Bài tập toán cao cấp Tập 1 part 7 pps

4.3. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t 167 cu ’ ahˆe . phu . o . ng tr`ınh (4.10) d u . o . . cgo . il`ahˆe . nghiˆe . mco . ba ’ n cu ’ an´onˆe ´ u mˆo ˜ i nghiˆe . mcu ’ ahˆe . (4.10) d ˆe ` u l`a tˆo ’ ho . . p tuyˆe ´ nt´ınh cu ’ a c´ac nghiˆe . m e 1 ,e 2 , ,e m . D - i . nh l´y (vˆe ` su . . tˆo ` nta . ihˆe . nghiˆe . mco . ba ’ n). Nˆe ´ uha . ng cu ’ a ma trˆa . n cu ’ ahˆe . (4.10) b´eho . nsˆo ´ ˆa ’ nth`ıhˆe . (4.10) c´o hˆe . nghiˆe . mco . ba ’ n. 168 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh Phu . o . ng ph´ap t`ım hˆe . nghiˆe . mco . ba ’ n 1) D ˆa ` u tiˆen cˆa ` n t´ach ra hˆe . ˆa ’ nco . so . ’ (gia ’ su . ’ d ´ol`ax 1 , ,x r )v`athu du . o . . chˆe . a 11 x 1 + ···+ a 1r x r = −a 1r+1 x r+1 −···−a 1n x n , a r1 x 1 + ···+ a rr x r = −a rr+1 x r+1 −···−a rn x n .      (4.12) 2) Gia ’ su . ’ hˆe . (4.12) c´o nghiˆe . ml`a x i =  α (i) 1 ,α (i) 2 , ,α (i) r ; x r+1 , ,x n )  ; i = 1,r. Cho c´ac ˆa ’ ntu . . do c´ac gi´a tri . x r+1 =1,x r+2 =0, ,x n =0 ta thu du . o . . c e 1 =  α (1) 1 ,α (1) 2 , ,α (1) r ;1, 0, ,0  Tu . o . ng tu . . ,v´o . i x r+1 =0,x r+2 =1,x r+3 =0, ,x n = 0 ta c´o e 2 =  α (2) 1 , ,α (2) r ;0, 1, 0, ,0  , v`a sau c`ung v´o . i x r+1 =0, ,x n−1 =0,x n =1tathudu . o . . c e k =(α (k) 1 , ,α (k) r , 0, ,1),k= n − r. Hˆe . c´ac nghiˆe . m e 1 ,e 2 , ,e k v`u . athudu . o . . cl`ahˆe . nghiˆe . mco . ba ’ n. C ´ AC V ´ IDU . V´ı d u . 1. T`ım nghiˆe . mtˆo ’ ng qu´at v`a hˆe . nghiˆe . mco . ba ’ ncu ’ ahˆe . phu . o . ng tr`ınh 2x 1 + x 2 − x 3 + x 4 =0, 4x 1 +2x 2 + x 3 − 3x 4 =0.  4.3. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t 169 Gia ’ i. 1) V`ı sˆo ´ phu . o . ng tr`ınh b´eho . nsˆo ´ ˆa ’ n nˆen tˆa . pho . . p nghiˆe . mcu ’ a hˆe . l`a vˆo ha . n. Hiˆe ’ n nhiˆen ha . ng cu ’ a ma trˆa . ncu ’ ahˆe . b˘a ` ng2v`ı trong c´ac d i . nh th´u . c con cˆa ´ p2c´odi . nh th´u . c con      2 −1 41      =0. Do vˆa . yhˆe . d ˜a cho tu . o . ng d u . o . ng v´o . ihˆe . 2x 1 − x 3 = −x 1 − x 4 , 4x 1 + x 3 = −2x 2 +3x 4 . T`u . d´o suy ra x 1 = −3x 2 +2x 4 6 ,x 3 = 5 3 x 4 . (4.13) Do d ´otˆa . pho . . p nghiˆe . mcu ’ ahˆe . c´o da . ng  −3α +2β 6 ; α; 5 3 β; β   ∀α, β ∈ R  (*) 2) Nˆe ´ u trong (4.13) ta cho c´ac ˆa ’ ntu . . do bo . ’ i c´ac gi´a tri . lˆa ` nlu . o . . t b˘a ` ng c´ac phˆa ` ntu . ’ cu ’ a c´ac cˆo . tdi . nh th´u . c      10 01      (=0) th`ı thu d u . o . . c c´ac nghiˆe . m e 1 =  − 1 2 ;1;0;0  v`a e 2 =  1 3 ;0; 5 3 ;1  . D ´ol`ahˆe . nghiˆe . mco . ba ’ ncu ’ ahˆe . phu . o . ng tr`ınh d˜a cho v`a nghiˆe . mtˆo ’ ng qu´at cu ’ ahˆe . d ˜a cho c´o thˆe ’ biˆe ’ udiˆe ˜ ndu . ´o . ida . ng X = λe 1 + µe 2 = λ  − 1 2 ;1;0;0  + µ  1 3 ;0; 5 3 ;1  170 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh trong d´o λ v`a µ l`a c´ac h˘a ` ng sˆo ´ t`uy ´y: X =  −3λ +2µ 6 ; λ; 5 3 µ; µ   ∀λ, µ ∈ R  . Khi cho λ v`a µ c´ac gi´a tri . sˆo ´ kh´ac nhau ta s˜e thu d u . o . . c c´ac nghiˆe . m riˆeng kh´ac nhau.  V´ı d u . 2. Gia ’ ihˆe . x 1 +2x 2 − x 3 =0, −3x 1 − 6x 2 +3x 3 =0, 7x 1 +14x 2 − 7x 3 =0.      Gia ’ i. Hˆe . d ˜a cho tu . o . ng d u . o . ng v´o . iphu . o . ng tr`ınh x 1 +2x 2 − x 3 =0. T`u . d ´o suy ra nghiˆe . mcu ’ ahˆe . l`a: x 1 = −2x 2 + x 3 , x 2 = x 2 , x 3 = x 3 ; x 2 v`a x 3 t`uy ´y, hay du . ´o . ida . ng kh´ac e =(−2x 2 + x 3 ; x 2 ; x 3 ). Cho x 2 =1,x 3 = 0 ta c´o e 1 =(−2; 1; 0), la . ichox 2 =0,x 3 =1tathudu . o . . c e 2 =(1, 0, 1). Hai h`ang e 1 v`a e 2 l`a dˆo . clˆa . p tuyˆe ´ n t´ınh v`a mo . i nghiˆe . mcu ’ ahˆe . dˆe ` uc´o da . ng X = λe 1 + µe 2 =(−2λ + µ; λ; µ) 4.3. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t 171 trong d´o λ v`a µ l`a c´ac sˆo ´ t`uy ´y.  V´ı du . 3. T`ım nghiˆe . mtˆo ’ ng qu´at v`a hˆe . nghiˆe . mco . ba ’ ncu ’ ahˆe . phu . o . ng tr`ınh x 1 +3x 2 +3x 3 +2x 4 +4x 5 =0, x 1 +4x 2 +5x 3 +3x 4 +7x 5 =0, 2x 1 +5x 2 +4x 3 + x 4 +5x 5 =0, x 1 +5x 2 +7x 3 +6x 4 +10x 5 =0.          Gia ’ i. B˘a ` ng c´ac ph´ep biˆe ´ nd ˆo ’ iso . cˆa ´ p, dˆe ˜ d`ang thˆa ´ yr˘a ` ng hˆe . d˜acho c´o thˆe ’ d u . avˆe ` hˆe . bˆa . c thang sau d ˆay x 1 +3x 2 +3x 3 +2x 4 +4x 5 =0, x 2 +2x 3 + x 4 +3x 5 =0, x 4 =0.      Ta s˜e cho . n x 1 , x 2 v`a x 4 l`am ˆa ’ nco . so . ’ ; c`on x 3 v`a x 5 l`am ˆa ’ ntu . . do. Ta c´o hˆe . x 1 +3x 2 +2x 4 = −3x 3 −4x 5 , x 2 + x 4 = −2x 3 −3x 5 , x 4 =0.      Gia ’ ihˆe . n`ay ta thu d u . o . . c nghiˆe . mtˆo ’ ng qu´at l`a x 1 =3x 3 +5x 5 , x 2 = −2x 3 −3x 5 , x 4 =0. Cho c´ac ˆa ’ ntu . . do lˆa ` nlu . o . . t c´ac gi´a tri . b˘a ` ng x 3 =1,x 5 = 0 (khi d´o x 1 =3,x 2 =2,x 3 =1,x 4 =0,x 5 = 0) v`a cho x 3 =0,x 5 = 1 (khi d´o x 1 =5,x 2 =3,x 3 =0,x 4 =0,x 5 = 1) ta thu du . o . . chˆe . nghiˆe . mco . ba ’ n e 1 = (3; −2; 1; 0; 0), e 2 = (5; −3; 0; 0; 1). 172 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh T`u . d ´o nghiˆe . mtˆo ’ ng qu´at c´o thˆe ’ viˆe ´ tdu . ´o . ida . ng X = λ(3; −2; 1; 0; 0) + µ(5; −3; 0; 0; 1) =(3λ +5µ; −2λ −3µ; λ;0;µ); ∀λ, µ ∈ R. B˘a ` ng c´ach cho λ v`a µ nh˜u . ng gi´a tri . sˆo ´ kh´ac nhau ta thu du . o . . c c´ac nghiˆe . m riˆeng kh´ac nhau. D ˆo ` ng th`o . i, mo . i nghiˆe . m riˆeng c´o thˆe ’ thu d u . o . . ct`u . d ´ob˘a ` ng c´ach cho . n c´ac hˆe . sˆo ´ λ v`a µ th´ıch ho . . p.  4.3. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t 173 B ` AI T ˆ A . P Gia ’ ic´achˆe . phu . o . ng tr`ınh thuˆa ` n nhˆa ´ t 1. x 1 +2x 2 +3x 3 =0, 2x 1 +3x 2 +4x 3 =0, 3x 1 +4x 2 +5x 3 =0.      . (D S. x 1 = α, x 2 = −2α, x 3 = α, ∀α ∈ R) 2. x 1 + x 2 + x 3 =0, 3x 1 − x 2 − x 3 =0, 2x 1 +3x 2 + x 3 =0.      .(D S. x 1 = x 2 = x 3 =0) 3. 3x 1 − 4x 2 + x 3 −x 4 =0, 6x 1 − 8x 2 +2x 3 +3x 4 =0.  (DS. x 1 = 4α −β 3 , x 2 = α, x 3 = β, x 4 =0;α, β ∈ R t`uy ´y) 4. 3x 1 +2x 2 − 8x 3 +6x 4 =0, x 1 − x 2 +4x 3 − 3x 4 =0.  (DS. x 1 =0,x 2 = α, x 3 = β, x 4 = −α +4β 3 ; α, β ∈ R t`uy ´y) 5. x 1 − 2x 2 +3x 3 − x 4 =0, x 1 + x 2 − x 3 +2x 4 =0, 4x 1 − 5x 2 +8x 3 + x 4 =0.      (D S. x 1 = − 1 4 α, x 2 = α, x 3 = 3 4 α, x 4 =0;α ∈ R t`uy ´y) 6. 3x 1 − x 2 +2x 3 + x 4 =0, x 1 + x 2 −x 3 − x 4 =0, 5x 1 + x 2 − x 3 =0.      (D S. x 1 = − α 4 , x 2 = 5α 4 + β, x 3 = α, x 4 = β; α, β ∈ R t`uy ´y) 7. 2x 1 + x 2 + x 3 =0, 3x 1 +2x 2 − 3x 3 =0, x 1 +3x 2 − 4x 3 =0, 5x 1 + x 2 − 2x 3 =0.          174 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh (DS. x 1 = α 7 , x 2 = 9α 7 , x 3 = α; α ∈ R t`uy ´y) 4.3. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t 175 T`ım nghiˆe . mtˆo ’ ng qu´at v`a hˆe . nghiˆe . mco . ba ’ ncu ’ a c´ac hˆe . phu . o . ng tr`ınh 8. 9x 1 +21x 2 − 15x 3 +5x 4 =0, 12x 1 +28x 2 − 20x 3 +7x 4 =0.  (D S. Nghiˆe . mtˆo ’ ng qu´at: x 1 = − 7 3 x 2 + 5 3 x 3 , x 4 =0. Hˆe . nghiˆe . mco . ba ’ n e 1 =(−7, 3, 0, 0), e 2 =(5, 0, 3, 0)) 9. 14x 1 +35x 2 −7x 3 − 63x 4 =0, −10x 1 − 25x 2 +5x 3 +45x 4 =0, 26x 1 +65x 2 − 13x 3 − 117x 4 =0.      (D S. Nghiˆe . mtˆo ’ ng qu´at: x 3 =2x 1 +5x 2 − 9x 3 . Hˆe . nghiˆe . mco . ba ’ n: e 1 =(1, 0, 2, 0); e 2 =(0, 1, 5, 0); e 3 = (0, 0, −9, 1)) 10. x 1 +4x 2 +2x 3 − 3x 5 =0, 2x 1 +9x 2 +5x 3 +2x 4 + x 5 =0, x 1 +3x 2 + x 3 − 2x 4 −9x 5 =0.      (D S. Nghiˆe . mtˆo ’ ng qu´at: x 1 =2x 3 +8x 4 , x 2 = −x 2 − 2x 4 ; x 5 =0. Hˆe . nghiˆe . mco . ba ’ n: e 1 =(2, −1, 1, 0, 0); e 2 =(8, −2, 0, 1, 0) 11. x 1 +2x 2 +4x 3 − 3x 4 =0, 3x 1 +5x 2 +6x 3 −4x 4 =0, 4x 1 +5x 2 − 2x 3 +3x 4 =0, 3x 1 +8x 2 +24x 3 −19x 4 =0.          (D S. Nghiˆe . mtˆo ’ ng qu´at: x 1 =8x 3 − 7x 4 , x 2 = −6x 3 +5x 4 . Hˆe . nghiˆe . mco . ba ’ n: e 1 =(8, −6, 1, 0), e 2 =(−7, 5, 0, 1)) 12. x 1 +2x 2 − 2x 3 + x 4 =0, 2x 1 +4x 2 +2x 3 − x 4 =0, x 1 +2x 2 +4x 3 − 2x 4 =0, 4x 1 +8x 2 − 2x 3 + x 4 =0.          (D S. Nghiˆe . mtˆo ’ ng qu´at x 1 = −2x 2 , x 4 =2x 3 . Hˆe . nghiˆe . mco . ba ’ n: e 1 =(−2, 1, 0, 0), e 2 =(0, 0, 1, 2)) 176 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh 13. x 1 +2x 2 +3x 3 +4x 4 +5x 5 =0, 2x 1 +3x 2 +4x 3 +5x 4 + x 5 =0, 3x 1 +4x 2 +5x 3 + x 4 +2x 5 =0, x 1 +3x 2 +5x 3 +12x 4 +9x 5 =0, 4x 1 +5x 2 +6x 3 − 3x 4 +3x 5 =0.                (D S. Nghiˆe . mtˆo ’ ng qu´at x 1 = x 3 +15x 5 , x 2 = −2x 3 −12x 5 , x 4 = x 5 . Hˆe . nghiˆe . mco . ba ’ n: e 1 =(1, −2, 1, 0, 0), e 2 = (15, −12, 0, 1, 1)) [...]... + a2 + a3 − 3a4 − 2a5 = θ; a1, a2, a3, a4) 2) a1 = (1, 1, 1, 1) , a2 = (2, 0, 1, 1) , a3 = (3, −4, 0, 1) , a4 = (13 , 10 , 3, −2) (DS 2a1 + a2 + 3a3 − a4 = θ; a1, a2, a3) 3) a1 = (1, 1, 1, 1) , a2 = (2, 0, 1, 1) , a3 = (3, 1, 1, 1) , ´ e o a e ınh) a4 = (4, −2, 1, −2) (DS Hˆ dˆc lˆp tuyˆn t´ 4) a1 = (1, 2, −2, 1) , a2 = ( 1, 0, 2, 1) , a3 = (0, 1, 0, 1) , ´ e o a e ınh) a4 = (3, 6, 0, 4) (DS Hˆ dˆc... a o a 1) a1 = (1, 2, 3, 4), a2 = (1, 2, 3, 4) (DS Pttt) 2) a1 = (1, 2, 3, 4), a2 = (1, −2, −3, −4) (DS Pttt) 3) a1 = (1, 2, 3, 4), a2 = (3, 6, 9, 12 ) (DS Pttt) 4) a1 = (1, 2, 3, 4), (a2 = (1, 2, 3, 5) (DS Dltt) 5) a1 = (1, 0, 0, 0), a2 = (0, 1, 0, 0), a3 = (0, 0, 1, 0), a4 = (0, 0, 0, 1) u ´ ’ v` a l` vecto t`y y cua R4 (DS Pttt) a a 6) a1 = (1, 1, 1, 1) , a2 = (0, 1, 1, 1) , a3 = (0, 0, 1, 1) , a4... V´ du 7 T` hang cua hˆ vecto trong R4 ı ım a1 = (1, 1, 1, 1) ; a2 = (1, 2, 3, 4); a3 = (2, 3, 2, 3); a4 = (2, 4, 5, 6) ’ Giai Ta lˆp ma trˆn c´c toa dˆ v` t`m hang a a a o a ı    1 1 1 1 1 1 1 0 1 2 1 2 3 4 h − h → h  1   2 2 −→  A=  0 1 0 2 3 2 3 h3 − 2h1 → h3 0 1 2 3 4 5 6 h4 − 3h1 → h4   1 1 1 1 0 1 2 3   −→   0 0 −2 −3 0 0 0 0 ’ o cua n´ Ta c´ o  1 3  →  1 h3... trong co so E1 , E2 , E3 Ta c´ y e     3 x1    1  x2 = T  1 x3 0  V` T 1 ı  1 −2 1   =  1 4 e 2  nˆn 1 1 1        1 −2 1 x1 3 5        2   1 =  7 x2  =  1 4 x3 1 1 1 0 −4 ’ o o Vˆy trong co so m´.i E1 , E2 , E3 ta c´ a x = (5, 7, −4) ’ a a V´ du 7 Trong khˆng gian R2 cho co so E1 , E2 v` c´c vecto E1 = ı o e1 − 2e2, E2 = 2e1 + e2 , x = 3e1 − 4e2 ... 2) a1 = (1, 1, 0); a2 = (1, 0, 1) ; a3 = (1, −2, 0) (DS Dltt) 3) a1 = (1, 3, 3); a2 = (1, 1, 1) ; a3 = (−2, −4, −4) (DS Pttt) 4) a1 = 1, −3, 0); a2 = (3, −3, 1) ; a3 = (2, 0, 1) (DS Pttt) 5) a1 = (2, 3, 1) ; a2 = (1, 1, 1) ; a3 = (1, 2, 0) (DS Pttt) ` ´ ’ ’ a e 5 Gia su v1, v2 v` v3 l` hˆ dˆc lˆp tuyˆn t´ a a e o a e ınh Ch´.ng minh r˘ng hˆ u sau dˆy c˜ng l` dltt: a u a 1) a1 = v1 + v2 ; a2 = v1 + v3;... (3, 7, 4, 1, 7) , a5 = (0, 11 , −5, 4, −4) (DS r = 3 hˆ pttt) e 4) a1 = (2, 1, 4, −4, 17 ), a2 = (0, 0, 5, 7, 9), a3 = (2, 1, −6, 10 , 11 ), e a4 = (8, 4, 1, 5, 11 ), a5 = (2, 2, 9, 11 , 10 ) (DS r = 5, hˆ dltt) 5.2 ’ ’ -o ’ Co so Dˆi co so - ` ’ Dinh ngh˜ 5.2 .1 Hˆ vecto E1 , E2, , En gˆm n vecto cua khˆng ıa e o o Rn du.o.c goi l` mˆt co so cua n´ nˆu ´ ’ ’ o e gian vecto a o a e 1) hˆ E1 ,... biˆu diˆn a3 v` a4 qua a1 v` a2 ım a e u ´ Ta viˆt e a3 = 1 a1 + ξ2 a2 hay l` a (1, 9, 4, 2) = 1 · (1, 4, 1, 1) + ξ2 · (2, 3, 1, 1) ⇒ (1, 9, 4, 2) = ( 1 + 2ξ2 , 4 1 + 3ξ2 , 1 − ξ2 , 1 + ξ2 ) -i ` a o o a e ’ ` 5 .1 D nh ngh˜ khˆng gian n-chiˆu v` mˆt sˆ kh´i niˆm co ban vˆ vecto ıa o e e 18 5 ´ v` thu du.o.c hˆ phu.o.ng tr` a ınh e 1 + 2ξ2 4 1 + 3ξ2 1 − ξ2 1 + ξ2  = 1,    = 9, = 4,  ... Euclide o 18 8 Rn ’ e 10 T´ hang r cua hˆ vecto v` chı r˜ hˆ d˜ cho l` pttt hay dltt: ınh a a ’ o e a 1) a1 = (1, −2, 2, −8, 2), a2 = (1, −2, 1, 5, 3), a3 = (1, −2, 4, 7, 0) ´ (DS r = 3, hˆ dˆc lˆp tuyˆn t´ e o a e ınh) 2) a1 = (2, 3, 1, 1) , a2 = (3, 1, 4, 2), a3 = (1, 2, 3, 1) , e a4 = (1, −4, 7, 5) (DS r = 3, hˆ pttt) 3) a1 = (2, 1, −3, 2, −6), a2 = (1, 5, −2, 3, 4), a3 = (3, 4, 1, 5, 7) , a4... l´ 1 n´ lˆp th`nh mˆt co ’ y o a a o l` dˆc lˆp tuyˆn t´ a o a ’ ’ ’ ım o o a V´ du 4 Gia su trong co so E1 , E2 vecto x c´ toa dˆ l` 1; −2 T` ı d´ trong co so E1 = E1 , E2 = E1 + E2 ’ toa dˆ cua vecto o o ’ ’ ´ ´ ` ’ ’ Giai Dˆu tiˆn ta viˆt ma trˆn chuyˆn t` co so E1 , E2 dˆn E1 , E2 a e e a e u e Ta c´ o E1 = 1 · e1 + 0 · e2, E2 = 1 · e1 + 1 · e2 Do d´ o T = 1 1 1 1 ⇒ T 1 = 0 21 0 1. .. a3 = (1, 1, 1) ’ ’ ´ ´ o e a o e ınh ’ a 2 H˜y x´c dinh sˆ λ dˆ vecto x ∈ R3 l` tˆ ho.p tuyˆn t´ cua c´c a a ´ e vecto a1 , a2, a3 ∈ R3 nˆu: 1) x = (1, 3, 5); a1 = (3, 2, 5); a2 = (2, 4, 7) ; a3 = (5, 6, λ) 18 6 Chu.o.ng 5 Khˆng gian Euclide o Rn (DS λ = 12 ) 2) x = (7, −2, λ); a1 = (2, 3, 5); a2 = (3, 7, 8); a3 = (1, −6, 1) (DS λ = 15 ) 3) x = (5, 9, λ); a1 = (4, 4, 3); a2 = (7, 2, 1) ; a3 = (4, 1, 6) . =      11 11 1234 2323 3456      h 2 − h 1 → h  2 h 3 − 2h 1 → h  3 h 4 − 3h 1 → h  4 −→      11 11 012 3 010 1 012 3      h 3 − h 2 → h  3 h 4 − h 2 → h  4 → −→      11 1 1 01. niˆe . mco . ba ’ nvˆe ` vecto . 18 3 Khi d´ot`u . d˘a ’ ng th´u . c vecto . x 1 a 1 + x 2 a 2 + ···+ x n a n + x n +1 a n +1 = θ suy ra a 11 x 1 + a 12 x 2 + ···+ a 1n +1 x n +1 =0, a n1 x 1 + a n2 x 2 + ···+ a nn +1 x n +1 =0.      D´ol`ahˆe . thuˆa ` n. a 4 qua a 1 v`a a 2 . Ta viˆe ´ t a 3 = ξ 1 a 1 + ξ 2 a 2 hay l`a (1, 9, 4, 2) = ξ 1 · (1, 4, 1, 1) + ξ 2 · (2, 3, 1, 1) ⇒ (1, 9, 4, 2) = (ξ 1 +2ξ 2 , 4ξ 1 +3ξ 2 ,ξ 1 − ξ 2 ,ξ 1 + ξ 2 ) 5 .1. D - i . nh