Chu . o . ng 5 Khˆong gian Euclide R n 5.1 D - i . nh ngh˜ıa khˆong gian n-chiˆe ` uv`amˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 177 5.2 Co . so . ’ .D - ˆo ’ ico . so . ’ 188 5.3 Khˆong gian vecto . Euclid. Co . so . ’ tru . . c chuˆa ’ n201 5.4 Ph´ep biˆe ´ nd ˆo ’ i tuyˆe ´ nt´ınh .213 5.4.1 D - i . nhngh˜ıa 213 5.4.2 Ma trˆa . ncu ’ aph´epbdtt .213 5.4.3 C´ac ph´ep to´an . . . . . . . . . . . . . . . . 215 5.4.4 Vecto . riˆeng v`a gi´a tri . riˆeng . . . . . . . . . 216 5.1 D - i . nh ngh˜ıa khˆong gian n-chiˆe ` uv`a mˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 1 ◦ . Gia ’ su . ’ n ∈ N.Tˆa . pho . . pmo . ibˆo . c´o thˆe ’ c´o (x 1 ,x 2 , .,x n )gˆo ` m n sˆo ´ thu . . c (ph´u . c) d u . o . . cgo . il`akhˆong gian thu . . c (ph´u . c) n-chiˆe ` u v`a d u . o . . c 178 Chu . o . ng 5. Khˆong gian Euclide R n k´yhiˆe . ul`aR n (C n ). Mˆo ˜ ibˆo . sˆo ´ d´odu . o . . cchı ’ bo . ’ i x =(x 1 ,x 2 , .,x n ) v`a d u . o . . cgo . il`ad iˆe ’ m hay vecto . cu ’ a R n (C n ). C´ac sˆo ´ x 1 , .,x n du . o . . c go . il`ato . ad ˆo . cu ’ adiˆe ’ m (cu ’ a vecto . ) x hay c´ac th`anh phˆa ` ncu ’ a vecto . x. Hai vecto . x =(x 1 , .,x n )v`ay =(y 1 , .,y n )cu ’ a R n du . o . . c xem l`a b˘a ` ng nhau nˆe ´ u c´ac to . ad ˆo . tu . o . ng ´u . ng cu ’ ach´ung b˘a ` ng nhau x i = y i ∀ i = 1,n. C´ac vecto . x =(x 1 , .,x n ), y =(y 1 , .,y n ) c´o thˆe ’ cˆo . ng v´o . i nhau v`a c´o thˆe ’ nhˆan v´o . i c´ac sˆo ´ α,β, . l`a sˆo ´ thu . . cnˆe ´ u khˆong gian d u . o . . cx´et l`a khˆong gian thu . . cv`al`asˆo ´ ph´u . cnˆe ´ u khˆong gian d u . o . . cx´et l`a khˆong gian ph´u . c. Theo d i . nh ngh˜ıa: 1 + tˆo ’ ng cu ’ a vecto . x v`a y l`a vecto . x + y def =(x 1 + y 1 ,x 2 + y 2 , .,x n + y n ). (5.1) 2 + t´ıch cu ’ a vecto . x v´o . isˆo ´ α hay t´ıch sˆo ´ α v´o . i vecto . x l`a vecto . αx = xα def =(αx 1 ,αx 2 , .,αx n ). (5.2) Hai ph´ep to´an 1 + v`a 2 + tho ’ a m˜an c´ac t´ınh chˆa ´ t (tiˆen dˆe ` ) sau dˆay I. x + y = y + x, ∀ x, y ∈ R n (C n ), II. (x + y)+z = x +(y + z) ∀ x, y, z ∈= R n (C n ), III. Tˆo ` nta . i vecto . - khˆong θ =(0, 0, .,0    n ) ∈ R n sao cho x + θ = θ + x = x, IV. Tˆo ` nta . i vecto . d ˆo ´ i −x =(−1)x =(−x 1 ,−x 2 , .,−x n ) sao cho x +(−x)=θ, V. 1 · x = x, 5.1. D - i . nh ngh˜ıa khˆong gian n-chiˆe ` uv`amˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 179 VI. α(βx)=(αβ)x, α, β ∈ R (C), VII. (α + β)x = αx + βx, VIII. α(x + y)=αx + αy trong d ´o α v`a β l`a c´ac sˆo ´ , c`on x, y ∈ R n (C n ). D - i . nh ngh˜ıa 5.1.1. 1 + Gia ’ su . ’ V l`a tˆa . pho . . p khˆong rˆo ˜ ng t`uy ´y v´o . i c´ac phˆa ` ntu . ’ d u . o . . ck´yhiˆe . ul`ax,y,z, . Tˆa . pho . . p V d u . o . . cgo . i l`a khˆong gian tuyˆe ´ n t´ınh (hay khˆong gian vecto . ) nˆe ´ u ∀ x, y ∈Vx´ac d i . nh du . o . . c phˆa ` n tu . ’ x + y ∈V(go . i l`a tˆo ’ ng cu ’ a x v`a y)v`a∀ α ∈ R (C)v`a∀ x ∈Vx´ac d i . nh du . o . . c phˆa ` ntu . ’ αx ∈V(go . i l`a t´ıch cu ’ asˆo ´ α v´o . i phˆa ` ntu . ’ x) sao cho c´ac tiˆen d ˆe ` I-VIII du . o . . c tho ’ a m˜an. Khˆong gian tuyˆe ´ n t´ınh v´o . i ph´ep nhˆan c´ac phˆa ` ntu . ’ cu ’ an´ov´o . i c´ac sˆo ´ thu . . c (ph´u . c) d u . o . . cgo . i l`a khˆong gian tuyˆe ´ n t´ınh thu . . c (tu . o . ng ´u . ng: ph´u . c). Khˆong gian R n c´o thˆe ’ xem nhu . mˆo . tv´ıdu . vˆe ` khˆong gian tuyˆe ´ n t´ınh, c´ac v´ı du . kh´ac s˜e d u . o . . cx´et vˆe ` sau. V`a trong gi´ao tr`ınh n`ay ta luˆon gia ’ thiˆe ´ tr˘a ` ng c´ac khˆong gian d u . o . . cx´et l`a nh˜u . ng khˆong gian thu . . c. 2 ◦ . Cho hˆe . gˆo ` m m vecto . n-chiˆe ` u x 1 ,x 2 , .,x m . (5.3) Khi d ´o vecto . da . ng y = α 1 x 1 + α 2 x 2 + ···+ α m x m ; α 1 ,α 2 , .,α m ∈ R. d u . o . . cgo . il`atˆo ’ ho . . p tuyˆe ´ nt´ınh cu ’ a c´ac vecto . d ˜a cho hay vecto . y biˆe ’ u diˆe ˜ n tuyˆe ´ n t´ınh d u . o . . c qua c´ac vecto . (5.3). D - i . nh ngh˜ıa 5.1.2. 1 + Hˆe . vecto . (5.3) d u . o . . cgo . il`ahˆe . d ˆo . clˆa . p tuyˆe ´ n t´ınh (d ltt) nˆe ´ ut`u . d ˘a ’ ng th´u . c vecto . λ 1 x 1 + λ 2 x 2 + ···+ λ m x m = θ (5.4) k´eo theo λ 1 = λ 2 = ··· = λ m =0. 180 Chu . o . ng 5. Khˆong gian Euclide R n 2 + Hˆe . (5.3) go . il`ahˆe . phu . thuˆo . c tuyˆe ´ n t´ınh (pttt) nˆe ´ utˆo ` nta . i c´ac sˆo ´ λ 1 ,λ 2 , .,λ m khˆong dˆo ` ng th`o . ib˘a ` ng 0 sao cho d ˘a ’ ng th´u . c (5.4) d u . o . . c tho ’ a m˜an. Sˆo ´ nguyˆen du . o . ng r d u . o . . cgo . il`aha . ng cu ’ ahˆe . vecto . (5.3) nˆe ´ u a) C´o mˆo . ttˆa . pho . . p con gˆo ` m r vecto . cu ’ ahˆe . (5.3) lˆa . p th`anh hˆe . d ltt. b) Mo . itˆa . p con gˆo ` m nhiˆe ` uho . n r vecto . cu ’ ahˆe . (5.3) d ˆe ` u phu . thuˆo . c tuyˆe ´ n t´ınh. D ˆe ’ t`ım ha . ng cu ’ ahˆe . vecto . ta lˆa . p ma trˆa . n c´ac to . ad ˆo . cu ’ an´o A =       a 11 a 12 . a 1n a 21 a 22 . a 2n . . . . . . . . . . . . a m1 a m2 . a mn       D - i . nh l´y. Ha . ng cu ’ ahˆe . vecto . (5.3) b˘a ` ng ha . ng cu ’ a ma trˆa . n A c´ac to . a d ˆo . cu ’ a n´o. T`u . d ´o, dˆe ’ kˆe ´ t luˆa . nhˆe . vecto . (5.3) d ltt hay pttt ta cˆa ` nlˆa . p ma trˆa . n to . ad ˆo . A cu ’ ach´ung v`a t´ınh r(A): 1) Nˆe ´ u r(A)=m th`ı hˆe . (5.3) d ˆo . clˆa . p tuyˆe ´ n t´ınh. 2) Nˆe ´ u r(A)=s<mth`ı hˆe . (5.3) phu . thuˆo . c tuyˆe ´ n t´ınh. C ´ AC V ´ IDU . V´ı d u . 1. Ch´u . ng minh r˘a ` ng hˆe . vecto . a 1 ,a 2 , .,a m (m>1) phu . thuˆo . c tuyˆe ´ n t´ınh khi v`a chı ’ khi ´ıt nhˆa ´ tmˆo . t trong c´ac vecto . cu ’ ahˆe . l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . c`on la . i. Gia ’ i. 1 + Gia ’ su . ’ hˆe . a 1 ,a 2 , .,a m phu . thuˆo . c tuyˆe ´ n t´ınh. Khi d´o tˆo ` nta . i c´ac sˆo ´ α 1 ,α 2 , .,α m khˆong dˆo ` ng th`o . ib˘a ` ng 0 sao cho α 1 a 1 + α 2 a 2 + ···+ α m a m = θ. Gia ’ su . ’ α m = 0. Khi d´o a m = β 1 a 1 + β 2 a 2 + ···+ β m−1 a m−1 ,β i = α i α m 5.1. D - i . nh ngh˜ıa khˆong gian n-chiˆe ` uv`amˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 181 t´u . cl`aa m biˆe ’ udiˆe ˜ n tuyˆe ´ n t´ınh qua c´ac vecto . c`on la . i. 2 + Ngu . o . . cla . i, ch˘a ’ ng ha . nnˆe ´ u vecto . a m biˆe ’ udiˆe ˜ n tuyˆe ´ n t´ınh qua a 1 ,a 2 , .,a m−1 a m = β 1 a 1 + β 2 a 2 + ···+ β m−1 a m−1 th`ı ta c´o β 1 a 1 + β 2 a 2 + ···+ β m−1 a m−1 +(−1)a m = θ. Do d ´ohˆe . d˜a cho phu . thuˆo . c tuyˆe ´ n t´ınh v`ı trong d˘a ’ ng th´u . ctrˆenc´ohˆe . sˆo ´ cu ’ a a m l`a kh´ac 0 (cu . thˆe ’ l`a = −1).  V´ı d u . 2. Ch´u . ng minh r˘a ` ng mo . ihˆe . vecto . c´o ch´u . a vecto . -khˆong l`a hˆe . phu . thuˆo . c tuyˆe ´ n t´ınh. Gia ’ i. Vecto . - khˆong luˆon luˆon biˆe ’ udiˆe ˜ nd u . o . . cdu . ´o . ida . ng tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . a 1 ,a 2 , .,a m : θ =0· a 1 +0· a 2 + ···+0· a m Do d´o theo di . nh ngh˜ıa hˆe . θ, a 1 , .,a m phu . thuˆo . c tuyˆe ´ n t´ınh (xem v´ı du . 1).  V´ı d u . 3. Ch´u . ng minh r˘a ` ng mo . ihˆe . vecto . c´o ch´u . a hai vecto . b˘a ` ng nhau l`a hˆe . phu . thuˆo . c tuyˆe ´ n t´ınh. Gia ’ i. Gia ’ su . ’ trong hˆe . a 1 ,a 2 , .,a n c´o hai vecto . a 1 = a 2 . Khi d´o ta c´o thˆe ’ viˆe ´ t a 1 =1· a 2 +0· a 3 + ···+0· a m t´u . c l`a vecto . a 1 cu ’ ahˆe . c´o thˆe ’ biˆe ’ udiˆe ˜ ndu . ´o . ida . ng tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . c`on la . i. Do d ´o h ˆe . phu . thuˆo . c tuyˆe ´ n t´ınh (v´ı du . 1).  V´ı d u . 4. Ch´u . ng minh r˘a ` ng nˆe ´ uhˆe . m vecto . a 1 ,a 2 , .,a m dˆo . clˆa . p tuyˆe ´ n t´ınh th`ı mo . ihˆe . con cu ’ ahˆe . d ´oc˜ung dˆo . clˆa . p tuyˆe ´ n t´ınh. Gia ’ i. D ˆe ’ cho x´ac di . nh ta x´et hˆe . con a 1 ,a 2 , .,a k , k<mv`a ch´u . ng minh r˘a ` ng hˆe . con n`ay d ˆo . clˆa . p tuyˆe ´ n t´ınh. 182 Chu . o . ng 5. Khˆong gian Euclide R n Gia ’ su . ’ ngu . o . . cla . i: hˆe . con a 1 ,a 2 , .,a k phu . thuˆo . c tuyˆe ´ n t´ınh. Khi d ´o ta c´o c´ac d˘a ’ ng th´u . c vecto . α 1 a 1 + α 2 a 2 + ···+ α k a k = θ trong d ´o c´o ´ıt nhˆa ´ tmˆo . t trong c´ac hˆe . sˆo ´ α 1 ,α 2 , .,α k kh´ac 0. Ta viˆe ´ t d ˘a ’ ng th´u . cd ´odu . ´o . ida . ng α 1 a 1 + α 2 A 2 + ···+ α k a k + α k+1 a k+1 + ···+ α m a m = θ trong d ´o ta gia ’ thiˆe ´ t α k+1 =0, .,α m =0. D˘a ’ ng th´u . c sau c`ung n`ay ch´u . ng to ’ hˆe . a 1 ,a 2 , .,a m phu . thuˆo . c tuyˆe ´ n t´ınh. Mˆau thuˆa ˜ n.  V´ı d u . 5. Ch´u . ng minh r˘a ` ng hˆe . vecto . cu ’ a khˆong gian R n e 1 =(1, 0, .,0), e 2 =(0, 1, .,0), . . . . e n =(0, .,0, 1) l`a d ˆo . clˆa . p tuyˆe ´ n t´ınh. Gia ’ i. T`u . d ˘a ’ ng th´u . c vecto . α 1 e 1 + α 2 e 2 + ···+ α n e n = θ suy ra r˘a ` ng (α 1 ,α 2 , .,α n )=(0, 0, .,0) ⇒ α 1 = α 2 = ···= α n =0. v`a do d ´ohˆe . e 1 ,e 2 , .,e n dˆo . clˆa . p tuyˆe ´ n t´ınh.  V´ı d u . 6. Ch´u . ng minh r˘a ` ng mo . ihˆe . gˆo ` m n + 1 vecto . cu ’ a R n l`a hˆe . phu . thuˆo . c tuyˆe ´ n t´ınh. Gia ’ i. Gia ’ su . ’ n + 1 vecto . cu ’ ahˆe . l`a: a 1 =(a 11 ,a 21 , .,a n1 ) a 2 =(a 12 ,a 22 , .,a n2 ) . . . . a n+1 =(a 1,n+1 ,a 2,n+1 , .,a n,n+1 ). 5.1. D - i . nh ngh˜ıa khˆong gian n-chiˆe ` uv`amˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 183 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 nhˆa ´ t n phu . o . ng tr`ınh v´o . i(n +1) ˆa ’ n nˆen hˆe . c´o nghiˆe . m khˆong tˆa ` mthu . `o . ng v`a (x 1 ,x 2 , .,x n ,x n+1 ) =(0, 0, .,0). Do d ´o theo di . nh ngh˜ıa hˆe . d˜a x´et l`a phu . thuˆo . c tuyˆe ´ n t´ınh.  V´ı d u . 7. T`ım ha . ng cu ’ ahˆe . vecto . trong R 4 a 1 =(1, 1, 1, 1); a 2 =(1, 2, 3, 4); a 3 =(2, 3, 2, 3); a 4 =(2, 4, 5, 6). Gia ’ i. Ta lˆa . p ma trˆa . n c´ac to . ad ˆo . v`a t`ım ha . ng cu ’ a n´o. Ta c´o A =      1111 1234 2323 3456      h 2 − h 1 → h  2 h 3 − 2h 1 → h  3 h 4 − 3h 1 → h  4 −→      1111 0123 0101 0123      h 3 − h 2 → h  3 h 4 − h 2 → h  4 → −→      11 1 1 01 2 3 00−2 −3 00 0 0      . T`u . d ´o suy r˘a ` ng r(A) = 3. Theo di . nh l´yd˜a nˆeu ha . ng cu ’ ahˆe . vecto . b˘a ` ng 3.  184 Chu . o . ng 5. Khˆong gian Euclide R n V´ı d u . 8. Kha ’ o s´at su . . phu . thuˆo . c tuyˆe ´ n t´ınh gi˜u . a c´ac vecto . cu ’ a R 4 : a 1 =(1, 4, 1, 1); a 2 =(2, 3,−1, 1); a 3 =(1, 9, 4, 2); a 4 =(1,−6,−5,−1). Gia ’ i. Lˆa . p ma trˆa . n m`a c´ac h`ang cu ’ a n´o l`a c´ac vecto . d ˜a cho v`a t`ım ha . ng cu ’ an´o S =      1411 23−11 1942 1 −6 −5 −1      ⇒ r(A)=2. Do d ´oha . ng cu ’ ahˆe . vecto . b˘a ` ng 2. V`ı c´ac phˆa ` ntu . ’ cu ’ ad i . nh th´u . c con ∆=      14 23      = −5 =0 n˘a ` mo . ’ hai h`ang d ˆa ` unˆena 1 v`a a 2 dˆo . clˆa . p tuyˆe ´ n t´ınh, c`on a 3 v`a a 4 biˆe ’ u diˆe ˜ n tuyˆe ´ n t´ınh qua a 1 v`a a 2 . [Lu . u´yr˘a ` ng mo . ic˘a . p vecto . cu ’ ahˆe . d ˆe ` u d ˆo . clˆa . p tuyˆe ´ n t´ınh v`ı ta c´o c´ac di . nh th´u . c con cˆa ´ p hai sau d ˆay =0:      14 19      ,      14 1 −6      ,      23 19      ,      23 1 −6      ,      19 1 −6      .] Ta t`ım c´ac biˆe ’ uth´u . cbiˆe ’ udiˆe ˜ n a 3 v`a 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 ngh˜ıa khˆong gian n-chiˆe ` uv`amˆo . tsˆo ´ kh´ai niˆe . mco . ba ’ nvˆe ` vecto . 185 v`a thu du . o . . chˆe . phu . o . ng tr`ınh ξ 1 +2ξ 2 =1, 4ξ 1 +3ξ 2 =9, ξ 1 − ξ 2 =4, ξ 1 + ξ 2 =2.          Ta ha . n chˆe ´ hai phu . o . ng tr`ınh d ˆa ` u. Di . nh th´u . ccu ’ ac´achˆe . sˆo ´ cu ’ a hai phu . o . ng tr`ınh n`ay ch´ınh l`a d i . nh th´u . c ∆ chuyˆe ’ nvi . .V`ı∆= 0 nˆen hˆe . hai phu . o . ng tr`ınh ξ 1 +2ξ 2 =1 4ξ 1 +3ξ 2 =9 c´o nghiˆe . m duy nhˆa ´ tl`aξ 1 =3,ξ 2 = −1. Do d´o a 3 =3a 1 − a 2 . Tu . o . ng tu . . ta c´o a 4 =2a 2 − 3a 1 .  B ` AI T ˆ A . P 1. Ch´u . ng minh r˘a ` ng trong khˆong gian R 3 : 1) Vecto . (x, y, z) l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . e 1 =(1, 0, 0), e 2 =(0, 1, 0), e 3 =(0, 0, 1). 2) Vecto . x =(7, 2, 6) l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . a 1 = (−3, 1, 2), a 2 =(−5, 2, 3), a 3 =(1,−1, 1). 2. H˜ay x´ac d i . nh sˆo ´ λ dˆe ’ vecto . x ∈ R 3 l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . a 1 ,a 2 ,a 3 ∈ R 3 nˆe ´ u: 1) x =(1, 3, 5); a 1 =(3, 2, 5); a 2 =(2, 4, 7); a 3 =(5, 6,λ). 186 Chu . o . ng 5. Khˆong gian Euclide R n (DS. λ = 12) 2) x =(7,−2,λ); a 1 =(2, 3, 5); a 2 =(3, 7, 8); a 3 =(1,−6, 1). (D S. λ = 15) 3) x =(5, 9,λ); a 1 =(4, 4, 3); a 2 =(7, 2, 1); a 3 =(4, 1, 6). (D S. ∀ λ ∈ R) 3. Ch´u . ng minh r˘a ` ng trong khˆong gian R 3 : 1) Hˆe . ba vecto . e 1 =(1, 0, 0), e 2 =(0, 1, 0), e 3 =(0, 0, 1) l`a hˆe . dltt. 2) Nˆe ´ u thˆem vecto . x ∈ R 3 bˆa ´ tk`y v`ao hˆe . th`ı hˆe . {e 1 ,e 2 ,e 3 ,x} l`a phu . thuˆo . c tuyˆe ´ n t´ınh. 3) Hˆe . gˆo ` mbˆo ´ n vecto . bˆa ´ tk`ycu ’ a R 3 l`a pttt. 4. C´ac hˆe . vecto . sau d ˆay trong khˆong gian R 3 l`a dltt hay pttt: 1) a 1 =(1, 2, 1); a 2 =(0, 1, 2); a 3 =(0, 0, 2). (DS. Dltt) 2) a 1 =(1, 1, 0); a 2 =(1, 0, 1); a 3 =(1,−2, 0). (DS. Dltt) 3) a 1 =(1, 3, 3); a 2 =(1, 1, 1); a 3 =(−2,−4,−4). (DS. Pttt) 4) a 1 =1,−3, 0); a 2 =(3,−3, 1); a 3 =(2, 0, 1). (DS. Pttt) 5) a 1 =(2, 3, 1); a 2 =(1, 1, 1); a 3 =(1, 2, 0). (DS. Pttt) 5. Gia ’ su . ’ v 1 , v 2 v`a v 3 l`a hˆe . dˆo . clˆa . p tuyˆe ´ n t´ınh. Ch´u . ng minh r˘a ` ng hˆe . sau d ˆay c˜ung l`a dltt: 1) a 1 = v 1 + v 2 ; a 2 = v 1 + v 3 ; a 3 = v 1 − 2v 2 . 2) a 1 = v 1 + v 3 ; a 2 = v 3 − v 1 ; a 3 = v 1 + v 2 − v 3 . 6. Ch´u . ng minh r˘a ` ng c´ac hˆe . vecto . sau d ˆay l`a phu . thuˆo . c tuyˆe ´ n t´ınh. D ˆo ´ iv´o . ihˆe . vecto . n`ao th`ı vecto . b l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh cu ’ a c´ac vecto . c`on la . i? 1) a 1 =(2, 0,−1), a 2 =(3, 0,−2), a 3 =(−1, 0, 1), b =(1, 2, 0). (D S. b khˆong l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh) 2) a 1 =(−2, 0, 1), a 2 =(1,−1, 0), a 3 =(0, 1, 2); b =(2, 3, 6). (D S. b l`a tˆo ’ ho . . p tuyˆe ´ n t´ınh) [...]... ´ (IV) x, x > 0 nˆu x = θ e ´ o o a Trong khˆng gian vecto Rn dˆi v´.i c˘p vecto a = (a1, a2, , an ), o 201 o Chu.o.ng 5 Khˆng gian Euclide 202 Rn ´ ´ b = (b1, b2, , bn ) th` quy t˘c tu.o.ng u.ng ı a n ai bi = a1 b1 + a2 b2 + · · · + an bn a, b = (5.12) i=1 ’ a s˜ x´c dinh mˆt t´ vˆ hu.´.ng cua hai vecto a v` b e a o ıch o o vˆy khˆng gian Rn v´.i t´ch vˆ hu.´.ng x´c dinh theo cˆng Nhu a... 5.2.1 Trong khˆng gian Rn : y o ´ ´ ’ a 1) Toa dˆ cua mˆt vecto dˆi v´.i mˆt co so l` duy nhˆt o a o ’ o o o ’ ’ -o ’ 5.2 Co so Dˆi co so 189 ` ` a ’ ’ o 2) Moi hˆ dltt gˆm n vecto dˆu lˆp th`nh co so cua khˆng gian o e a e n R ’ ´ e ’ Ta x´t vˆn dˆ: Khi co so thay dˆi th` toa dˆ cua mˆt vecto trong e a ` o ı o ’ o n ’i thˆ n`o ? ´ a khˆng gian R thay dˆ e o o trong khˆng gian Rn c´ hai co so... phu thuˆc tuyˆn t´ u a b 2 V´ du 4 Hˆ c´c vecto do.n vi trong Rn v´.i t´ vˆ hu.´.ng (5.12) ı e a o ıch o o e1 = (1, 0, 0, , 0) e2 = (0, 1, 0, , 0) en = (0, 0, 0, , 1) o Chu.o.ng 5 Khˆng gian Euclide 206 Rn ’ ’ ’ a a ’ a l` mˆt v´ du vˆ co so tru.c chuˆn trong Rn Co so n`y goi l` co so a o ı ` e ´ ch´nh t˘c trong Rn ı a ’ ’ Giai Hiˆn nhiˆn ei , ej = 0 ∀ i = j, ej = 1 ∀ j = 1,... khˆng gian u o Euclide l` t`.ng dˆi mˆt tru.c giao th` a u ı o o a1 + a2 + · · · + am 2 = a1 2 + a2 2 + · · · + am 2 ˜ ’ a Chı dˆ n X´t t´ vˆ hu.´.ng e ıch o o a1 + a2 + · · · + am , a1 + a2 + · · · + am ´ ´ o o o a e 8 Ap dung qu´ tr` tru.c giao h´a dˆi v´.i c´c hˆ vecto sau dˆy cua a ınh a ’ Rn : 1) a1 = (1, −2, 2), a2 = (−1, 0, −1), a3 = (5, −3, −7) 212 o Chu.o.ng 5 Khˆng gian Euclide Rn 2... Rn du.o.c goi l` mˆt co so cua n´ nˆu ´ ’ ’ o e gian vecto a o a e 1) hˆ E1 , E2 , , En l` hˆ dltt; e x ∈ Rn dˆu biˆu diˆn tuyˆn t´ du.o.c qua c´c vecto ’ ˜ ´ ` 2) moi vecto a e e e e ınh ’ e cua hˆ E1 , , En ´ ’ ’ Ch´ y r˘ng co so cua Rn l` mˆt hˆ c´ th´ tu bˆt k` gˆm n vecto u´ ` a o a o e o u a y ` ´ o a e ınh ’ o dˆc lˆp tuyˆn t´ cua n´ ` Diˆu kiˆn 2) c´ ngh˜ r˘ng ∀ x ∈ Rn. .. ’ ta lˆp ma trˆn m` cˆt th´ i cua n´ l` c´c toa dˆ cua vecto th´ i cua a a a o u u so m´.i trong co so c˜ D´ ch´ l` ma trˆn chuyˆn ’ ’ u o ınh a co ’ o a e o Chu.o.ng 5 Khˆng gian Euclide 190 Rn ’ ’ Gia su vecto a ∈ Rn v` a a = x1 ε1 + x2ε2 + · · · + xn εn , a = y1 E1 + y2E2 + · · · + yn En ´ ’ Khi d´ quan hˆ gi˜.a c´c toa dˆ cua c`ng mˆt vecto dˆi v´.i hai co so o e u a o ’ u o o o o ’ kh´c... moi khˆng gian Euclid n-chiˆu dˆu tˆn tai co y o e e o tru.c chuˆn ’ ’ a so ’ ’ e o e o ` o e o e ’ Dˆ c´ diˆu d´ ta c´ thˆ su dung ph´p tru.c giao h´a Gram-Smidth a mˆt co so vˆ co so tru.c chuˆn Nˆi dung cua thuˆt to´n d´ nhu ’ ’ ` ’ ’ du o e a o a a o sau ´ ´ ’ ’ e a e o e Gia su E1 = a1 Tiˆp d´ ph´p du.ng du.o.c tiˆn h`nh theo quy nap 203 o Chu.o.ng 5 Khˆng gian Euclide 204 Rn ’ ´ ´... −4 1 −2 5 2 11 ’ trong co so E2 , E1 Do d´ x1 = , x2 = o 5 5 ’ a o a V´ du 8 Trong khˆng gian R3 cho co so E1 , E2 , E3 n`o d´ v` trong ı o so d´ c´c vecto E1 , E2 , E3 v` x c´ toa dˆ l` E1 = (1, 1, 1); E2 = co ’ o a a o o a (1, 2, 2), E3 = (1, 1, 3) v` x = (6, 9, 14) a o Chu.o.ng 5 Khˆng gian Euclide 196 Rn ` ’ a 1+ Ch´.ng minh r˘ng E1 , E2 , E3 c˜ng lˆp th`nh co so trong R3 u u a a ’ 2+ T`... 17, 6) Nhu vˆy ta d˜ bˆ sung thˆm u o a a o e x3, x4 v` thu du.o.c hˆ vecto tru.c giao x1, x2, x3 , x4 trong hai vecto e a ` u D´ l` co so tru.c giao ’ khˆng gian 4-chiˆ o e o a ` ˆ BAI TAP 209 o Chu.o.ng 5 Khˆng gian Euclide 210 Rn ’ ’ u ´ ’ 1 Gia su a = (a1, a2), b = (b1 , b2) l` nh˜.ng vecto t`y y cua R2 Trong a u ´ ´ c´c quy t˘c sau dˆy, quy t˘c n`o x´c dinh t´ vˆ hu.´.ng trˆn R2 : a a... (1, 1, , 0), En = (1, 1, , 1) ’ l` mˆt co so trong Rn a o ` a e 4 Ch´.ng minh r˘ng hˆ vecto u E1 = (1, 2, 3, , n − 1, n), E2 = (1, 2, 3, , n − 1, 0), En = (1, 0, 0, , 0, 0) ’ o lˆp th`nh co so trong khˆng gian Rn a a ’ ˜ e ’ 5 H˜y kiˆm tra xem mˆ i hˆ vecto sau dˆy c´ lˆp th`nh co so trong a e o a o a a khˆng gian R4 khˆng v` t` c´c toa dˆ cua vecto x = (1, 2, 3, 4) trong . . cnˆe ´ u khˆong gian d u . o . . cx´et l`a khˆong gian thu . . cv`al`asˆo ´ ph´u . cnˆe ´ u khˆong gian d u . o . . cx´et l`a khˆong gian ph´u . c. Theo. c) d u . o . . cgo . il`akhˆong gian thu . . c (ph´u . c) n-chiˆe ` u v`a d u . o . . c 178 Chu . o . ng 5. Khˆong gian Euclide R n k´yhiˆe . ul`aR n (C