5.2. Co . so . ’ .D - ˆo ’ ico . so . ’ 195 1 + Ch´u . ng minh r˘a ` ng E 1 ,E 2 lˆa . p th`anh co . so . ’ cu ’ a R 2 . 2 + T`ım to . adˆo . vecto . x trong co . so . ’ E 1 ,E 2 . 3 + T`ım to . adˆo . cu ’ a vecto . x trong co . so . ’ E 2 ,E 1 . Gia ’ i. 1 + Ta lˆa . p ma trˆa . n c´ac to . adˆo . cu ’ a E 1 v`a E 2 : A =  1 −2 21  ⇒ detA =5=0. Do d ´ohˆe . hai vecto . E 1 ,E 2 l`a dltt trong khˆong gian 2-chiˆe ` u R 2 nˆen n´o lˆa . p th`anh co . so . ’ . 2 + Trong co . so . ’ d˜a cho vecto . x c´o to . adˆo . l`a (3, −4). Gia ’ su . ’ trong co . so . ’ E 1 ,E 2 vecto . x c´o to . adˆo . (x 1 ,x 2 ). Ta lˆa . p ma trˆa . n chuyˆe ’ nt`u . co . so . ’ E 1 , E 2 dˆe ´ nco . so . ’ E 1 ,E 2 : T =  12 −21  ⇒ T −1 = 1 5  12 −21  Khi d ´o  x 1 x 2  = T −1  3 −4  ⇒  x 1 x 2  = 1 5  1 −2 21  3 −4  = 1 5  11 2  =    11 5 2 5    . Vˆa . y x 1 = 11 5 , x 2 = +2 5 . 3 + V`ı E 1 ,E 2 l`a co . so . ’ cu ’ a R 2 nˆen E 2 ,E 1 c˜ung l`a co . so . ’ cu ’ a R 2 .Ma trˆa . n chuyˆe ’ nt`u . co . so . ’ E 1 , E 2 dˆe ´ nco . so . ’ E 2 ,E 1 c´o da . ng A ∗ =  21 1 −2  ,A ∗ −1 = − 1 5  −2 −1 −12  3 −4  = − 1 5  −2 −11  =    2 5 11 5    Do d ´o x 1 = 2 5 , x 2 = 11 5 trong co . so . ’ E 2 ,E 1 . V´ı d u . 8. Trong khˆong gian R 3 cho co . so . ’ E 1 , E 2 , E 3 n`ao d´o v`a trong co . so . ’ d´o c´ac vecto . E 1 ,E 2 ,E 3 v`a x c´o to . adˆo . l`a E 1 =(1, 1, 1); E 2 = (1, 2, 2), E 3 =(1, 1, 3) v`a x =(6, 9, 14). 196 Chu . o . ng 5. Khˆong gian Euclide R n 1 + Ch´u . ng minh r˘a ` ng E 1 ,E 2 ,E 3 c˜ung lˆa . p th`anh co . so . ’ trong R 3 . 2 + T`ım to . adˆo . cu ’ a x trong co . so . ’ E 1 ,E 2 ,E 3 . Gia ’ i. 1 + tu . o . ng tu . . nhu . trong v´ı du . 7, ha . ng cu ’ ahˆe . ba vecto . E 1 ,E 2 ,E 3 b˘a ` ng 3 nˆen hˆe . vecto . d ´odˆo . clˆa . p tuyˆe ´ n t´ınh trong khˆong gian 3-chiˆe ` u nˆen n´o lˆa . p th`anh co . so . ’ cu ’ a R 3 . 2+ Dˆe ’ t`ım to . adˆo . cu ’ a x trong co . so . ’ E 1 ,E 2 ,E 3 ta c´o thˆe ’ tiˆe ´ n h`anh theo hai phu . o . ng ph´ap sau. (I) V`ı E 1 ,E 2 ,E 3 lˆa . p th`anh co . so . ’ cu ’ a R 3 nˆen x = x 1 E 1 + x 2 E 2 + x 3 E 3 ⇒ (6, 9, 14) = x 1 (1, 1, 1) + x 2 (1, 2, 2) + x 3 (1, 1, 3) v`a do d´o x 1 ,x 2 ,x 3 l`a nghiˆe . mcu ’ ahˆe . x 1 + x 2 + x 3 =6, x 1 +2x + x 3 =9, x 1 +2x 2 +3x 3 =14.      ⇒ x 1 = 1 2 ,x 2 =3,x 3 = 5 2 · (I I) Lˆa . p ma trˆa . n chuyˆe ’ nt`u . co . so . ’ E 1 , E 2 , E 3 sang co . so . ’ E 1 ,E 2 ,E 3 : T EE =    111 121 123    ⇒ T −1 EE = 1 2    4 −1 −1 −22 0 0 −11    . Do d´o    x 1 x 2 x 3    = T −1 EE    6 9 14    = 1 2    1 6 5    =      1 2 3 5 2      v`a thu du . o . . ckˆe ´ t qua ’ nhu . tronng (I).  B ` AI T ˆ A . P 5.2. Co . so . ’ .D - ˆo ’ ico . so . ’ 197 1. Ch´u . ng minh r˘a ` ng c´ac hˆe . vecto . sau dˆay l`a nh˜u . ng co . so . ’ trong khˆong gian R 4 : 1) e 1 =(1, 0, 0, 0); e 2 =(0, 1, 0, 0); e 3 =(0, 0, 1, 0); e 4 =(0, 0, 0, 1). 2) E 1 =(1, 1, 1, 1); E 2 =(0, 1, 1, 1); E 3 =(0, 0, 1, 1); E 4 =(0, 0, 0, 1). 2. Ch´u . ng minh r˘a ` ng hˆe . vecto . do . nvi . : e 1 =(1, 0, ,0    n−1 ); e 2 =(0, 1, 0, ,0), ,e n =(0, 0, ,0    n−1 , 1) lˆa . p th`anh co . so . ’ trong R n .Co . so . ’ n`ay du . o . . cgo . il`aco . so . ’ ch´ınh t˘a ´ c. 3. Ch´u . ng minh r˘a ` ng hˆe . vecto . E 1 =(1, 0, ,0), E 2 =(1, 1, ,0), E n =(1, 1, ,1) l`a mˆo . tco . so . ’ trong R n . 4. Ch´u . ng minh r˘a ` ng hˆe . vecto . E 1 =(1, 2, 3, ,n− 1,n), E 2 =(1, 2, 3, ,n− 1, 0), E n =(1, 0, 0, ,0, 0) lˆa . p th`anh co . so . ’ trong khˆong gian R n . 5. H˜ay kiˆe ’ m tra xem mˆo ˜ ihˆe . vecto . sau dˆa y c ´o l ˆa . p th`anh co . so . ’ trong khˆong gian R 4 khˆong v`a t`ım c´ac to . adˆo . cu ’ a vecto . x =(1, 2, 3, 4) trong mˆo ˜ ico . so . ’ d´o. 1) a 1 =(0, 1, 0, 1); a 2 =(0, 1, 0, −1); a 3 =(1, 0, 1, 0); a 4 =(1, 0, −1, 0). (DS. 3, −1, 2, −1) 2) a 1 =(1, 2, 3, 0); a 2 =(1, 2, 0, 3); a 3 =(1, 0, 2, 3); 198 Chu . o . ng 5. Khˆong gian Euclide R n a 4 =(0, 1, 2, 3). (DS. 2 3 , − 1 6 , 1 2 , 1) 3) a 1 =(1, 1, 1, 1); a 2 =(1, −1, 1, −1); a 3 =(1, −1, 1, 1); a 4 =(1, −1, −1, −1). (DS. 3 2 , − 1 2 , 1, −1) 4) a 1 =(1, −2, 3, −4); a 2 =(−4, 1, −2, 3); a 3 =(3, −4, 1, −2); a 4 =(−2, 3, −4, 1). (DS. − 13 10 , − 7 10 , − 13 10 , − 17 10 ) Nhˆa . nx´et. Ta nh˘a ´ cla . ir˘a ` ng c´ac k´y hiˆe . u e 1 ,e 2 , ,e n du . o . . cd`ung dˆe ’ chı ’ c´ac vecto . do . nvi . cu ’ a tru . c x i (i =1, 2, ,n): e i =(1, 0, ,0    n−1 ),e 2 =(0, 1, 0, ,0), ,e n =(0, ,0    n−1 , 1) 6. T`ım ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ e 2 ,e 3 ,e 1 . (D S.    001 100 010    ) 7. T`ım ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 ,e 4 dˆe ´ nco . so . ’ e 3 ,e 4 ,e 2 ,e 1 . (D S.      0001 0010 1000 0100      ) 8. Cho ma trˆa . n  −11 20  l`a ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 dˆe ´ nco . so . ’ E 1 , E 2 .T`ım to . adˆo . cu ’ a vecto . E 1 , E 2 . (DS. E 1 =(−1, 2); E 2 =(1, 0)) 9. Gia ’ su . ’    12−1 31 0 20 1    5.2. Co . so . ’ .D - ˆo ’ ico . so . ’ 199 l`a ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ E 1 , E 2 , E 3 .T`ım to . adˆo . cu ’ a vecto . E 2 trong co . so . ’ e 1 ,e 2 ,e 3 .(DS. E 2 =(2, 1, 0)) 10. T`ım ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ E 1 =2e 1 − e 3 + e 2 ; E 2 =3e 1 − e 2 + e 3 ; E 3 = e 3 . (D S.    230 1 −10 −111    ) 11. T`ım ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ E 1 = e 2 + e 3 ; E 2 = −e 1 +2e 3 ; E 3 = e 1 + e 2 . (DS.    0 −11 101 120    ) 12. T`ım ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 ,e 4 dˆe ´ nco . so . ’ E 1 =2e 2 +3e 3 + e 4 ; E 2 = e 1 − 2e 2 +3e 3 − e 4 ; E 3 = e 1 + e 4 ; E 4 =2e 1 + e 2 −e 3 + e 4 . (DS.      0112 2 −20 1 330−1 1 −11 1      ) 13. Cho  21 −12  l`a ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 dˆe ´ nco . so . ’ E 1 , E 2 .T`ım to . adˆo . cu ’ a c´ac vecto . e 1 , e 2 trong co . so . ’ E 1 , E 2 . (DS. e 1 =  2 5 , 1 5  . e 2 =  − 1 5 , 2 5  ) Chı ’ dˆa ˜ n. T`u . ma trˆa . nd˜a cho t`ım khai triˆe ’ n E 1 , E 2 theo co . so . ’ e 1 ,e 2 . T`u . d´o t`ım khai triˆe ’ n e 1 ,e 2 theo co . so . ’ E 1 , E 2 . 200 Chu . o . ng 5. Khˆong gian Euclide R n 14. Cho ma trˆa . n    1 −13 51 2 14−1    l`a ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ E 1 , E 2 , E 3 .T`ım to . adˆo . vecto . e 2 trong co . so . ’ E 1 , E 2 , E 3 . (DS. e 2 =  11 41 , − 4 41 , − 5 41  ) 15. Cho ma trˆa . n    101 002 −131    l`a ma trˆa . n chuyˆe ’ nt`u . co . so . ’ e 1 ,e 2 ,e 3 dˆe ´ nco . so . ’ E 1 , E 2 , E 3 .T`ım to . adˆo . c´ac vecto . e 1 ,e 2 ,e 3 trong co . so . ’ E 1 , E 2 , E 3 . (D S. e 1 =  1, 1 3 , 0  , e 2 =  − 1 2 , − 1 3 , 1 2  , e 3 =  0, 1 3 , 0  ) 16. Trong co . so . ’ e 1 ,e 2 vecto . x c´o to . ad ˆo . l`a (1; 2). T`ım to . adˆo . cu ’ a vecto . d ´o trong co . so . ’ E 1 = e 1 +2e 2 ; E 2 = −e 1 + e 2 . (D S. x =  − 1 3 , − 4 3  ) 17. Trong co . so . ’ e 1 ,e 2 vecto . x c´o to . ad ˆo . l`a ( −3; 1). T`ım to . adˆo . cu ’ a vecto . d´o trong co . so . ’ E 1 = −2e 1 + e 2 ; E 2 = e 2 . (D S. x =  3 2 , − 1 2  ) 18. Trong co . so . ’ e 1 ,e 2 ,e 3 vecto . x c´o to . ad ˆo . l`a (−1; 2; 0). T`ım to . adˆo . cu ’ a vecto . d´o trong co . so . ’ E 1 =2e 1 − e 2 +3e 3 , E 2 = −3e 1 + e 2 − 2e 3 ; E 3 =4e 2 +5e 3 .(DS. (−0, 68; −0, 12; 0, 36)) 19. Trong co . so . ’ e 1 ,e 2 ,e 3 vecto . x c´o to . ad ˆo . l`a (1, −1, 0). T`ım to . adˆo . cu ’ a vecto . d´o trong co . so . ’ : E 1 =3e 1 + e 2 +6e 3 , E 2 =5e 1 − 3e 2 +7e 3 , E 3 = −2e 1 +2e 2 − 3e 3 . 5.3. Khˆong gian vecto . Euclid. Co . so . ’ tru . . cchuˆa ’ n 201 (DS. x =(−0, 6; 1, 2; 1, 6)) 20. Trong co . so . ’ e 1 ,e 2 ,e 3 vecto . x c´o to . adˆo . l`a (4, 0, −12). T`ım to . a dˆo . cu ’ a vecto . d ´o trong co . so . ’ E 1 = e 1 +2e 2 + e 3 , E 2 =2e 1 +3e 2 +4e 3 , E 3 =3e 1 +4e 2 +3e 3 . (D S. x =(−4, −8, 8)) 21. Trong khˆong gian v´o . imˆo . tco . so . ’ l`a e 1 ,e 2 ,e 3 cho c´ac vecto . E 1 = e 1 + e 2 , E 2 =2e 1 − e 2 + e 3 , E 3 = e 2 − e 3 . 1) Ch´u . ng minh r˘a ` ng E 1 , E 2 , E 3 lˆa . p th`anh co . so . ’ . 2) T`ım to . ad ˆo . cu ’ a vecto . x = e 1 +8e 2 − 5e 3 trong co . so . ’ E 1 , E 2 , E 3 . (D S. x =(3, −1, 4)) 22. Trong co . so . ’ e 1 ,e 2 ,e 3 cho c´ac vecto . a =(1, 2, 3), b =(0, 3, 1), c =(0, 0, 2), d =(4, 3, 1). Ch´u . ng minh r˘a ` ng c´ac vecto . a, b, c lˆa . p th`anh co . so . ’ v`a t`ım to . adˆo . cu ’ a vecto . d trong co . so . ’ d´o. (D S. d  4, − 5 3 , − 14 3  ) 5.3 Khˆong gian vecto . Euclid. Co . so . ’ tru . . c chuˆa ’ n Khˆong gian tuyˆe ´ n t´ınh thu . . c V du . o . . cgo . i l`a khˆong gian Euclid nˆe ´ u trong V du . o . . c trang bi . mˆo . t t´ıch vˆo hu . ´o . ng, t ´u . cl`anˆe ´ uv´o . imˆo ˜ ic˘a . p phˆa ` ntu . ’ x, y ∈Vd ˆe ` utu . o . ng ´u . ng v´o . imˆo . tsˆo ´ thu . . c (k´y hiˆe . ul`a x, y) sao cho ∀x, y, z ∈Vv`a sˆo ´ α ∈ R ph´ep tu . o . ng ´u . ng d ´o tho ’ a m˜an c´ac tiˆen dˆe ` sau dˆa y (I) x, y = y,x; (I I)  x + y,z = x, z + y,z; (I II) αx, y = αx, y; (IV) x, x > 0nˆe ´ u x = θ. Trong khˆong gian vecto . R n dˆo ´ iv´o . ic˘a . p vecto . a =(a 1 ,a 2 , ,a n ), 202 Chu . o . ng 5. Khˆong gian Euclide R n b =(b 1 ,b 2 , ,b n ) th`ı quy t˘a ´ ctu . o . ng ´u . ng a, b = n  i=1 a i b i = a 1 b 1 + a 2 b 2 + ···+ a n b n (5.12) s˜e x´ac di . nh mˆo . t t´ıch vˆo hu . ´o . ng cu ’ a hai vecto . a v`a b. Nhu . vˆa . y khˆong gian R n v´o . it´ıchvˆohu . ´o . ng x´ac di . nh theo cˆong th ´u . c (5.12) tro . ’ th`anh khˆong gian Euclid. Do d ´o khi n´oi vˆe ` khˆong gian Euclid R n ta luˆon luˆon hiˆe ’ u l`a t´ıch vˆo hu . ´o . ng trong d´o x´ac di . nh theo (5.12). Gia ’ su . ’ x ∈ R n . Khi d´osˆo ´  x, x du . o . . cgo . il`adˆo . d`ai (hay chuˆa ’ n) cu ’ a vecto . x v`a d u . o . . ck´yhiˆe . ul`ax.Nhu . vˆa . y x def =  x, x (5.13) Vecto . x v´o . id ˆo . d`ai = 1 du . o . . cgo . il`ad u . o . . c chuˆa ’ n h´oa hay vecto . d o . n vi . .D ˆe ’ chuˆa ’ n h´oa mˆo . t vecto . kh´ac θ bˆa ´ tk`y ta chı ’ cˆa ` n nhˆan n´o v´o . isˆo ´ λ = 1 x . Dˆo . d`ai c´o c´ac t´ınh chˆa ´ t 1 + x =0⇔ x = θ. 2 + λx = |λ|x, ∀λ ∈ R. 3 + |x, y|  xy (bˆa ´ td˘a ’ ng th ´u . c Cauchy-Bunhiakovski) 4 + x + y  x + y (bˆa ´ td˘a ’ ng th´u . c tam gi´ac hay bˆa ´ td˘a ’ ng th ´u . c Minkovski). T`u . bˆa ´ td˘a ’ ng th´u . c3 + suy r˘a ` ng v´o . i hai vecto . kh´ac θ bˆa ´ tk`yx, y ∈ R n ta dˆe ` uc´o |x, y| xcos y  1 ⇔−1  x, y xy  1. Sˆo ´ x, y xy c´o thˆe ’ xem nhu . cosin cu ’ a g´oc ϕ n`ao d´o. G´oc ϕ m`a cos ϕ = x, y xy , 0  ϕ  π (5.14) 5.3. Khˆong gian vecto . Euclid. Co . so . ’ tru . . cchuˆa ’ n 203 du . o . . cgo . il`ag´oc gi˜u . a hai vecto . x v`a y. Hai vecto . x, y ∈ R n du . o . . cgo . il`avuˆong g´oc hay tru . . c giao nˆe ´ ut´ıch vˆo hu . ´o . ng cu ’ ach´ung b˘a ` ng 0: x, y =0. Hˆe . vecto . a 1 ,a 2 , ,a m ∈ R n du . o . . cgo . il`ahˆe . tru . . c giao nˆe ´ uch´ung tru . . c giao t`u . ng dˆoi mˆo . t, t´u . cl`anˆe ´ u a i ,a j  =0∀i = j. Hˆe . vecto . a 1 ,a 2 , ,a m ∈ R n du . o . . cgo . il`ahˆe . tru . . c giao v`a chuˆa ’ n h´oa (hay hˆe . tru . . c chuˆa ’ n)nˆe ´ u a i ,a i  = δ ij =    0nˆe ´ u i = j 1nˆe ´ u i = j D - i . nh l ´y 5.3.1. Mo . ihˆe . tru . . c giao c´ac vecto . kh´ac khˆong d ˆe ` ul`ahˆe . dˆo . c lˆa . p tuyˆe ´ n t´ınh. Hˆe . gˆo ` m n vecto . E 1 , E 2 , ,E n ∈ R n du . o . . cgo . il`aco . so . ’ tru . . c giao nˆe ´ u n´o l`a mˆo . tco . so . ’ gˆo ` m c´ac vecto . tru . . c giao t`u . ng dˆoi mˆo . t. Trong khˆong gian R n tˆo ` nta . inh˜u . ng co . so . ’ d˘a . cbiˆe . ttiˆe . nlo . . idu . o . . c go . i l`a nh˜u . ng co . so . ’ tru . . c chuˆa ’ n (vai tr`o nhu . co . so . ’ D ˆec´ac vuˆong g´oc trong h`ınh ho . c gia ’ i t´ıch). Hˆe . gˆo ` m n vecto . E 1 , E 2 , ,E n ∈ R n du . o . . cgo . i l`a mˆo . t co . so . ’ tru . . c chuˆa ’ n cu ’ a R n nˆe ´ u c´ac vecto . n`ay t`u . ng d ˆoi mˆo . t tru . . c giao v`a d ˆo . d`ai cu ’ a mˆo ˜ i vecto . cu ’ ahˆe . dˆe ` ub˘a ` ng 1, t´u . cl`a (E i , E k )=    0nˆe ´ u i = k, 1nˆe ´ u i = k. D - i . nh l´y 5.3.2. Trong mo . i khˆong gian Euclid n-chiˆe ` udˆe ` utˆo ` nta . ico . so . ’ tru . . c chuˆa ’ n. Dˆe ’ c´o diˆe ` ud´o ta c´o thˆe ’ su . ’ du . ng ph´ep tru . . c giao h´oa Gram-Smidth du . amˆo . tco . so . ’ vˆe ` co . so . ’ tru . . cchuˆa ’ n. Nˆo . i dung cu ’ a thuˆa . t to´an d´o n h u . sau Gia ’ su . ’ E 1 = a 1 .Tiˆe ´ pd´o ph´ep du . . ng du . o . . ctiˆe ´ n h`anh theo quy na . p. 204 Chu . o . ng 5. Khˆong gian Euclide R n Nˆe ´ u E 1 , E 2 , ,E i d˜a d u . o . . cdu . . ng th`ı E i+1 c´o thˆe ’ lˆa ´ y E i+1 = a i+1 + i  j=1 α j a j , trong d´o α j = − a i+1 , E j  E j , E j  ,j= 1,i du . o . . ct`ımt`u . d iˆe ` ukiˆe . n E i+1 tru . . c giao v´o . imo . i vecto . E 1 , E 2 , ,E i . C ´ AC V ´ IDU . 1. Trong c´ac ph´ep to´an du . ´o . idˆay ph´ep to´an n`ao l`a t´ıch vˆo hu . ´o . ng cu ’ a hai vecto . x =(x 1 ,x 2 ,x 3 ), y =(y 1 ,y 2 ,y 3 ) ∈ R 3 : 1) x, y = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 ; 2) x, y = x 1 y 1 +2x 2 y 2 +3x 3 y 3 ; 3) x, y = x 1 y 1 + x 2 y 2 −x 3 y 3 . Gia ’ i. 1) Ph´ep to´an n`ay khˆong l`a t´ıch vˆo hu . ´o . ng v`ı n´o khˆong tho ’ a m˜an tiˆen d ˆe ` III cu ’ a t´ıch vˆo hu . ´o . ng: αx, y = α 2 x 2 1 y 2 1 + α 2 x 2 2 y 2 2 + α 2 x 2 3 y 2 3 = α(x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 ) 2) Ph´ep to´an n`ay l`a t´ıch vˆo hu . ´o . ng. Thˆa . tvˆa . y, hiˆe ’ n nhiˆen c´ac tiˆen d ˆe ` I v`a I I tho ’ a m˜an. Ta kiˆe ’ m tra c´ac tiˆen dˆe ` III v`a IV. Gia ’ su . ’ x  =(x  1 ,x  2 ,x  3 ), x  =(x  1 ,x  2 ,x  3 ) ∈ R 3 . Khi d´o x  + x  ,y =(x  1 + x  1 )y 1 +2(x  2 + x  2 )y 2 +3(x  3 + x  3 )y 3 =(x  1 y 1 +2x  2 y 2 +3x  3 y 3 )+(x  1 y 1 +2x  2 y 2 +3x  3 y 3 ) = x  ,y+ x  ,y. Tiˆe ´ p theo ta x´et x, x = x 2 1 +2x 2 2 +3x 2 3  0v`a x, x =0⇔ x 2 1 +2x 2 2 +3x 2 3 =0⇔ x 1 = x 2 = x 3 =0⇔ x = θ. [...]... a 1) a1 = (1, 1, 1, 1) , a2 = (1, 1, 1, 1) 1 1 1 1 1 1 1 1 (DS E1 = , − , , − , E2 = , , , , E3 = − 2 2 2 2 2 2 2 2 1 1 √ , 0, √ , 0 , 2 2 ’ ´ o ´ 5.4 Ph´p biˆn d ˆi tuyˆn t´ e e e ınh 213 1 1 E4 = 0, − √ , 0, √ ) 2 2 2) a1 = (1, 1, 1, 3), a2 = (1, 1, −3, 1) √ 1 3 1 1 √ ,− √ ,− √ , , 2 3 2 3 2 3 2 = √ 1 1 1 3 √ , √ ,− ,− √ , 2 2 3 2 3 2 3 2 1 1 E3 = √ , √ , √ , 0 , E4 = 6 6 6 (DS 5.4 5.4 .1 E1 E2... gian R4 : a ’ o a o a e 1) a1 = (1, 1, 1, 1) , a2 = (1, 1, −3, −3), a3 = (4, 3, 0, 1) 1 1 1 1 1 1 1 1 (DS E1 = , , , , E2 = , , − , − , E3 = 2 2 2 2 2 2 2 2 1 1 1 1 ,− , ,− ) 2 2 2 2 2) a1 = (1, 2, 2, 0), a2 = (1, 1, 3, 5), a3 = (1, 0, 1, 0) 1 5 1 1 2 2 (DS E1 = , , , 0 , E2 = 0, − √ , √ , √ , 3 3 3 3 3 3 3 3 3 17 8 5 6 E3 = √ , − √ , √ , − √ ) 78 3 78 3 78 3 78 ’ ` ’ a 10 Ch´.ng to r˘ng c´c hˆ vecto... Chu.o.ng 5 Khˆng gian Euclide o Rn 2 1 2 (DS E1 = a1 = (1, −2, 2); E2 = − , − , − ; E3 = (6, −3, −6)) 3 3 3 2) a1 = (1, 1, 1, 1) , a2 = (3, 3, 1, 1) , a3 = (−2, 0, 6, 8) (DS E1 = a1 = (1, 1, 1, 1) ; E2 = (2, 2, −2, −2), E3 = ( 1, 1, 1, 1) ) 3) a1 = (1, 1, 1, 1) ; a2 = (3, 3, 1, 1) ; a3 = ( 1, 0, 3, 4) − (DS E1 = a1 = (1, 1, 1, 1) , E2 = (2, 2, −2, −2), E3 = 7 7 1 1 , ,− , ) 2 2 2 2 ’ 9 Tru.c chuˆn h´a... − βy1 = αx1 , α(x2 − x1), α(x3 − x1) + βy1, β(y2 − y1), β(y3 − y2) = α(x1, x2 − x1 , x3 − x1 ) + β(y1, y2 − y1, y3 − y1 ) = αL(x) + βL(y) ´ ’ ´ Vˆy L l` ph´p biˆn dˆi phˆn tuyˆn t´ a a e e o a e ınh ’ ´ ’ e ı a ’ o o o 2+ Dˆ t`m ma trˆn cua L dˆi v´.i co so E1 , E2 , E3 ta c´ L(E1 ) = L (1, 1, 1) = (1, 1 − 1, 1 − 1) = (1, 0, 0) = E1 + 0 · E2 + 0 · E3 , L(E2 ) = L(0, 1, 1) = (0, 1, 1) = 0 · E1 + 1 ·... E1 = E1 = (1, 2, 1) Tiˆp theo d˘t ´ ´ ’ Giai 1) Tru o e e a c l` u E2 = E2 + λE1 sao cho E2 , E1 = 0, t´ a E2 , E1 = E1 , E2 + λ E1 , E1 = 0 ’ ’ 5.3 Khˆng gian vecto Euclid Co so tru.c chuˆn o a 207 ’ Nhu.ng E1 , E1 = 0 (cu thˆ l` > 0) v` E1 = E1 = θ Do d´ ı o e a λ=− (1, 2, 1) , (1, 1, 0) E1 , E2 1 =− =− · 2 + 22 + 12 E1 , E1 1 2 Do d´ o 1 1 1 , 0, − E2 = (1, 1, 0) − (1, 2, 1) = 2 2 2 ´ Tiˆp... + x2  = 1 1 0 x2  y3 x1 + x2 + x3 x3 1 1 1 v` do d´ a o   1 0 0   A = 1 1 0 1 1 1 ’ ´ o ´ 5.4 Ph´p biˆn d ˆi tuyˆn t´ e e e ınh ’ ’ ’ Gia su B l` ma trˆn cua L theo co so E1 , E2, E3 Khi d´ o a a ’     1 −2 1 1 0 0 2 1 0     1 B = TeE ATeE =  1 4 2  1 1 0  1 0 1 1 1 1 1 1 1 3 1 2   −4 −3 1   =  10 7 −2 3 2 0 ´ ’ ’ ’ a a e e o V´ du 4 Gia su L : R3 → R3 v` L∗... trˆn R2 : a a 1) a, b = a1b1 + a2b2 2) a, b = ka1b1 + a2b2 , k, = 0 3) a, b = a1b1 + a1b2 + a2 b1 4) a, b = 2a1b1 + a1b2 + a2b1 + a2b2 5) a, b = 3a1b1 + a1b2 + a2b1 − a2 b2 a a ıch o o (DS 1) , 2) v` 4) x´c dinh t´ vˆ hu.´.ng ıch o o 3) v` 5) khˆng x´c dinh t´ vˆ hu.´.ng) a o a 2 Trong khˆng gian Euclide R4, x´c dinh g´c gi˜.a c´c vecto.: o a o u a 5 1) a = (1, 1, 1, 1) , b = (3, 5, 1, 1) (DS arccos... ’ Giai 1 Ma trˆn chuyˆn t` a e u e l`: a    1 −2 1 2 1 0    1  =  1 0 1 ⇒ TeE  1 4 2  1 1 1 3 1 2  TeE ´ ’ ’ 1 2 3 a o ’ ’ Gia su (x∗ , x∗, x∗) l` toa dˆ cua x dˆi v´.i co so {E1 , E2 , E3 } Khi o o o d´          x∗ x1 1 −2 1 3 5 1  ∗      1   2   1 = −7 x2 = TeE x2  =  1 4 x∗ x3 1 1 1 0 −4 3 ´ ’ o o a a o Vˆy toa dˆ cua x dˆi v´.i co so E1 , E2... L(e1 ) v` L(e2 ) theo co so ch´ t˘c Ta c´ L(e1 ) = L (1, 0) = (1, 2 · 1) = L (1, 2) = 1 · e1 + 2 · e2, L(e2 ) = L(0, 1) = (1, 2 · 0) = L (1, 0) = 1 · e1 + 0 · e2 T` d´ thu du.o.c u o A= 1 1 2 0 ’ V´ du 2 X´t khˆng gian R3 v´.i co so E: E1 = (1, 1, 1) , E2 = (0, 1, 1) , ı e o o ’ ´ ’ ’ a a e e o a u E3 = (0, 0, 1) v` ph´p biˆn dˆi L : R3 → R3 x´c dinh bo.i d˘ng th´.c L[(u1 , u2, u3)] = (u1, u2 − u1 ,... e a o th`nh co so tru.c giao: ’ ’ sung cho c´c hˆ d´ dˆ tro a a e o e ’ 1) a1 = (1, −2, 1, 3), a2 = (2, 1, −3, 1) ’ a a (DS Ch˘ng han, c´c vecto a3 = (1, 1, 1, 0), a4 = ( 1, 1, 0, 1) ) 2) a1 = (1, 1, 1, −3), a2 = (−4, 1, 5, 0) ’ (DS Ch˘ng han, c´c vecto a3 = (2, 3, 1, 0) v` a4 = (1, 1, 1, 1) ) a a a ’ ` ’ a a a o a a 11 Ch´.ng to r˘ng c´c vecto sau dˆy trong R4 l` tru.c giao v` bˆ sung u ’ ’ . E 1 = a 1 = (1, 1, 1, 1) ; E 2 =(2, 2, −2, −2), E 3 =( 1, 1, 1, 1) ) 3) a 1 = (1, 1, 1, 1) ; a 2 =(3, 3, 1, 1) ; a 3 =( 1, 0, 3, 4). (DS. E 1 = a 1 = (1, 1, 1, 1) , E 2 =(2, 2, −2, −2), E 3 =  − 1 2 , 1 2 ,. 3); 19 8 Chu . o . ng 5. Khˆong gian Euclide R n a 4 =(0, 1, 2, 3). (DS. 2 3 , − 1 6 , 1 2 , 1) 3) a 1 = (1, 1, 1, 1) ; a 2 = (1, 1, 1, 1) ; a 3 = (1, 1, 1, 1) ; a 4 = (1, 1, 1, 1) . (DS. 3 2 , − 1 2 ,. h´oa c´ac co . so . ’ d ´o 1) a 1 = (1, 1, 1, 1) , a 2 = (1, 1, 1, 1) . (D S. E 1 =  1 2 , − 1 2 , 1 2 , − 1 2  , E 2 =  1 2 , 1 2 , 1 2 , 1 2  , E 3 =  − 1 √ 2 , 0, 1 √ 2 , 0  , 5.4. Ph´ep