − z − t, z − t, z, t ; z, t ∈ R 2 2 2 2 1 1 1 3 − , , 1, 0 , − , − , 0, 1 2 2 2 2 x−y+z−t=0 2x + z + t = 0 = − 21 , 12 , 1, 0 , − 12 , − 32 , 0, 1 ✱ ❞❡ ♠❛♥✲ ❡r❛ q✉❡ ❡❧ s✉❜❡s♣❛❝✐♦ t✐❡♥❡ ❞✐♠❡♥s✐ó♥ ✷✱ s✐❡♥❞♦ ❜❛s❡ ❞❡ ❞✐❝❤♦ s✉❜❡s♣❛❝✐♦✳ − 12 , 12 , 1, 0 , − 12 , − 32 , 0, 1 ✉♥❛ ❜✮ ❈❛❧❝✉❧❛ ❜❛s❡ ② ❞✐♠❡♥s✐ó♥ ❞❡ (1, 2, 1, 1), (1, 1, 1, 2), (3, 2, 3, 2) ❡♥ Z45 ✳ s♦❧✉❝✐ó♥✿ ❱❡❛♠♦s s✐ ❧♦s ✈❡❝t♦r❡s q✉❡ ❣❡♥❡r❛♥ ❡❧ s✉❜❡s♣❛❝✐♦ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s ♦ ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡s✳ P❛r❛ ❡❧❧♦✱ ❡s❝r✐❜✐♠♦s ❧♦s ✈❡❝t♦r❡s ❝♦♠♦ ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② ❝❛❧❝✉❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③ A✿  1 A= 1 3 2 1 2 1 1 3   E2 + 4E1 1 2 1 1  0 4 0 2  ∼ 2 E3 + 2E1 0 1 0   1 2 1 1  0 4 0 1 =U 0 0 0 0  1 E + E2 1  3 ∼ 4 ❈♦♠♦ ❧❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ A ② ❧❛s ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ❣❡♥❡r❛♥ ❡❧ ♠✐s♠♦ s✉❜❡s♣❛❝✐♦ ❞❡ Z45 ✱ ❡s ❞❡❝✐r✱ ❝♦♠♦✿ (1, 2, 1, 1), (1, 1, 1, 2), (3, 2, 3, 2) = (1, 2, 1, 1), (0, 4, 0, 1) ✽ r❡s✉❧t❛ q✉❡ ❡❧ s✉❜❡s♣❛❝✐♦ q✉❡ ♥♦s ❤❛♥ ❞❛❞♦ t✐❡♥❡ ❞✐♠❡♥s✐ó♥ ✷✱ s✐❡♥❞♦ {(1, 2, 1, 1), (0, 4, 0, 1)} ✉♥❛ ❜❛s❡ ❞❡ ❞✐❝❤♦ s✉❜❡s❛❝✐♦✳ ✺✳✲ P❛r❛ ❧♦s s✐❣✉✐❡♥t❡s ♣❛r❡s ❞❡ s✉❜❡s♣❛❝✐♦s ❝❛❧❝✉❧❛ s✉ s✉♠❛ ② s✉ ✐♥t❡rs❡❝❝✐ó♥✿ x−y+z−t+u=0 ② 2x + z + t − u = 0  x−y+z−t+u=0  2x + z + t − u = 0 ❡♥ Z55  x+t+u=0 ❛✮ (x, y, z, t, u)/   (x, y, z, t, u)/  s♦❧✉❝✐ó♥✿ ▲❧❛♠❡♠♦s✱ W1 = ② (x, y, z, t, u)/   W2 = (x, y, z, t, u)/  x−y+z−t+u=0 2x + z + t − u = 0  x−y+z−t+u=0  2x + z + t − u = 0  x+t+u=0 ❛ ❧❛ ✈✐st❛ ❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡ W1 ② W2 ❡s ❡✈✐❞❡♥t❡ q✉❡ W2 ⊆ W1 ② ❡♥t♦♥❝❡s W1 ∩ W2 = W2 W1 ∪ W2 = W1 ② ❝♦♠♦ ♣♦r ♦tr♦ ❧❛❞♦ W1 + W2 = W1 ∪ W2 ✱ r❡s✉❧t❛ q✉❡ W1 + W2 = W1 ✳ ❊♥ r❡s✉♠❡♥✿ W1 ∩ W 2 = W2 W1 + W2 = W1 ❜✮ ❈❛❧❝✉❧❛ ❧❛ s✉♠❛ ② ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ ❧♦s s✐❣✉✐❡♥t❡s s✉❜❡s♣❛❝✐♦s✿ (1, 2, 1, 1) , (1, 1, 1, 2) , (3, 1, 2, 1) , (2, 0, 1, −1) ② (2, 3, −1, 2) ❡♥ R4 ✳ s♦❧✉❝✐ó♥✿ ▲❧❛♠❡♠♦s✱ W1 = (1, 2, 1, 1) , (1, 1, 1, 2) , (3, 1, 2, 1) , (2, 0, 1, −1) ② W2 = (2, 3, −1, 2) ❱❡❛♠♦s ❡♥ ♣✐♠❡r ❧✉❣❛r s✐ ❧♦s ✈❡❝t♦r❡s q✉❡ ❣❡♥❡r❛♥ ❛❧ ♣r✐♠❡r s✉❜❡s♣❛❝✐♦ s♦♥ ❧✐♥❡❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ P❛r❛ ❡❧❧♦✱ ❝♦♠♦ s✐❡♠♣r❡✱ ❡s❝r✐❜✐♠♦s ❧♦s ✈❡❝t♦r❡s ❝♦♠♦ ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② ❝❛❧❝✉❧❛♠♦s ❧❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ❞❡ ❞✐❝❤❛ ♠❛tr✐③ A✿  1  1   3 2 2 1 1 0 1 1 2 1  1 E2 − E1 2  E  3 − 3E1 1  ∼ −1 E4 − 2E1  1  0   0 0 ✾  2 1 1 −1 0 1   −E2 −5 −1 −2  ∼ −4 −1 −3  1  0   0 0  2 1 1 E3 + 5E2 1 0 −1   ∼ −5 −1 −2  E4 + 4E2 −4 −1 −3  1 2 1  0 1 0   0 0 −1 0 0 0  1  0   0 0 2 1 0 0  1 1 0 −1   E4 − E3 −1 −7  ∼ −1 −7  1 −1  =U −7  0 ❞❡ ♠❛♥❡r❛ q✉❡ W1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) s✐❡♥❞♦ {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} ✉♥❛ ❜❛s❡ ❞❡ W1 ✳ ❈♦♠♦ W2 = (2, 3, −1, 2) ✱ ❡s ❡✈✐❞❡♥t❡ q✉❡ {(2, 3, −1, 2)} ❡s ✉♥❛ ❜❛s❡ ❞❡ W2 . ❊♥t♦♥❝❡s✱ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s ❢♦r♠❛❞♦ ♣♦r ❧❛ ✉♥✐ó♥ ❞❡ ❧❛s ❞♦s ❜❛s❡s ❣❡♥❡r❛rá ❛❧ s✉❜❡s♣❛✲ ❝✐♦ W1 + W2 ✳ ❊s ❞❡❝✐r✿ W1 + W2 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2) ❱❡❛♠♦s✱ ❛❤♦r❛✱ s✐ ❡st♦s ✈❡❝t♦r❡s s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✿  1  0 A=  0 2  2 1 0 3 1 2  0 1   0 0 0 0   1 1 1  E − 2E 0 −1  1  0  4  0 ∼ −1 −7  0 −1 2   1 1 1  0 −1   E4 − 3E3  0  0 ∼ −1 −7  0 −3 −1  2 1 1 1 0 −1   E4 + E2 ∼ 0 −1 −7  −1 −3 0  2 1 1 1 0 −1  =U 0 −1 −7  0 0 20 ❝♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ♥♦ ❤❛② ♥✐♥❣✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s✱ ❧♦s ✈❡❝t♦r❡s ✜❧❛ ❞❡ ❧❛ ♠❛tr✐③ A s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ❉❡ ♠❛♥❡r❛ q✉❡ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s B = {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2)} ❢♦r♠❛♥ ✉♥❛ ❜❛s❡ ❞❡❧ s✉❜❡s♣❛❝✐♦ W1 + W2 ✳ ❈♦♠♦ ❡♥ ❧❛ ❜❛s❡ B ❞❡ W1 + W2 ❤❛② ❝✉❛tr♦ ✈❡❝t♦r❡s✱ r❡s✉❧t❛ q✉❡✿ ❞✐♠ (W1 + W2 ) = 4 W1 + W2 ❡s s✉❜❡s♣❛❝✐♦ ❞❡ R4 ⇒ W 1 + W 2 = R4 P♦r ♦tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡ ❞✐♠ (W1 + W2 ) = ❞✐♠ W1 + ❞✐♠ W2 − ❞✐♠ (W1 ∩ W2 ) ❞❡ ❞♦♥❞❡ ❞✐♠ (W1 ∩ W2 ) = ❞✐♠ W1 + ❞✐♠ W2 − ❞✐♠ (W1 + W2 ) ✶✵ =3+1−4=0 ❉❡ ♠❛♥❡r❛ q✉❡ W1 ∩ W2 = {(0, 0, 0, 0)}✳ ❈♦♥❝❧✉✐♠♦s ♣✉❡s q✉❡✿ W1 ⊕ W2 = R4 ✳ ❖❜s❡r✈❛❝✐ó♥ ✿ ❉❛❞♦s ❞♦s s✉❜❡s♣❛❝✐♦s W1 ② W2 ❞❡ ✉♥ ❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧✱ s✐ ♥♦s ♣✐❞❡♥ q✉❡ ❝❛❧❝✉❧❡♠♦s ❡❧ s✉❜❡s♣❛❝✐♦ s✉♠❛ W1 + W2 ② ❡❧ s✉❜❡s♣❛❝✐♦ ✐♥t❡r✲ s❡❝❝✐ó♥ W1 ∩W2 ✱ ❧❛ ❢♦r♠❛ ♠ás ❞✐r❡❝t❛ ❞❡ ♣r♦❝❡❞❡r✱ s✐♥ ♥❡❝❡s✐❞❛❞ ❞❡ ❤❛❝❡r r❡❢❡r❡♥❝✐❛ ❛ ❢ór♠✉❧❛s s♦❜r❡ ❞✐♠❡♥s✐♦♥❡s ❞❡❧ t✐♣♦✱ ❞✐♠ (W1 + W2 ) = ❞✐♠ W1 + ❞✐♠ W2 − ❞✐♠ (W1 ∩ W2 ) ❡s ❧❛ s✐❣✉✐❡♥t❡✳ • P❛r❛ ❝❛❧❝✉❧❛r W1 + W2 s❡ ✉♥❡♥ ❧♦s ✈❡❝t♦r❡s ❞❡ ✉♥❛ ❜❛s❡ ❞❡ W1 ❝♦♥ ❧♦s ✈❡❝t♦r❡s ❞❡ ✉♥❛ ❜❛s❡ ❞❡ W2 ② ❛ ❝♦♥t✐♥✉❛❝✐ó♥ s❡ ❝❛❧❝✉❧❛ ❡❧ s✉❜❡s♣❛❝✐♦ ❣❡♥❡r❛❞♦ ♣♦r ❞✐❝❤♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s✱ s✐❡♥❞♦ ❞✐❝❤♦ s✉❜❡s♣❛❝✐♦ ✐❣✉❛❧ ❛❧ s✉❜❡s♣❛❝✐♦ W1 + W2 ❜✉s❝❛❞♦✳ • P❛r❛ ❝❛❧❝✉❧❛r W1 ∩ W2 s❡ ❥✉♥t❛♥ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s q✉❡ ❞❡✜♥❡♥ ❛ ❛♠❜♦s s✉❜❡s♣❛❝✐♦s ② s❡ ❝❛❧❝✉❧❛ ❡❧ s✉❜❡s♣❛❝✐♦ ❝✉②❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s s♦♥ ❧❛ ✉♥✐ó♥ ❞❡ t♦❞❛s ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s✳ ❆sí✱ s✐ ❡♥ ❡❧ ♣r♦❜❧❡♠❛ ❛♥t❡r✐♦r q✉❡r❡♠♦s ❝❛❧❝✉❧❛r ❡❧ s✉❜❡s♣❛❝✐♦ W1 ∩W2 s✐❣✉✐❡♥❞♦ ❧❛ ♦❜s❡r✈❛❝✐ó♥ ❤❡❝❤❛✱ ❤❛rí❛♠♦s ❧♦ s✐❣✉✐❡♥t❡✿ ❊♥ ♣r✐♠❡r ❧✉❣❛r✱ ❡s❝r✐❜✐rí❛♠♦s ❡❧ s✉❜❡s♣❛❝✐♦ W1 ❡♥ ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s✱ ❡s ❞❡❝✐r✱ ❝♦♠♦ W1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) s✐❡♥❞♦ {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} ✉♥❛ ❜❛s❡ ❞❡ W1 ✳ ❘❡s✉❧t❛ q✉❡✱ ❉❛❞♦ (x, y, z, t) ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ R4 ✱ ❡♥t♦♥❝❡s s❛❜❡♠♦s q✉❡ (x, y, z, t) ∈ (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) ⇐⇒ ❡①✐st❡♥ ❡s❝❛❧❛r❡s x1 , x2 , x3 ∈ R t❛❧❡s q✉❡ (x, y, z, t) = x1 · (1, 2, 1, 1) + x2 · (0, 1, 0, −1) + x3 · (0, 0, −1, −7) ⇐⇒  x1 = x    2x1 + x2 = y ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s ❡s ❝♦♠♣❛t✐✲ x1 − x3 = z    x1 − x2 − 7x3 = t   ✳✳ 1 0 0 ✳ x     ✳✳  2 1 0 ✳ y   ❜❧❡ ⇐⇒ ❝✉❛♥❞♦ tr❛♥s❢♦r♠❡♠♦s ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛    ✳✳  1  0 −1 ✳ z   1 ✶✶ −1 −7 ✳✳ ✳ t ❛ ✉♥❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♦❝✉rr❛ q✉❡ ❡♥ ❞✐❝❤❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♥♦ ♥♦s q✉❡❞❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❝♦♠♦ ❝♦❧✉♠♥❛ ♣✐✈♦t❡✳     ✳✳ ✳✳ 1 0 0 ✳ x 1 0 0 ✳ x   E2 − 2E1       ✳✳ ✳✳    2 1 0 ✳ y − 2x  1 0 ✳ y  E3 − E1  0  E4 + E2      ✳ ✳ ∼ ∼ ✳ ✳   1 0 −1 ✳ z − x  0 −1 ✳ z    E4 − E1  0  ✳ ✳ 0 −1 −7 ✳✳ t − x 1 −1 −7 ✳✳ t   1 0   0 1    0 0  ✳ 0 ✳✳ ✳ 0 ✳✳    1 0   0 1    0 0  x  x  ✳ ✳  E4 − 7E3 0 ✳✳ y − 2x 0 ✳✳ y − 2x   ✳ ✳✳ ∼ ✳  −1 ✳ z−x −1 ✳ z−x  ✳✳ ✳✳ 0 0 0 ✳ 4x + y − 7z + t 0 0 −7 ✳ −3x + y + t ② ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ♣❛r❛ q✉❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ ♥♦ s❡❛ ❝♦❧✉♠♥❛ ♣✐✈♦t❡ ❤❛ ❞❡ ♦❝✉rr✐r q✉❡ 4x + y − 7z + t = 0✳ ▲✉❡❣♦✱         W1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) = (x, y, z, t) ∈ R4 /4x + y − 7z + t = 0 ❆❤♦r❛✱ t❛♠❜✐é♥ ❡s❝r✐❜✐rí❛♠♦s ❡❧ s✉❜❡s♣❛❝✐♦ W2 = (2, 3, −1, 2) ❡♥ ❡❝✉❛✲ ❝✐♦♥❡s ✐♠♣❧í❝✐t❛s✱ ♣❛r❛ ❡❧❧♦✱ ❛❧ ✐❣✉❛❧ q✉❡ ❛♥t❡s✱ ❉❛❞♦ (x, y, z, t) ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ R4 ✱ ❡♥t♦♥❝❡s s❛❜❡♠♦s q✉❡ (x, y, z, t) ∈ (2, 3, −1, 2) ⇐⇒ ❡①✐st❡ ❛❧❣✉♥ ❡s❝❛❧❛r x1 ∈ R t❛❧❡s q✉❡ (x, y, z, t) = x1 · (2, 3, −1, 2) ⇐⇒  2x1 = x    3x1 = y ❡s ❝♦♠♣❛t✐❜❧❡ ⇐⇒ ❝✉❛♥❞♦ ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s −x1 = z    2x1 = t   ✳✳ 2 ✳ x     ✳  3 ✳✳ y    ❛ ✉♥❛ ❢♦r♠❛ tr❛♥s❢♦r♠❡♠♦s ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛    −1 ✳✳✳ z    ✳✳ ✳ t ❡s❝❛❧♦♥❛❞❛ ♦❝✉rr❛ q✉❡ ❡♥ ❞✐❝❤❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♥♦ ♥♦s q✉❡❞❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❝♦♠♦ ❝♦❧✉♠♥❛ ♣✐✈♦t❡✳     ✳✳ ✳✳ 2 ✳ x −1 ✳ z     E2 + 3E1     ✳ ✳  3 ✳✳ y  E1 ↔ E3  3 ✳✳ y  E3 + 2E1         ∼ ∼  −1 ✳✳✳ z   2 ✳✳✳ x      E4 + 2E1 ✳ ✳ 2 ✳✳ t 2 ✳✳ t 2 ✶✷  ✳✳ ✳ z   ✳✳ ✳ y + 3z    ✳✳ ✳ x + 2z   ✳ 0 ✳✳ t + 2z ② ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ♣❛r❛ q✉❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛  ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛   −1   0    0  y + 3z = 0  ♥♦ s❡❛ ❝♦❧✉♠♥❛ ♣✐✈♦t❡ ❤❛ ❞❡ ♦❝✉rr✐r q✉❡✱ x + 2z = 0 2z + t = 0 ✱ ❧✉❡❣♦     y + 3z = 0  W2 = (2, 3, −1, 2) = (x, y, z, t) ∈ R4 / x + 2z = 0   2z + t = 0 ❊♥t♦♥❝❡s✱ ❝♦♠♦ W1 = (1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) = (x, y, z, t) ∈ R4 /4x + y − 7z + t = 0 ②  y + 3z = 0  W2 = (2, 3, −1, 2) = (x, y, z, t) ∈ R4 / x + 2z = 0   2z + t = 0   4x + y − 7z + t = 0       y + 3z = 0 ✳ ❉❡ ♠❛♥✲ r❡s✉❧t❛ q✉❡✱ W1 ∩ W2 = (x, y, z, t) ∈ R4 / x + 2z = 0       2z + t = 0   ❡r❛✱ q✉❡ ✈❛♠♦s ❛ r❡s♦❧✈❡r ❡❧ s✐st❡♠❛✱   4 1   0 1    1 0  0 0  1   0    0  0 0 0  1 1 ✳✳ ✳ ✳✳ 3 0 ✳ ✳ 2 0 ✳✳ ✳ 2 1 ✳✳ −7 1  0   0    0   4x + y − 7z + t = 0 y + 3z = 0 x + 2z = 0 2z + t = 0   1 0  E1 ↔ E3   0 1  ∼  4 1  0  ✳ 0 ✳✳ 0   ✳ 3 0 ✳✳ 0   E3 − E2  ✳✳ ∼ −15 1 ✳ 0   2 2 ✳ 1 ✳✳ 0 0 0        2 3 0 ✶✸ 0 ✳ 0 ✳✳  0   0   E3 − 4E1  ∼ 0   ✳✳ ✳ ✳ 2 1 ✳✳ 0 −7   1 0   0 1    0 0  ✳ 0 ✳✳ 2 3 −18 2 1 ✳ 0 ✳✳ ✳ 0 ✳✳ ✳ 1 ✳✳ ✳ 1 ✳✳  0   0   E3 ↔ E4  ∼ 0   0 ✳✳ ✳ ✳✳ 0 ✳ 3 ✳ 1 ✳✳ 2 ✳ ✳ 0 0 0 10 ✳✳ 0 0 −18 1 ✳✳ 0 ❉❡ ♠❛♥❡r❛ q✉❡ ♥♦s q✉❡❞❛ ❡❧ s✐❣✉✐❡♥t❡ s✐st❡♠❛ ❡q✉✐✈❛❧❡♥t❡✱   1 0   0 1    0 0  2  ✳ 0 ✳✳ 0   ✳ 0 ✳✳ 0   E4 + 9E3  ✳✳ ∼ 1 ✳ 0   x y +2z +3z 2z   1 0 2   0 1 3    0 0 2  = = +t = 10t = 0 0 0 0 0  0   0    0   0        q✉❡ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r t✐❡♥❡ ❝♦♠♦ s♦❧✉❝✐ó♥ ú♥✐❝❛✱ (x, y, z, t) = (0, 0, 0, 0)✳ ❉❡ ♠❛♥❡r❛ q✉❡✱ W1 ∩ W2 = {(0, 0, 0, 0)}✳ ❈♦♠♦ ②❛ s❛❜✐❛♠♦s✳ ✻✳✲ P❛r❛ ❧♦s s✐❣✉✐❡♥t❡s s✉❜❡s♣❛❝✐♦s✱ ❡st✉❞✐❛ s✐ s♦♥ ✐❣✉❛❧❡s ②✱ ❝❛s♦ ❞❡ ♥♦ s❡r❧♦✱ ❡st✉❞✐❛ s✐ ✉♥♦ ❞❡ ❧♦s ❞♦s ❡stá ✐♥❝❧✉✐❞♦ ❡♥ ❡❧ ♦tr♦✿ ❛✮ (x, y, z, t) / x−y+z−t=0 2x + z + t = 0 ② (2, 3, −1, 2) ❡♥ R4 ✳ s♦❧✉❝✐ó♥✿ ▲❧❛♠❡♠♦s W1 = (x, y, z, t) / x−y+z−t=0 2x + z + t = 0 ② W2 = (2, 3, −1, 2) ❊❧ s✉❜❡s♣❛❝✐♦ W1 ❤❛ s✐❞♦ ❡st✉❞✐❛❞♦ ❡♥ ❡❧ ♣r♦❜❧❡♠❛ ✹❛✮ ② s❛❜❡♠♦s q✉❡ ❞✐♠ (W1 ) = 2✳ ❊♥t♦♥❝❡s✱ W1 = W2 ✱ ♣✉❡s ❞✐♠ (W2 ) = 1✳ ❱❡❛♠♦s s✐ ❡❧ ✈❡❝t♦r (2, 3, −1, 2) ♣❡rt❡♥❡❝❡ ❛ W1 ✳ ❈♦♠♦ 2 − 3 + (−1) − 2 = −4 = 0 ⇒ (2, 3, −1, 2) ∈ / W1 ✳ ▲✉❡❣♦✱ W2 W1 ✳ ❖❜✈✐❛♠❡♥t❡ W1 W2 ✳ x−y+z−t+u=0 ② 2x + z + t − u = 0  x−y+z−t+u=0  2x + z + t − u = 0 ❡♥ Z55 ✳  x+t+u=0 ❜✮ W1 = (x, y, z, t, u)/   W2 = (x, y, z, t, u)/  s♦❧✉❝✐ó♥✿ ❆ ❧❛ ✈✐st❛ ❞❡ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧✐❝✐t❛s q✉❡ ❞❡✜♥❡♥ ❛ ❧♦s s✉❜❡s♣❛✲ ❝✐♦s ❡s ❡✈✐❞❡♥t❡ q✉❡ W2 ⊆ W1 ✳ P❛r❛ ❝♦♠♣r♦❜❛r q✉❡ W1 W2 ❜❛st❛ ❝♦♥ ✈❡r q✉❡ ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛ ❤♦♠♦❣é♥❡♦ q✉❡ ❞❡✜♥❡ ❛ W2 ❛❧ ♣♦♥❡rs❡ ❡♥ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s ♥♦ ❞❡❥❛ ♥✐♥❣✉♥❛ ✜❧❛ ♥✉❧❛✳ ✳ 1 ✳✳ 0  E2 + 3E1  1 −1 1 −1   ✳ ∼  2 0 1 1 −1 ✳✳ 0    E −E 3 1 ✳✳ 1 0 0 1 1 ✳ 0   ✶✹ ✳✳  1 −1 1 −1 1 ✳ 0    ✳  0 2 4 3 2 ✳✳ 0    ✳✳ 0 1 4 2 0 ✳ 0   E2 ↔ E3 ∼  1   0  −1 1 −1 1 4 0 2 4  1   0  −1 1 1 4 0 0 1 1 ✳✳ ✳ 0   ✳ 2 0 ✳✳ 0   ✳ 4 2 ✳✳ 0   E3 + 3E2 ∼ ✳✳ ✳ 0   ✳ 2 0 ✳✳ 0   ✳✳ 3 2 ✳ 0   −1 1 ❝✮ W1 = (1, 2, 1, 1), (1, 1, 1, 2), (3, 1, 2, 1), (2, 0, 1, −1) ② W2 = (1, 1, 1, 1), (2, 3, −1, 2) ❡♥ R4 ✳ s♦❧✉❝✐ó♥✿ P♦r ❡❧ ♣r♦❜❧❡♠❛ ✺❝✮ s❛❜❡♠♦s q✉❡ ✉♥❛ ❜❛s❡ ❞❡ W1 ❡s B = {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7)} ♣♦r ♦tr♦ ❧❛❞♦✱ ❡s ❡✈✐❞❡♥t❡ q✉❡ E = {(1, 1, 1, 1), (2, 3, −1, 2)} ❡s ✉♥❛ ❜❛s❡ ❞❡ W2 ✳ ▲✉❡❣♦ W1 = W2 ✳ P♦r ♦tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡ ❞✐♠ (W1 + W2 ) = ❞✐♠ W1 + ❞✐♠ W2 − ❞✐♠ (W1 ∩ W2 ) ② ❝♦♠♦ W1 + W2 = R4 ✳ P✉❡s✱ ❝♦♠♦ ✈✐♠♦s ❡♥ ❡❧ ♣r♦❜❧❡♠❛ ✺❝✮ ❧♦s ✈❡❝t♦r❡s {(1, 2, 1, 1) , (0, 1, 0, −1) , (0, 0, −1, −7) , (2, 3, −1, 2)} s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s ② ♣♦r t❛♥t♦ ✉♥❛ ❜❛s❡ ❞❡ R4 ✳ ❊♥t♦♥❝❡s r❡s✉❧t❛ q✉❡✱ 4 = 3 + 2 − ❞✐♠ (W1 ∩ W2 ) ⇒ ❞✐♠ (W1 ∩ W2 ) = 1 ⇒ W2 W1 ✭♣✉❡s s✐ ❢✉❡r❛ W2 ⊆ W1 ⇒ W1 ∩ W2 = W2 ② s✉❝❡❞❡rí❛ q✉❡ ❞✐♠ (W1 ∩ W2 ) = ❞✐♠ W2 = 2✮✳ ❖❜s❡r✈❛❝✐ó♥ ✿ ❉❛❞♦s ❞♦s s✉❜❡s♣❛❝✐♦s W1 ② W2 ❞❡ ✉♥ ❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧✱ s✐ q✉❡r✲ ❡♠♦s ❛✈❡r✐❣✉❛r s✐ ✉♥♦ ❡stá ❝♦♥t❡♥✐❞♦ ❡♥ ♦tr♦✱ ♣♦r ❡❥❡♠♣❧♦✱ s✐ q✉✐s✐ér❛♠♦s s❛❜❡r s✐ W1 ⊆ W2 ✱ ❡♥t♦♥❝❡s ❧♦ ♠ás ❝♦♠♦❞♦ ❡s ❝♦♥♦❝❡r ✉♥❛ ❜❛s❡ ❞❡❧ s✉❜❡✲ s♣❛❝✐♦ W1 ② ✉♥❛s ❡❝✉❛✐♦♥❡s ✐♠♣❧í❝✐t❛s q✉❡ ❞❡✜♥❛♥ ❛❧ s✉❜❡s♣❛❝✐♦ W2 ✱ ♣✉❡s ❡♥ ❡s❡ ❝❛s♦ ❜❛st❛ ❝♦♥ ❝♦♠♣r♦❜❛r s✐ ❧♦s ✈❡❝t♦r❡s ❞❡ ❧❛ ❜❛s❡ ❞❡ W1 ❡stá♥ ❡♥ ❡❧ s✉❜❡s♣❛❝✐♦ W2 ✭❡s ❞❡❝✐r✱ ❝✉♠♣❧❡♥ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ W2 ✮✳ ❨❛ q✉❡ ❡♥t♦♥❝❡s✱ s✉❝❡❞❡ q✉❡✿ • ❙✐ t♦❞♦s ❧♦s ✈❡❝t♦r❡s ❞❡ ✉♥❛ ❜❛s❡ ❞❡ W1 ♣❡rt❡♥❡❝❡♥ ❛ W2 ✭❡s ❞❡❝✐r✱ ❝✉♠♣❧❡♥ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ W2 ✮✱ ❡♥t♦♥❝❡s W1 ⊆ W2 ✳ • ❙✐✱ ♣♦r ❡❧ ❝♦♥tr❛r✐♦✱ ❛❧❣ú♥ ✈❡❝t♦r ❞❡ ✉♥❛ ❜❛s❡ ❞❡ W1 ✱ ♥♦ ♣❡rt❡♥❡❝❡ ❛ W2 ✭❡s ❞❡❝✐r✱ ♥♦ ❝✉♠♣❧❡ ❛❧❣✉♥❛ ❞❡ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ W2 ✮✱ ❡♥t♦♥❝❡s ♦❜✈✐❛♠❡♥t❡ W1 W2 ✳ ✶✺ ❙✐ t❛♠❜✐é♥ q✉✐s✐ér❛♠♦s s❛❜❡r s✐ W2 ⊆ W1 ♣r♦❝❡❞❡rí❛♠♦s ❞❡ ✐❣✉❛❧ ❢♦r♠❛✱ ❡s ❞❡❝✐r ❜✉s❝❛rí❛♠♦s ✉♥❛ ❜❛s❡ ❞❡ W2 ② ✉♥❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ W1 ✱ ♣❛r❛ ❞❡s♣✉és ❝♦♠♣r♦❜❛r ❧♦ ✐♥❞✐❝❛❞♦ ❛♥t❡r✐♦r♠❛♥t❡✳ P♦r ú❧t✐♠♦✱ s✉❝❡❞❡rá q✉❡✱   W 1 ⊆ W 2 W1 = W2 ⇔ y   W2 ⊆ W1 ❊♥ r❡s✉♠❡♥✱ s✐ ❤❡♠♦s ❞❡ ❛✈❡r✐❣✉❛r s✐ ❞♦s s✉❜❡s♣❛❝✐♦s s♦♥ ✐❣✉❛❧❡s✱ ❧♦s ♠❡❥♦r ❡s ❝♦♥♦❝❡r ✉♥❛ ❜❛s❡ ② ✉♥❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ ❝❛❞❛ s✉❜❡s♣❛❝✐♦✳ ✼✳✲ ❊♥❝✉❡♥tr❛ ❜❛s❡s ❝♦♥ ❧❛s s✐❣✉✐❡♥t❡s ❝♦♥❞✐❝✐♦♥❡s✿ ❛✮ ❊♥ R3 ❞❡ ❢♦r♠❛ q✉❡ ✉♥ ✈❡❝t♦r ❡sté ❡♥ ❧❛ r❡❝t❛ x−y+z =0 2x − z = 0 ② ♦tr♦s ❞♦s ❡♥ ❡❧ ♣❧❛♥♦ x + y + z = 0✳ s♦❧✉❝✐ó♥✿ ❘❡s♦❧✈❛♠♦s ❡❧ s✐st❡♠❛ ❤♦♠♦❣é♥❡♦ q✉❡ ❞❡✜♥❡ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠✲ ♣❧í❝✐t❛s ❞❡ ❧❛ r❡❝t❛✿ ✳ 1 1 ✳✳ 0  E2 − 2E1  1 −1  1 −1 ✳✳ ∼ 0 2 −3 2 0 −1 ✳ 0    ✳✳ ✳ 0  ✳✳ ✳ 0  ❞❡ ♠❛♥❡r❛ q✉❡ ❡❧ s✐st❡♠❛ ❡s ❡q✉✐✈❛❧❡♥t❡ ❛❧ s✐❣✉✐❡♥t❡✿ x − y 2y + − z 3z = = 0 0 ←→ x − y 2y = = −z 3z ❛ ❧❛ ✈✐st❛ ❞❡❧ ú❧t✐♠♦ s✐st❡♠❛ ✈❡♠♦s q✉❡ ❧❛s ✈❛r✐❛❜❧❡s x ❡ y s♦♥ ♣r✐♥❝✐♣❛❧❡s ✭♦ ❜ás✐❝❛s✮✱ ❧❛s q✉❡ s❡ ❝♦rr❡s♣♦♥❞❡♥ ❝♦♥ ❧❛s ❝♦❧✉♠♥❛s ♣✐✈♦t❡ ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥✲ ❛❞❛✱ ② q✉❡ ❧❛ ✈❛r✐❛❜❧❡ z ❡s ❧✐❜r❡✳ ❊♥t♦♥❝❡s✱ ❞❡s♣❡❥❛♥❞♦ y ❞❡ ❧❛ s❡❣✉♥❞❛ ❡❝✉❛❝✐ó♥ ② s✉st✐t✉②é♥❞♦❧❛ ❡♥ ❧❛ ♣r✐♠❡r❛✱ ♦❜t❡♥❡♠♦s ❧❛s ✐♥✜♥✐t❛s s♦❧✉❝✐♥❡s ❞❡ ❡st❡ s✐st❡♠❛ ❝♦♠♣❛t✐❜❧❡ ✐♥❞❡t❡r♠✐♥❛❞♦✱ q✉❡ s♦♥✿ x y z  1 =  2z 3 = z 2  ∈ R ✭❧✐❜r❡✮ ▲✉❡❣♦ ❧❛ r❡❝t❛ ❡s✱ 1 3 z, z, z /z ∈ R 2 2 = z 1 3 , , 1 /z ∈ R 2 2 = 1 3 , ,1 2 2 = (1, 3, 2) ❈❛❧❝✉❧❡♠♦s ❛❤♦r❛ ✉♥❛ ❜❛s❡ ❞❡❧ ♣❧❛♥♦ r❡s♦❧✈✐❡♥❞♦ ❧❛ ❡❝✉❛❝✐ó♥ ✐♠♣❧í❝✐t❛ q✉❡ ❧♦ ❞❡✜♥❡ x + y + z = 0 ↔ x = −y − z ✶✻ ❧❛ ✈❛r✐❛❜❧❡ x ❡s ♣r✐♥❝✐♣❛❧ ② ❧❛s ✈❛r✐❛❜❧❡s y ② z s♦♥ ❧✐❜r❡s✳ ❙✐❡♥❞♦ ❧❛s ✐♥✜♥✐t❛s s♦❧✉❝✐♦♥❡s✱  x = −y − z  y ∈ R ✭❧✐❜r❡✮ .
 z ∈ R ✭❧✐❜r❡✮ ▲✉❡❣♦ ❡❧ ♣❧❛♥♦ ❡s✱ {(−y − z, y, z) /y, z ∈ R} = {y · (−1, 1, 0) + z · (−1, 0, 1) /y, z ∈ R} = (−1, 1, 0) , (−1, 0, 1) .
▲❛ ❜❛s❡ ❜✉s❝❛❞❛ ❡s B = {(−1, 1, 0) , (−1, 0, 1) , (1, 3, 2)}✳ ❜✮ ❊♥ Z45 ❞❡ ❢♦r♠❛ q✉❡ ❤❛②❛ ❡①❛❝t❛♠❡♥t❡ ✉♥ ✈❡❝t♦r ❡♥ ❡❧ s✉❜❡s♣❛❝✐♦ W = {(x, y, z, t) /x − y + z − t = 0}✳ s♦❧✉❝✐ó♥✿ ❘❡s♦❧✈❡♠♦s ❧❛ ❡❝✉❛❝✐ó♥ ✐♠♣❧í❝✐t❛ q✉❡ ❞❡✜♥❡ ❛❧ s✉❜❡s♣❛❝✐♦✱ x − y + z − t = 0 ↔ x = y − z + t (= y + 4z+) ❧❛ ✈❛r✐❛❜❧❡ x ❡s ♣r✐♥❝✐♣❛❧ ② ❧❛s ✈❛r✐❛❜❧❡s y ✱ z ② t s♦♥ ❧✐❜r❡s✳ ❙✐❡♥❞♦ ❧❛s ✐♥✜♥✐t❛s s♦❧✉❝✐♦♥❡s✱  x = y + 4z + t    y ∈ R ✭❧✐❜r❡✮ .
z ∈ R ✭❧✐❜r❡✮    t ∈ R ✭❧✐❜r❡✮ ▲✉❡❣♦ ❡❧ s✉❜❡s♣❛❝✐♦ ❡s✱ W = (y + 4z + t, y, z, t) /y, z, t ∈ Z5 = {y · (1, 1, 0, 0) + z · (4, 0, 1, 0) + t · (1, 0, 0, 1) /y, z, t ∈ Z5 } = (1, 1, 0, 0) , (4, 0, 1, 0) , (1, 0, 0, 1) .
P✉❡s ❜✐❡♥✱ ❝♦❥❛♠♦s ❡❧ ✈❡❝t♦r (1, 1, 0, 0) ∈ W ② ❧♦s ✈❡❝t♦r❡s✱  /W  (0, 1, 0, 0) ∈ (0, 0, 1, 0) ∈ /W  (0, 0, 0, 1) ∈ /W ❖❜✈✐❛♠♥t❡✱ B = {(1, 1, 0, 0) , (0, 1, 0, 0) , (0, 0, 1, 0) , (0, 0, 0, 1)} ❡s ✉♥❛ ❜❛s❡ ❞❡ Z55 ✱ ❝✉♠♣❧✐❡♥❞♦ ❧♦ q✉❡ q✉❡rí❛♠♦s✳ ❝✮ ❊♥❝✉❡♥tr❛ ✉♥❛ ❜❛s❡ ❡♥ Z47 ❞❡ ❢♦r♠❛ q✉❡ ❤❛②❛ ❞♦s ✈❡❝t♦r❡s ❡♥ ❡❧ ♣❧❛♥♦ x−y+z−t=0 ✱ ♦tr♦ ❡♥ ❡❧ ♣❧❛♥♦ 2x − z = 0 ❡❧❧♦s✳ x−y+z =0 ② ❡❧ ú❧t✐♠♦ ❢✉❡r❛ ❞❡ 2x − z + t = 0 s♦❧✉❝✐ó♥✿ ❘❡s♦❧✈❛♠♦s ❡❧ ♣r✐♠❡r s✐st❡♠❛✱   1 2 ✳✳ ✳ ✳ 0  E2 + 5E1  1 −1 1 −1 ✳✳ 0  ✳ ✳ ∼ 0 ✳✳ 0 0 2 4 2 ✳✳ 0  −1  1 −1 0 −1 ✶✼  ❞❡ ♠❛♥❡r❛ q✉❡ ❡❧ s✐st❡♠❛ ❡s ❡q✉✐✈❛❧❡♥t❡ ❛❧ s✐❣✉✐❡♥t❡✿ x − y 2y + + − + z 4z t = 2t = 0 0 ↔ − x y 2y −z + t −4z − 2t = = ❞♦♥❞❡ ❧❛s ✈❛r✐❛❜❧❡s x ❡ y s♦♥ ♣r✐♥❝✐♣❛❧❡s ② ❧❛s ✈❛r✐❛❜❧❡s z ② t s♦♥ ❧✐❜r❡s✳ ❉❡s♣❡✲ ❥❛♠♦s y ❞❡ ❧❛ s❡❣✉♥❞❛ ❡❝✉❛❝✐ó♥✱ 2y = −4z − 2t = 3z + 5t → y = 5z + 6t ↓ r❡❝♦r❞❛❞ q✉❡ ❡st❛♠♦s ❡♥ Z7 s✉st✐t✉✐♠♦s ❡❧ ✈❛❧♦r ❞❡ y ❡♥ ❧❛ ♣r✐♠❡r❛ ❡❝✉❛❝✐ó♥ ② ❞❡s♣❡❥❛♠♦s x✱ x − 5z − 6t = −z + t → x = 4z ▲✉❡❣♦ ❧❛s s♦❧✉❝✐♦♥❡s s♦♥✱ x = y ∈ z ∈ t ∈ 4z 5z + 6t Z7 ✭❧✐❜r❡✮ Z7 ✭❧✐❜r❡✮     .
   ❊♥t♦♥❝❡s✱ ❡❧ ♣r✐♠❡r ♣❧❛♥♦ ❡s✱ {(4z, 5z + 6t, z, t) /z, t ∈ Z7 } = {z · (4, 5, 1, 0) + t · (0, 6, 0, 1) /z, t ∈ Z7 } = (4, 5, 1, 0) , (0, 6, 0, 1) .
❘❡♣✐t✐❡♥❞♦ ❡❧ ♠✐s♠♦ ♣r♦❝❡❞❡r ❝♦♥ ❡❧ s❡❣✉♥❞♦ s✐st❡♠❛ ♦❜t❡♥❡♠♦s✱   1 2 −1 1 0 −1   ✳ 0 ✳✳ 0  E2 + 5E1  1 ✳ ∼ 0 1 ✳✳ 0 ✳✳ ✳ 0  ✳✳ 2 4 1 ✳ 0  −1 1 0 ❞❡ ♠❛♥❡r❛ q✉❡ ❡❧ s✐st❡♠❛ ❡s ❡q✉✐✈❛❧❡♥t❡ ❛❧ s✐❣✉✐❡♥t❡✿ x − y 2y + + z 4z = + t = 0 0 ↔ x − y 2y = = −z −4z − t ❧❛s ✈❛r✐❛❜❧❡s x ❡ y s♦♥ ♣r✐♥❝✐♣❛❧❡s ② ❧❛s ✈❛r✐❛❜❧❡s z ② t s♦♥ ❧✐❜r❡s✳ ❉❡s♣❡❥❛♠♦s y ❞❡ ❧❛ s❡❣✉♥❞❛ ❡❝✉❛❝✐ó♥✱ 2y = −4z − t = 3z + 6t → y = 5z + 3t ↓ r❡❝♦r❞❛❞ q✉❡ ❡st❛♠♦s ❡♥ Z7 s✉st✐t✉✐♠♦s ❡❧ ✈❛❧♦r ❞❡ y ❡♥ ❧❛ ♣r✐♠❡r❛ ❡❝✉❛❝✐ó♥ ② ❞❡s♣❡❥❛♠♦s x✱ x − 5z − 3t = −z → x = 4z + 3t ✶✽ ▲✉❡❣♦ ❧❛s s♦❧✉❝✐♦♥❡s s♦♥✱ x = 4z + 3t y ∈ 5z + 3t z ∈ Z7 ✭❧✐❜r❡✮ t ∈ Z7 ✭❧✐❜r❡✮     .
   ❊♥t♦♥❝❡s✱ ❡❧ s❡❣✉♥❞♦ ♣❧❛♥♦ ❡s✱ {(4z + 3t, 5z + 3t, z, t) /z, t ∈ Z7 } = {z · (4, 5, 1, 0) + t · (3, 3, 0, 1) /z, t ∈ Z7 } = (4, 5, 1, 0) , (3, 3, 0, 1) .
❊s ❞❡❝✐r ❧♦s ♣❧❛♥♦s s♦♥✿ (4, 5, 1, 0) , (0, 6, 0, 1) (4, 5, 1, 0) , (3, 3, 0, 1) P✉❡s ❜✐❡♥✱ t♦♠❡♠♦s ❡❧ s✐❣✉✐❡♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s ❞❡ Z47 ✱ B = {(4, 5, 1, 0) , (0, 6, 0, 1) , (3, 3, 0, 1) , (0, 0, 0, 1)} , ❞♦♥❞❡ ❧♦s ❞♦s ♣r✐♠❡r♦s ❡stá♥ ❡♥ ❡❧ ♣r✐♠❡r ♣❧❛♥♦✱ ❡❧ t❡r❝❡r♦ ❡stá ❡♥ ❡❧ s❡❣✉♥❞♦ ♣❧❛♥♦ ② ❡❧ ❝✉❛rt♦ ♥♦ ♣❡rt❡♥❡❝❡ ❛ ♥✐♥❣✉♥♦ ❞❡ ❧♦s ❞♦s ♣❧❛♥♦s✱ ② ❝♦♠♣r♦❜❡♠♦s q✉❡ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ ❡♥ ❝✉②♦ ❝❛s♦ B s❡rí❛ ✉♥❛ ❜❛s❡ ❝♦♠♦ ❧❛ q✉❡ ❜✉s❝❛♠♦s✳  4  0   3 0 5 6 3 0 1 0 0 0  0 1   E3 + E1 1  ∼ 1  4  0   0 0 5 6 1 0 1 0 1 0  0 1   E3 + E2 1  ∼ 1  4  0   0 0 5 6 0 0 1 0 1 0  0 1   2  1 ❈♦♠♦ ❧♦s ✈❡❝t♦r❡s s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ B ❡s ✉♥❛ ❜❛s❡✳ ❞✮ ❊♥ R3 ❞❡ ❢♦r♠❛ q✉❡ ❡❧ ♣r✐♠❡r ✈❡❝t♦r ❡sté ❡♥ ❡❧ ❡❥❡ ❳✱ ❡❧ s❡❣✉♥❞♦ ❡♥ x−y+z =0 2x − z = 0 ❝♦♦r❞❡♥❛❞❛s (2, 1, 2)✳ ❧❛ r❡❝t❛ ②✱ ❝♦♥ r❡s♣❡❝t♦ ❛ ❡s❛ ❜❛s❡✱ ❡❧ ✈❡❝t♦r (1, 1, 1) t❡♥❣❛ s♦❧✉❝✐ó♥✿ ❘❡s♦❧✈❛♠♦s ❡❧ s✐st❡♠❛ ❤♦♠♦❣é♥❡♦ ❞❡ ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s q✉❡ ❞❡✜♥❡♥ ❛ ❧❛ r❡❝t❛ ✈❡❝t♦r✐❛❧✱ x−y+z =0 2x − z = 0 ✳ 1 ✳✳ 0  E2 − 2E1  1 −1 1  1 −1 ✳✳ ∼ 2 0 −1 ✳ 0 0 2 −3    ✶✾ ✳✳ ✳ 0  ✳✳ ✳ 0  ❞❡ ♠❛♥❡r❛ q✉❡ ❡❧ s✐st❡♠❛ ❡s ❡q✉✐✈❛❧❡♥t❡ ❛❧ s✐❣✉✐❡♥t❡✱ x − y 2y + z − 3z = = 0 0 ↔ − x y 2y = = −z 3z , ❞♦♥❞❡ ❧❛s ✈❛r✐❛❜❧❡s x ❡ y s♦♥ ♣r✐♥❝✐♣❛❧❡s ② ❧❛ ✈❛r✐❛❜❧❡s z ❡s ❧✐❜r❡✳ ❉❡s♣❡❥❛♠♦s y ❞❡ ❧❛ s❡❣✉♥❞❛ ❡❝✉❛❝✐ó♥ ② s✉st✐t✉♠♦s s✉ ✈❛❧♦r ❡♥ ❧❛ ♣r✐♠❡r❛✱ ♦❜t❡♥✐❡♥❞♦ ❝♦♠♦ s♦❧✉❝✐♦♥❡s✿  1 x y z =  2z 3 z = , 2  ∈ R ✭❧✐❜r❡✮ ❡♥t♦♥❝❡s ❧❛ r❡❝t❛ ❡s✱ 1 3 z, z, z /z, ∈ R 2 2 = = 1 3 , ,1 2 2 z· 1 3 , , 1 /z ∈ R 2 2 = (1, 3, 2) .
❈♦♠♦ ♥♦s ♣✐❞❡♥ ✉♥❛ ❜❛s❡ ❞❡ R3 ✱ t❛❧ q✉❡ ❡❧ ♣r✐♠❡r ✈❡❝t♦r ❡sté ❡♥ ❡❧ ❡❥❡ ❳✱ ❡❧ s❡❣✉♥❞♦ ❡♥ ❧❛ r❡❝t❛ ❛♥t❡r✐♦r✱ ② q✉❡ ❝♦♥r❡s♣❡❝t♦ ❛ ❡s❛ ❜❛s❡ ❡❧ ✈❡❝t♦r (1, 1, 1) t❡♥❣❛ ❝♦♦r❞❡♥❛❞❛s (2, 1, 2)✳ ▲❧❛♠❡♠♦s B = {(1, 0, 0) , (1, 3, 2) , (a, b, c)}✱ ❛ ❞✐❝❤❛ ❜❛s❡✳ ❊♥t♦♥❝❡s✱ ♣❛r❛ q✉❡ ❧❛s ❝♦♦r❞❡♥❛❞❛s ❞❡❧ ✈❡❝t♦r (1, 1, 1) r❡s♣❡❝t♦ ❞❡ ❧❛ ❜❛s❡ B s❡❛♥ (2, 1, 2)✱ ❤❛ ❞❡ s✉❝❡❞❡r q✉❡✱ (1, 1, 1) = (2, 1, 2)B ⇐⇒ (1, 1, 1) = 2 · (1, 0, 0) + 1 · (1, 3, 2) + 2 · (a, b, c)   a = −1  1 = 3 + 2a  1 1 = 3 + 2b ↔ b = −1 → (a, b, c) = −1, −1, − .
 2 1  1 = 2 + 2c c = −2 ❉❡ ♠❛♥❡r❛ q✉❡ ❧❛ ❜❛s❡ ❜✉s❝❛❞❛ ❡s✱ B= (1, 0, 0) , (1, 3, 2) , −1, −1, − 1 2 .
✽✳✲ P❛s❛ ❧♦s s✐❣✉✐❡♥t❡s ♣r♦❜❧❡♠❛s ❛❧ ❝♦rr❡s♣♦♥❞✐❡♥t❡ K n ② r❡s✉❡❧✈❡❧♦s✿ ❛✮ ❊st✉❞✐❛ s✐ s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ♦ ❜❛s❡✱ 1 − x + x2 , x + x2 , 2 − x + x2 ❡♥ Q2 [x] .
s♦❧✉❝✐ó♥✿ ❙❡❛ C = 1, x, x2 ❧❛ ❜❛s❡ ❝❛♥í♥✐❝❛ ❞❡ Q2 [x]✳ ❙✐ a + bx + cx2 ❡s ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ Q2 [x]✳ ❊♥t♦♥❝❡s✱ a + bx + cx2 = a · 1 + b · x + c · x2 ⇒ a + bx + cx2 = (a, b, c)C ✷✵ ❞❡ ♠❛♥❡r❛ q✉❡ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r Q2 [x] ❝♦♥ Q3 ❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦✱ ↔ Q3 ↔ (a, b, c) Q2 [x] a + bx + cx2 ❞❡ ♠❛♥❡r❛ q✉❡✱ ↔ Q3 ↔ (1, −1, 1) ↔ (0, 1, 1) ↔ (2, −1, 1) Q2 [x] 1 − x + x2 x + x2 2 − x + x2 ❧✉❡❣♦ ♥♦s ♣r❡❣✉♥t❛♠♦s s✐ {(1, −1, 1) , (0, 1, 1) , (2, −1, 1)} s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡✲ ♣❡♥❞✐❡♥t❡s✱ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ♦ ❜❛s❡ ❞❡❧ ❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ Q3 ✳ P❛r❛ r❡s♣♦♥❞❡r✱ ♣♦♥❡♠♦s ❡st♦s ✈❡❝t♦r❡s ❝♦♠♦ ❧❛s ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② tr❛♥s❢♦r♠❛♠♦s ést❛ ❛ ✉♥❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s U ✳     −1 1 1 −1 1 E − 2E E − E2 1  1 1  3 0 1 1  3 ∼ ∼ −1 1 0 1 −1   1 −1 1  0 1 1 =U 0 0 −2 1 A= 0 2 ❝♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ U ♥♦ ❤❛② ♥✐♥❣✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s✱ ❧❛s tr❡s ✜❧❛s ❞❡ A s♦♥ ❧✐♥❡❛❧✲ ♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ❉❡ ♠❛♥❡r❛ q✉❡ ❧♦s tr❡s ✈❡❝t♦r❡s✱ (1, −1, 1) , (0, 1, 1) , (2, −1, 1) ∈ Q3 s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✳ ❈♦♠♦ s♦♥ tr❡s ② ❝♦♠♦ ❞✐♠ Q3 = 3✱ ❞✐❝❤♦s ✈❡❝t♦r❡s s❡rá♥ ❜❛s❡ ② ♣♦r ❧♦ t❛♥t♦ s✐st❡♠❛ ❣❡♥❡r❛❞♦r✳ ❊♥ r❡s✉♠❡♥✱ ❧♦s ✈❡❝t♦r❡s 1 − x + x2 , x + x2 , 2 − x + x2 s♦♥ ❜❛s❡ ✭❡s ❞❡❝✐r✱ s✐st❡♠❛ ❣❡♥❡r❛❞♦r ② ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✮ ❞❡ Q2 [x]✳ ❜✮ ❘❡❞✉❝❡ ❡❧ s✐❣✉✐❡♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s ❤❛st❛ q✉❡ s❡❛ ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡✲ ♣❡♥❞✐❡♥t❡ ② ❛♠♣❧✐❛❧♦ ❤❛st❛ ❢♦r♠❛r ✉♥❛ ❜❛s❡ 1 2 • 1 1 1 2 , 0 1 s♦❧✉❝✐ó♥✿ ❙❡❛ C = ❝❛♥ó♥✐❝❛ ❞❡ M2×2 (Z5 )✳ ❙✐ t♦♥❝❡s✱ a c b d =a· 1 0 0 0 , 0 2 0 1 , 0 0 1 0 ❡♥ M2×2 (Z5 )✳ 1 0 0 1 0 0 0 0 , , , ❧❛ ❜❛s❡ 0 0 0 0 1 0 0 1 a b ❡s ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ M2×2 (Z5 )✳ ❊♥✲ c d +b· 0 0 1 0 ✷✶ +c· 0 1 0 0 +d· 0 0 0 1 ⇒ a c b d = (a, b, c, d)C ❞❡ ♠❛♥❡r❛ q✉❡ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r M2×2 (Z5 ) ❝♦♥ Z45 ❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦✱ M2×2 (Z5 ) a b c d ↔ M2×2 (Z5 ) 1 1 2 1 1 0 2 1 0 0 2 1 0 1 0 0 ↔ ❞❡ ♠❛♥❡r❛ q✉❡✱ Z45 ↔ (a, b, c, d) Z45 ↔ (1, 1, 2, 1) ↔ (1, 0, 2, 1) ↔ (0, 0, 2, 1) ↔ (0, 1, 0, 0) ❧✉❡❣♦ ❧♦ q✉❡ q✉❡r❡♠♦s ❡s✱ ❝♦♠♣r♦❜❛r s✐ ❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡s {(1, 1, 2, 1) , (1, 0, 2, 1) , (0, 0, 2, 1) , (0, 1, 0, 0)} ❞❡ Z45 ❡s ❧✐♥❡❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡ ②✱ ❡♥ ❝❛s♦ ❞❡ ♥♦ s❡r❧♦✱ r❡❞✉❝✐r❧♦ ❤❛st❛ q✉❡ s❡❛♥ ✐♥❞❡♣❡♥❞✐❡♥t❡s ② ❛♠♣❧✐❛r❧♦s ❤❛st❛ ❢♦r♠❛r ✉♥❛ ❜❛s❡ ❞❡ Z45 ✳ P❛r❛ r❡s♣♦♥❞❡r✱ ♣♦♥❡♠♦s ❡st♦s ✈❡❝t♦r❡s ❝♦♠♦ ❧❛s ✜❧❛s ❞❡ ✉♥❛ ♠❛tr✐③ A ② tr❛♥s❢♦r♠❛♠♦s ést❛ ❛ ✉♥❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣♦r ✜❧❛s U ✳  1  1 A=  0 0 1 0 0 1 2 2 2 0  1 1   E2 + 4E1 1  ∼ 0  1  0   0 0 1 4 0 0 2 0 2 0  1  0   0 0 1 4 0 1 2 0 2 0  1 0   E4 + 4E2 1  ∼ 0  1 0  =U 1  0 ❈♦♠♦ ❡♥ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ U ❛♣❛r❡❝❡ ✉♥❛ ✜❧❛ ❞❡ ❝❡r♦s ❝♦♥❝❧✉✐♠♦s q✉❡ ❧♦s ✈❡❝t♦r❡s q✉❡ ❢♦r♠❛♥ ❧❛s ✜❧❛s ❞❡ A s♦♥ ❧✐♥❡❛❧♠❡♥t❡ ❞❡♣❡♥❞✐❡♥t❡s✳ P♦r ♦tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡ ❧❛s ✜❧❛s ❞❡ A✱ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ✈❡❝t♦r❡s ❞❡ Z45 ✱ ② ❧❛s ✜❧❛s ❞❡ U ✱ t❛♠❜✐é♥ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ✈❡❝t♦r❡s ❞❡ Z45 ✱ ❣❡♥❡r❛♥ ❡❧ ♠✐s♠♦ s✉❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ ❞❡ Z45 ✳ ❉❡ ♠❛♥❡r❛ q✉❡ ❧❛ ❜❛s❡ ❞❡ Z45 ❜✉s❝❛❞❛ s❡rí❛✱ B = {(1, 1, 2, 1) , (0, 4, 0, 0) , (0, 0, 2, 1) , (0, 0, 0, 1)} ❞♦♥❞❡ ❤❡♠♦s ✐♥❝❧✉✐❞♦ ❡❧ ✈❡❝t♦r (0, 0, 0, 1) ♣❛r❛ ❝♦♠♣❧❡t❛r ❧❛ ❜❛s❡✳ ✷✷ ♦❜s❡r✈❛❝✐ó♥ ✿ ❝♦♠♦ ❞✉r❛♥t❡ ❡❧ ❡s❝❛❧♦♥❛♠✐❡♥t♦ ❞❡ ❧❛ ♠❛tr✐③ A ♥♦ ❤❡♠♦s ♣❡r♠✉✲ t❛❞♦ ♥✐♥❣✉♥❛ ✜❧❛✱ t❛♠❜✐é♥ ❧❛s tr❡s ♣r✐♠❡r❛s ✜❧❛s ❞❡ ❧❛ ♠❛tr✐③ A s♦♥ ❧✐♥❡❛❧✲ ♠❡♥t❡ ✐♥❞❡♣❡♥❞✐❡♥t❡s✱ ❞❡ ♠❛♥❡r❛ q✉❡ t❛♠❜✐é♥ ♣♦❞rí❛♠♦s ❤❛❜❡r ♣r♦♣✉❡st♦ ❝♦♠♦ ❜❛s❡ ❞❡ Z45 ❛✿ B˜ = {(1, 1, 2, 1) , (1, 0, 2, 1) , (0, 0, 2, 1) (0, 0, 0, 1)} ❉❡ ♠❛♥❡r❛ q✉❡ ♣♦❞❡♠♦s ❞❛r ❝♦♠♦ ❜❛s❡ ❞❡ M2×2 (Z5 ) ❛✱ B= 1 2 1 1 , 0 0 4 0 , 0 2 0 1 , 0 0 0 1 B˜ = 1 2 1 1 , 1 2 0 1 , 0 2 0 1 , 0 0 0 1 ♦ t❛♠❜✐é♥ ❛✱ .
❝✮ ❊♥❝✉❡♥tr❛ ❧❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❡ ✐♠♣❧í❝✐t❛s ❞❡ ❧♦s s✉❜❡s♣❛❝✐♦s✿    1 1 1 , 1 1 0   1 1 1 , 2 0 1 • W1 = 1  0 1 • W2 = P (x) = a + bx + cx2 /  2 1  0 P (2) = 0 P (1) = 0 ❡♥ M3×2 (Z5 )✳ ❡♥ (Z5 )2 [X]✳ ✭❞♦♥❞❡✱ (Z5 )2 [X] ❂ ❡s♣❛❝✐♦ ✈❡❝t♦r✐❛❧ ❞❡ ❧♦s ♣♦❧✐♥♦♠✐♦s ❞❡ ❣r❛❞♦ ♠❡♥♦r ♦ ✐❣✉❛❧ q✉❡ ❞♦s ❝♦♥ ❝♦❡✜❝✐❡♥t❡s ❡♥ ❡❧ ❝✉❡r♣♦ Z5 ✮✳ s♦❧✉❝✐ó♥✿ ❚❛❧ ❝♦♠♦ ❤✐❝✐♠♦s ❡♥ ❡❧ ❛♣❛rt❛❞♦ ❛♥t❡r✐♦r✱ ✐❞❡♥t✐✜❝❛♠♦s M2×3 (Z5 )  1 1  0 1   1 0  1 1  1 1   1 0  1 2  2 1  1 0 ↔ Z65 ↔ (1, 1, 0, 1, 1, 0) ↔ (1, 1, 1, 1, 1, 0) ↔ (1, 2, 2, 1, 1, 0) ❞❡ ♠❛♥❡r❛ q✉❡ ❜✉s❝❛♠♦s ❧❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❡ ✐♠♣❧í❝✐t❛s ❞❡❧ s✉❜❡✲ s♣❛❝✐♦✱ (1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) , ❞❡ Z65 ✳ ▲❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❞❡ ❡st❡ s✉❜❡s♣❛❝✐♦ ❞❡ Z65 s♦♥✱ a · (1, 1, 0, 1, 1, 0) + b · (1, 1, 1, 1, 1, 0) + c · (1, 2, 2, 1, 1, 0) ; ✷✸ a, b, c ∈ Z5 ❡s ❞❡❝✐r✱ a, b, c ∈ Z5 (a + b + c, a + b + 2c, b + 2c, a + b + c, a + b + c, 0) ; ▲✉❡❣♦ t❡♥❡♠♦s q✉❡✱   a+b+c W1 =  b + 2c  a+b+c   a + b + 2c  s✉❜❡s♣❛❝✐♦ ❡①♣r❡s❛❞♦ ❡♥ a + b + c  /a, b, c ∈ Z5  ecuaciones paramtricas 0 ♦❜s❡r✈❛❝✐ó♥ ✿ ▲❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❞❡❧ s✉❜❡s♣❛❝✐♦ W1 t❛♠❜✐é♥ s❡ ♣♦✲ ❞rí❛♥ ❤❛❜❡r ❡st❛❜❧❡❝✐❞♦ tr❛❜❛❥❛♥❞♦ ❞✐r❡❝t❛♠❡♥t❡ ❝♦♥ ❧❛s ♠❛tr✐❝❡s 3 × 2✱ ❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦✱  1 a· 0 1   1 1 1 +b· 1 0 1   1 1 1 +c· 2 0 1   2 a+b+c 1  =  b + 2c 0 a+b+c  a + b + 2c a+b+c  0 ❞❡ ✐❣✉❛❧ ❢♦r♠❛✱ ♦❜s❡r✈❛ q✉❡ t❛♠❜✐é♥ ♣♦❞❡♠♦s ❤❛❝❡r ❡❧ ❝❛♠✐♥♦ ✐♥✈❡rs♦✱ ❡s ❞❡❝✐r✱ ❞❛❞❛ ❧❛ ♠❛tr✐③  a+b+c  b + 2c a+b+c  a + b + 2c a+b+c  0 ❞❡s❝♦♠♣♦♥❡r❧❛ ❝♦♠♦ ❧❛ s✉♠❛✱  a+b+c  b + 2c a+b+c   1 a + b + 2c a + b + c  = a· 0 1 0   1 1 1 +b· 1 0 1   1 1 1 +c· 2 0 1  2 1  0 ❆❤♦r❛ ✈❛♠♦s ❛ ❡♥❝♦♥tr❛r ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡❧ s✉❜❡s♣❛❝✐♦ ❞❡ Z65 ✱ W = (1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) ❡q✉✐✈❛❧❡♥t❡ ✭✐s♦♠♦r❢♦✮ ❛ W1 ✳ ❙❡❛ (x, y, z, t, r, s) ✉♥ ✈❡❝t♦r ❝✉❛❧q✉✐❡r❛ ❞❡ Z65 ✱ ❡♥t♦♥❝❡s s❛❜❡♠♦s q✉❡ (x, y, z, t, r, s) ∈ (1, 1, 0, 1, 1, 0) , (1, 1, 1, 1, 1, 0) , (1, 2, 2, 1, 1, 0) ⇐⇒ ❡①✐st❡♥ ❡s❝❛❧❛r❡s x1 , x2 , x3 ∈ Z5 t❛❧❡s q✉❡ (x, y, z, t, r, s) = x1 · (1, 1, 0, 1, 1, 0) + x2 · (1, 1, 1, 1, 1, 0) + x3 · (1, 2, 2, 1, 1, 0) ⇐⇒ ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s✱ ❡s❝r✐t♦ ♠❛tr✐❝✐❛❧♠❡♥t❡✱         1 1 0 1 1 0 1 1 1 1 1 0 1 2 2 1 1 0         x1    ·  x2  =      x 3   ✷✹ x y z t r s         ❡s ❝♦♠♣❛t✐❜❧❡✳ ❊s ❞❡❝✐r✱ ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❧✐♥❡❛❧❡s x1 x1 + + x1 x1 + + x2 x2 x2 2x x2  = x    = y     = z = t    = r     = s + x3 + 2x3 + 2x3 + x3 + x3 0 ❡s ❝♦♠♣❛t✐❜❧❡ ⇐⇒ ❝✉❛♥❞♦ tr❛♥s❢♦r♠❡♠♦s ❧❛ ♠❛tr✐③ ❛♠♣❧✐❛❞❛ ❞❡❧ s✐st❡♠❛   1   1    0    1    1  1 1 1 2 1 2 1 1 1 1 0 0 0 ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳  x   y    z    t    r   s ❛ ✉♥❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛✱ ♦❝✉rr❛ q✉❡ ❡♥ ❞✐❝❤❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♥♦ ♥♦s q✉❡❞❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❝♦♠♦ ❝♦❧✉♠♥❛ ♣✐✈♦t❡✳   1   1    0    1    1  0 1 1 1 2 1 2 1 1 1 1 0 0 ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳  x   y   E2 − 2E  z   E4 − E1  ∼ t   E5 − E1  r     1   0    0    0    0  1 1 0 1 1 2 0 0 0 0 0 0 0 s   1   0    0    0    0  1 0 1 1 2 0 1 0 0 0 0 0 0 ✳✳ ✳ x ✳✳ ✳ z ✳✳ ✳ y−x ✳✳ ✳ t−x ✳✳ ✳ r−x ✳✳ ✳ s  ✳✳ ✳ x  ✳✳  ✳ y−x    ✳✳ ✳ z   E2 ↔ E3  ✳✳ ∼ ✳ t−x    ✳✳ ✳ r−x   ✳✳ ✳ s               ② ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ♣❛r❛ q✉❡ ❧❛ ú❧t✐♠❛ ❝♦❧✉♠♥❛ ❞❡ ❧❛ ♠❛tr✐③ ❡s❝❛❧♦♥❛❞❛ ♥♦ ✷✺ s❡❛ ❝♦❧✉♠♥❛ ♣✐✈♦t❡ ❤❛ ❞❡ ♦❝✉rr✐r q✉❡ t r − x − x s  0  4x 0 ↔ 4x  0 = = =  = 0  = 0  = 0 + t + r s ▲✉❡❣♦ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧✐❝✐t❛s ❞❡❧ s✉❜❡s♣❛❝✐♦ W s♦♥✱ W =   4x (x, y, z, t, r, s) ∈ Z65 / 4x   = 0  = 0  = 0 + t + r s P♦r t❛♥t♦✱ ❡❧ s✉❜❡s♣❛❝✐♦ W1 ❡①♣r❡s❛❞♦ ❡♥ ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s s❡rá✱   x W1 =  z  r  = 0  s✉❜❡s♣❛❝✐♦ ❡①♣r❡s❛❞♦ ❡♥ = 0  ecuaciones implcitas = 0  y 4x + t t  ∈ M3×2 (Z5 )/ 4x + r s s ❆❤♦r❛ ✈❛♠♦s ❛ ❜✉s❝❛r ❧❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❡ ✐♠♣❧í❝✐t❛s ❞❡ • W2 = P (x) = a + bx + cx2 / P (2) = 0 P (1) = 0 ❡♥ (Z5 )2 [X]✳ P (2) = 0 ⇐⇒ a + 2b + 4c = 0 P (x) = b + 2cx → P (1) = 0 ⇐⇒ b + 2c = 0 ▲✉❡❣♦✱ W2 = P (x) = a + bx + cx2 / a + 2b + 4c = b + 2c = s✉❜❡s♣❛❝✐♦ ❡①♣r❡s❛❞♦ ❡♥ 0 0 ecuaciones implcitas P❛r❛ ❝❛❧❝✉❧❛r ❧❛s ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s só❧♦ ❤❡♠♦s ❞❡ r❡s♦❧✈❡r ❡❧ s✐st❡♠❛ ❞❡ ❡❝✉❛❝✐♦♥❡s ❤♦♠♦❣é♥❡♦ q✉❡ ❞❡✜♥❡ ❧❛s ❡❝✉❛❝✐♦♥❡s ✐♠♣❧í❝✐t❛s ❞❡ W2 ✳ ❊s ❞❡❝✐r✱ ❤❡♠♦s ❞❡ r❡s♦❧✈❡r ❡❧ s✐❣✉✐❡♥t❡ s✐st❡♠❛ ❤♦♠♦❣é♥❡♦✱ ❡♥ Z5 ✿ a + 2b + 4c = b + 2c = 0 0 ↔ a + 2b = b = c 3c → a + 6c = c → a + c = c →a=0 ❞♦♥❞❡ ❧❛s ✐♥❝ó❣♥✐t❛s a ② b s♦♥ ♣r✐♥❝✐♣❛❧❡s ② ❧❛ c ❡s ❧✐❜r❡✳ ▲❛s s♦❧✉❝✐♦♥❡s s♦♥✱ a b c = = ∈  0  3c  Z5 ✭❧✐❜r❡✮ ▲✉❡❣♦✱ ❡❧ s✉❜❡s♣❛❝✐♦ W2 ❡①♣r❡s❛❞♦ ❡♥ ❡❝✉❛❝✐♦♥❡s ♣❛r❛♠étr✐❝❛s ❡s✱ W2 = P (x) = 3cx + cx2 /c ∈ Z5 = c · 3x + x2 /c ∈ Z5 = 3x + x2 ✷✻ s✉❜❡s♣❛❝✐♦ ❡①♣r❡s❛❞♦ ❡♥ ecuaciones paramtricas s✉❜❡s♣❛❝✐♦ ❡①♣r❡s❛❞♦ ♠❡❞✐❛♥t❡ ✉♥ sistema generador ...

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