DANG TOAN PHUONG - KHONG GIAN VEC TO OCLIT VA KHONG GIAN VEC TO UNITA
A. TOM TAT lit THUYET
DANG SONG TUYEN TINH DOI XUNG
VA DANG TOAN PHUONG
1. Dinh nghia
Gia sit V la khong gian vele to trial truang so th0c Wit dang song tuyeal tinh xac dinh tren V la mat tinh xa
0:VxV—>R (x, y) H) 0 (x, y) sao cho vdi 662 kjc x, y, z e V va 7. e 7f& ta ca:
0 (x + z, y) = O (x. y) + 0 (z, y) 0 (x, y + z) = 0 (x, y) + 0 (x. z) 0 (X x. y) 0 (x, y)
0 ( x, y) = 0 (x. y) 0 (x, y) = 0 (y, x)
Neu vdi moi x, y e V ta co 0(x, y) = 0(y, x) thi 0 (bloc gyi la dth
Ngu vdi moi x e V ta 66 0(x, x) = 0 till 0 dmic gni la phOn cl6i xung.
2. Bieu thik toa do cita clang song tuyen tinh
Gia sn dim V = n, e = (e) = 1, 2, n la cd sa cila V. Anh xa song tuy6n tinh B hohn toan (Liao the dinh bai ma trail A = (a 0), a do a i, = 0 (0,, e i).
Khi do voi x= , y= y, e„ , ta co 0 (x, Y)= Lao xi Y1 i=1
hay yiet dual (bong ma tran 0(x, y) = .A.Y, a do X, Y la the
ma ran tr neat cac tya dO cua x, y trong co sa (e i).
Gia s& e = (E,), j = 1, 2, n la mOt cd sa khac ciaa V ma
CiiCi . Dang song tuygn tinh 0 trong cd so e = (au có ma ,=1
trim A, va trong cd sa s = (8) có ma Ran A', to co A' = . A . C,
a do C = (c, J).
Rang song tuy6n tinh B la do'i xi:mg khi va chi khi trong co sa 0 = (e) nao do, 0 co ma tran del xiing.
3. Dang -than pinning
N6u B: V x V —> R la dang song tuyeh tinh del xung, thi H: V R, H (x) = B (x, x) dude goi la clang loan phudng le6t hop voi 0, con B dude goi la dang eqe vim H. Trong mot cd sa da chon, ma tran cua B cling dude goi la ma tran cua dang toan phudng H ling voi nO. N6u bi6t dang toan phudng H, thi clang cue 0 dm H hoan Wan dude the dinh, Cu thO:
•
0 (x, y)= 2 (H(x + y)-H(x)-HGTD
Gia sai trong co sd e = (a), 0 (x, y)= Za ii x i yi , thi H(4= Za ii x i yi a do (x„ xx ) va (y1 , yd IA toa dO cam x va y Wong N6u trong 100 cd sọ nao do a = (e), dang than phudng H co dang Ii(4= D i x?, LW cd sa a = (th dime goi la cd sd chinh tac d6i vdi H, va to not trong cd so e = (th, H co dang chinh the. Ta cling not cd sa e = (ci) a tren IA cd sd true giao (180 voi dang cuc 0. Ta có dinh 15/
Dinh 13i: Neu H IA mot dang than phuong bat kjI tren khang gian vec to thuc n chigu V thi trong V luon ton tai mot ea sa a = ( 0 a trong co stl do, H co dang chinh tac.
Chu $': TR. cling có dinh nghia dang toan phudng tren killing gian vec td V tren tthang K tuy s ', gan viii dang song tuy6n Huh d6i /ding the chi-1h tren V.
4. Rang va hach caa dang toan phudng
Cho clang Loan phudng H tren khang gian vec td thuc n chik. Gia six trong mot cd sa nao do, dang toan piniong H co ma tran A vol rang A = r. Trong mot cd sa khac vdi ma tran chuyk cd sa C, thi H có ma trait A' = . A . C, rang A = rang N = r. S6 r kh8ng phu thutic vac) co sa dang xet va dupe goi IA hang ciia dang toan phttong H, cling dupe goi IA hang ctia dang cuc 0 caa H.
Khi hang H = n = dim V, dang than phudng H dude goi IA khang suy
N6u V = V, S V2 ma O (x, y) = 0 vdi moi x E V, va y E V2 thi a n6i V la t6ng trttc ti6p trtIc giao cern V, va V2 (d6i vei 0) va ki riOu la V =V, ® V2 . T6ng quAt, to co khai niem tong trttc ti6p
rye giao: V = e V, e e r Vk
Ta goi hoch cim H (hay hack) cim 0) la tap V 0 = {x e (x, y) = 0 vdi moi y e V4 Day la mot khong gian con am V ma
ang H = dim V - dim Vo va vdi moi phCin bu tuy6n tinh W ctia trong V, thi 0 lion ch6 teen W la khong suy biers Va
3 = W
5. Dinh lY chi se quail tinh va dinh Hi Sylvester
Gia su fi la mot don ft 'man phvong teen 14.-khOng gian vec to V.
H &toe goi la xac dinh n6u Mix) * 0 v6i moi x s O.
H dti0c goi la xac dioh doting nen I4(x) > 0 \FM moi x x 0.
H dvoc goi IA xac dinh Am n6u 14(x) < 0 veli moi x* 0.
Dinh 19
Cho V la R. - khong gian vec to n chi6u. H la dung town hiving tren V thi V la tong true 061) true giao (doi v6i H) clia a khong gian Vo, V,, Vs
V = V. CS V. ff) Vo ma HIV, la xac (huh during, HIV_ la xac inh am, H I V0 = 0. Cach phan tich tren khong duy nhal nhvng
o luon ]a hod) cem H, dim V. = p, dim V. = q khong (16i; p, q leo OM to goi la chi s6 dtiong van 'al-1h, chi s6 am (man tinh Oa H (hay ens clang eve 0 cim no).
Dinh 19 tren dude goi IA dinh 19 chi s6 quan tinh.
Dinh ly Sylvester
Gia sit V la R - khong gian vec to n chigu. H la clang to pfuldng tren V, A la ma tran cua clang Loan phudng H tro mat cd sa nao do Goi Ak la ma trail con ;oiling cap k a goc tr ben trai cim ma trail A (Ak tao bai giao cua k clang <tau va k ( dau elm ma trail A). Khi do:
H Ea clang toan pinicing xac dinh dining khi va chi k det Ak > 0 vdi rani k = 1, 2, n.
H le dang toan phuong xac dinh am khi va chi khi det Ak %
vdi k than va det Ak < 0 vdi k Le.
Chu 9: Khi H có ma tran dti). 'clang A, n6u H )(lc dinh duct to cling noi ma trail A 'the dinh ducing.
§ 2 KH6NG WAN VEC TO OCLIT 1. Dinh nghia
Cho E la khong gian vec to tren truiing s6 tlnic R. Ta mat tich ve hudng a tren E la mkt anh xa song tuy6n tinh, xilng va xac dinh dining lien E, ki hiou ban <,> hay (. , .), ngh la to co: < >; E x E R la anh xa song tuy6n tinh
thOa man (x, x) > 0 vdi moi x E 1E va <x, x> = 0 suy ra x = 0.
Khong gian vec td E cling vdi mot tich va hung xac dii tren E dnpc goi la nEat khong gian vec Ed (Mit.
2. Mc).t so tinh chat
a) Ta goi la chudn elm x e 1E, kr hiku ricfi , s6 thcic khong x, >
Ta có halt clang theic sau, goi la bat clang these Cos Bunhiacopski:
<, >2 < N2 MYM2 •
b) Hai vec td x, y &tee goi IA true giao vol nhau n6u <x, y> = 0.
Khi do to co:
)1 2 = D1M 2 M2 •
3. Cd sa trip chua'n trong khong gian vec to dclit hitu han chi6u
Gia sit E IA khong gian vec to delft n chigu. it vec td e l , e2, ...,e„
1 i = j dupe goi la co so true chuitn cUa lE neu <e i e j > =
0 i # j
Dinh 19: 'Prong bluing gian vec td Gclit n chieu holt kY. loon ton tat mgt co sa true chuan.
Chung minh: gia sti la„ a,i l3 mot co sa nao do dm khong gian vec td (kilt E. Khi do co the ray dung mot en sa true chuan { e,, e2, ...e„} nlut sau:
c o
c o = a do = et, - <cti , ME2M
i.
e 3 = M
vei e 2 = e > e l - e9> e2 Mg3
n-1 a
en ka
= , d do a n =a n -I< a n ,e k >e k . k=1
Thutit Loan chi ra a day &talc goi la qua trinh Hate chuA' hoa Gram - Schmidt co so {a„ a„}. 1-56 thky khong gian sin bai {e l , , ek} trimg vat khong gian sinh bai vat (a l , ak} vi moi k = 1, ..., n.
Nigu {et,. ai=1,2 m) la co so true chuL x c V thi x vdi = (x, N6u co y =E yi e i thi <x,y > = Ex i yi .
1=1
Gia s& F la khfing gian vac to con eim khong gian yea t Oclit E. We to a E E goi lk true giao vdi F neru <a, p> = 0 vt
moi 13 E F. Hai khong gian con F, va F2 ena E goi la trip gia netu moi vec to cua F, trip giao vdi F1. WO khong gian con F ei
E, t*p Fi = E I a _L thanh khong gian con cam E v n6u dim E = n, dim F = m , thi dim F' = n - m. Khi do (F')' = va E = F F1 .
4. TV ding cau trip giao va tti ding eau dal xiing a) Dinh nghia 1: Anh xa tuyeM tinh f: E —> E' 6 do 1 the khong gian NT& to ()alit dine goi la inh xa tuy6n tinh trg giao n6u no bao ton tich vo bleing, nghia la vdi moi x, y e E, c6 <f(x), f(y)> = <x, y>.
Anh xa tuyeM tinh true giao tit E den E chicle goi la to don, caM trip giao cua IE.
b) Tinh chat cna tg thing au trite giao
+) Ty &Ong eau f: E -, E la true giao khi va chi khi no bign 1Cit ed ad true chud'n thanh ed so true chudn.
+) f I 'a td &Ong eau true giao khi va chi khi ma tr8n A cim f rong ed sa true chudn la met ma trail true giao, nghia la =
+) f la td citing eau true giao thi moi gia tri rieng mid f ddu dng 1 hoac -1.
+) Ngu f la to citing eau true giao, va W la met kheng gian on f bad Morn, thi Wi cling la khong gian con f &Kt bign.
e) Dinh nghia 2
Tv King cau f: E --) E cna kh8ng gian yea td dclit E dude Ri la d61 xfing (hay to lien hop) ndu voi moi vac td x, y e E co.
1(x), y> = <x, f(y)›.
d) Tinh eh/it dm ttl ddng edit ded 'ding
Tti (long cdu f: E E cim kheing gian vac td debt la ddi va chi khi ma tran f trong met cd se) true chudn toy a met ma trdn itol xding.
+) bigu f la tu (long cdu clod zing cim kitting gian vac td gait
a chigu E va IF la met khbng gian con f - bdt bign cna E, thi f I E ding la met to dOng cdu del 'ding, va tong la met khong gian vec td con f - bdt bidn.
+) Moi nghi8m (pink) cia da Odic ddc tiding ena tp ding .+6,81 dal Jiang f dgu la dup.
+) N6u f la to d6ng ciu del xeing cua khong gian vac to gclii luau han chiOu E thi trong E co nail co sa true chufin goer nhilng vec td rieng eim f.
+) Cac Mating gian con rieng ung via cae gia tri rieng phar, biet ena mot to d6ng c5u dal 'ding la true giao vdi nhau.
§3 KHONG GIAN VEC TO UNITA
Trong rule nay to xet cac kliling gian vec td tren twang se ph& C.
I Dinh nghia
Cho U Ea mat khong gian vec td tren truang C. Mlit dang Hecmit xac dinh during trim U la met anh xa <, >; U x U C thea man the tinh chit saw vdi moi x, y, z E U; C C to cd:
a) <Xx + rtz, y> = X <x, y> + p <z, y>
b) <x, Jay + rtz> = X <x, y> + à <z, z>
a do X rt la lien hdp phew clia X, u.
c) <x, y> = < y,x >
d) <x, x> 0 vdi moi x \TA <x, x> = 0 suy ra x = 0.
Khdng gian vec td U tren trtiang C ming vdi mot dang Hecmit the dinh dating tren U dtidc goi la kh8ng gian vac to Unita.
2 Tinh chift am khong gian vec td Unita a) Cid si U la khong gian vac to Unita n - chtiu.
Cd sa {e,, e 9 , e„} dude goi la ed sa trip chuan ngu 1 i= j
c,,e, >=8-• =
1} 0 i j
Ta có kgt qua: Vdi moi kheng gian vec td Unita dgu ton tai ed so true chufin.
b) Ta goi chuan cua \Tee td x trong }cluing gian vec td Unita
114= j< x, x > . Ta cd bat clang thilc Cosi - Bunhiacopski:
H X, y H2.11312 moi x, y E U
d do I <x, y> la modun dia s6phiic <x, y>.
Chung minh: Vdi X la s6 phuc Lily 9, to eo <2}x - y, Xx-y> z 0.
Binh nghia dang flecmit, La ce:
<Xx - y, Xx - y> = X.?, <x, x> - 1<x, y> - <y, x> + <y, y>
= I/1 < x, x > -2„(x, y) < x,y > + < Y>
do: 12}I < x, x > y) -X < x, y > + < y, y >20 (1) Dat: <x, y> = I <x, y> I (cos 0 + i sill 0),
de) arg <x, y> = 0.
Ta lay A= t (cos q'- i sin co) vdi t c R + tuy9.
Ta et; R I = t, k<x, y> = Iliac (1) co dang:
X.<x,y> = tl<x, y>I Bra clang
t2 x, x> - 2t <x, y> 1 + <y, y>
(2) xay ra vdi moi t 0. N6u to 1Ny X = t (-cos p + isinp) raj t 2(
thi I X I = t, X <x, y> = 7r < x,y > = -t 1<x, y> I ya bat dang thif (1) co clang:
t2 <x, x> + 2t Vx, y>I <y, y> 0 (3) vdi moi t> 0. Kat hop (2) va (3) to
t2 <x, x> + 2t 1<x, y>1 + <y, y> 0 vdi moi t e Tit do suy ra I <x, y> 12 < <x, x> . <y, y>. Ta co dieu phai chting minh.
c) Voi moi x, y thuec khong gian tree to Unita U, to co:
il x+ Yll = 11x11+11.
3. Toan ter tuy6n firth tit lien hdp a) Khdi niem Loin to (tit &Mg eau) lien hop
Cho f la to deng cau cem kheng gian vac td Unita U. Tt dOng eau g cem U dtioc goi la lien hop vdi 1, n6u yea moi x, y e U, to có <f(x), y> = <x, g(y)>.
b) Dinh 5i: MOi mot tv deng cau cUa khong gian vec tc Unita co duy nhlt met tit deng cau lien hop.
c) Tinh chat cua tit deng cau lien hop. KY hieu la to tieing call lien hop dm 1. Ta co
1°) Id* = Id.
2°) (f + f* + g*
3) (kg)* n. g*
4) (f*)* = f 5) (g•f)* = r . g*
d) Ti doting can f dna khong gian vec to Unita U die goi la td lien hop, nett f = f*.
e) Dinh lj: Gin sii f la t-L7 d6ng cdu cua khong gian vec to Unita U. Khi do to có f = f, + i fa, ado f, va fa la cac tg d6ng cdu WI lien hop, &too goi tticing nag la (than thnc va ph'Un a() cda td
d6ng cdu I
Chiing minh:
Goi PIA td d6ng cdu lien hop end f.
f f • Wit f
2 =-i(f-f*)
Khi do = , = fa va f = f, + if,
g) Dinh Gia sd A la ma tran ciaa tti d6ng cdu f e End(V) trong mot cd sa true chud'n cda V, a do V la khong gian vec to Unita. Khi do f la tti (long cdu tti lien hop khi va chi khi A t =:(c
Chti 9: Ma tran phirc A co tinh chdt A t =A throe goi la ma trdn Hecmit hay ma trail tn lion hop.
h) Dinh lj: Cdc gia riling cita tst d6ng cdu W lien hop dell tilde.
Chung minh: GM sd U la khong gian vec to Unita, f e End(U) la tq doing cdu td lien hop cua U, x e U la mat vec to riling cda f nng vdi gia tri rintralt
Ta c6 f(x) = Xx,
<x, f(x)> = <x, Xx> = X <x, x> =
= <f(x), x> = <Xx, x> = 1r <x, x>
Tit do: - X) <x, x> = 0. Do <x, x> > 0 nen I = X hay X thee.
i) Dinh 0: Cae vec td rieng Ung vat ode gia tri rieng phan biet caa met to deng au tai lien hdp la true giao vdi nhau.
°ding minh:
Gia sit f la met tp deng au W lien hdp caa 'cluing gian Unita U; x, y la hai vec to rieng Ung voi hai gia tri rieng phan biet X 1, X2 .
Ta co f(x) = f(y) = X2y
<f(x), y> = <X ix, y> = X i<x, y> =
= <x, f(y)> = <x, X2y) = X2<x, y>
Tit do (A1 - A2) <x, y> = 0 suy ra<x, y> = 0.
B. VI DV
Vi du 3.1: Gia s& E va F la hai khong gian vac to tren trtiong seithuc K.
1) Chung minh rAng n6u 9: E x E Fla anh xa song tuy6n tfnh thi 9 viet dude met each duy nhait a dang 9 = s + a, a do s:
E x Fla anh xa song tuyen tinh doi xiing, con a: E x E —> F la anh xa song tuygn tinh phan del xfing.
2) Ki hieu (E, It) la khong gian cac dang song tuyin tinh tren E, con S (tudng ling A) la khong gian eon am 2 22 (E, K) g6m eac dang song tuyen tinh dee xling (tudng Cing, phan del 'cling).
flay xac dinh s6chieu cua 292 (E, K), can S va am A.
Li gidi
1) Xac dinh s, a: E x E -> F hal tong thitc s(x, y) = -1
(y(x, y(y, x)) 2
a(x, = 2 y) - 9(Y, 9)
kieIm tra s la song tuy6n tinh d6i xUng, con a la song tuy6n tinh phan d6i xiing va y = s + a. De' chUng minh bik din do la duy nhA, gia sit cp = s' + a' trong do s' del 'clang va a' phan del xung.
Da't s - s' = a' - a = y. VI = s - s' nen W dea xUng, y = a a nen y phan dayi ximg. Vol moi (x, y) c E x E, to co
kv(x, y) = - 41(3', = - W(x, suy ra w(x, y)=0= = 0 to do s = s' va a = a'.
2) Ta bigt rang .2'2 (E, R) clang cku vdi khong gian cac ma trAn vuong cap n tren K, (n = dim E). Do do dim 9z(E, K) = n 2 . Vi S (ttiong ung A) dAng caIu vdi khong gian cox ma tram doi xiing (phan xiing) cap n. Do do
n(n +1) . n(n -1) dim S - , dim A =
2 2
Vi du 3.2: Gia sit E va F la hai khong gian vec to tren trueing se" that K. Dang song tuy6n tinh f tren E x IF (Woe g9i la suy bi6n trai (phai) netu có x e E, x # 0 (Wong ung y e ]F, y x 0) sao cho f(x, y) = 0 van moi y e ]F (Wong ung f(x, y) = 0 vdi moi
x e E). f &roc g9i la kheng suy bign ngu n6 khong suy bin trai va khong suy bign phai.
Oiling minh rang:
1) Ngu f kh6ng suy bign trai va F hi u han chigu thi E cling Mtn han chigu va dim ]E < dim F.
2) Neu f khong suy bign phi va E huu han chieu thi F cling huu han chigu va dimF < dimE.
3) Ngu f kheng suy bin va met trong hai khong gian ]E, có s6 chigu Min han, thi khong gian kia cling co s6 chi6u hilu han ve. dim E = dim F.
Lo gidi:
1) Ta chang minh Wang phan chetng. Gia sit f kheng suy bign trai, (yo yz, yo} la met cd set cua F va dim E > n. Khi do trong E et) he doe lap tuy6n anh gem n + 1 vac to {x„ x o , xo, xozz }.
Dat a o = yeti 1 < i < n+1, 1 < j < n.
He thuan nhal:
n+1
= 0
do s6 8n nhigu hon s6 plutdng trinh nen co nghiem khong tam
n+1
11111611g (C„ C„+1). Khi do x„ = Ec i x i la vac to khac kheng cua
E ma f(xo, = 0 vol nail j = 1, ..., n. Vi 1.Y1, --= yij la cd SO cua F, nen ax„ y) = 0 vdi Inca y e F. Digu nay trai vdi gia thigt f kheng
suy bign trai.
2) Chiang minh Wong tv nhtt phan 1.
3) Day la Re qua true ti6p ciao hai phan tren.
Vi du 3.3. Gia sit E la kh8ng gian vee to tren trthang sen thvc g la (tang song tuy6n tinh tren 1E va g (y, x) = 0 mot khi g (x, y) = 0. Chang minh rang g hoac del Ming, hoac phan d6i xting.
Ldi
Gia sii g khOng phan dal xang, khi do co x0 e E [le g(xo, x0) # 0.
Ta hay °hung minh g doi x3ng. Vin m6i x E E. do g (x o, xo) # 0 nen co a e Yb de g (x, xo) = a. g (xo, xo) . Khi do g (x - a xo, xo) = 0.
TIT gia thi6t suy ra g (xo, x - axo) = 0, do do g(xo , x) = g(xo, axo) = a g(xo , xo) = g(x, x„).
Bay girt lay x, y e E. N6u g(x, xo) # 0 thi ce aeRa g(x, y)
= a g (x, x o)
hay g (x, y - a xo) = 0 = g (y - a x o, x).
g(y, x) = a g(xo , x) = a g(x, x o) = g(x, y).
Tug:Mg -St neu g(xo , y) z 0 thi to cang c6 g(x, y) = g(y, x).
Cual cung, gia sit rang g(x, x o) = g(xo , y) = 0, khi g(x, y) = g(x, y + xo) va g(y, x) = g(y + xo, x).
Ta c6 g(xo , y + x0) = g(xo, xo) # 0 nen g(x, y) = a g (x o, y + xo)
= a g(xo , xo) = g(x, Y xo)
Suy ra g (x - a xo, y + x o) = 0
g(y + xo , x - a xo) = 0
g (y + x0, x) = a g (y + xo, x0)
= a g(x„, x0) = g(x,y).
NMI va.37, trong moi &Jiang hop to clgu co g(x, y) = g(y, x).
nghia la g dOl xQng.
Vi du 3.4. Cho E la khong gian vac td thuc n chieu VA co Fa Bang song toyan tinh dal xung xac dinh throng teen E. Gra sit x,, x2, xk la nhung vec td mia 1E. Dal aid = xi), 1 j s k.
Ta goi dinh Ulric det (a 1 ) la dinh thuc Cram (Ma cac vac td x,, xk va ki hiOu la Gr xk).
Chang minh rang Gr (x i, xk) 0 va Gr (x 1 , . xk) = 0 khi va chi khi xl , xk phu tha0c tuyeln tinh.
Lidi brick
Ta chgng to rang ngu xi , xk phu thuoc tuyan tinh thi Gr(x l , xk) = 0, con ngu , xk clOc lap tuygn tinh thi Gr(x l , xk) .> 0.
Gia su x,, xk phu thul)c tuy6n tinh, th6 thi co vac. td (2 < r 5 k) bigu thi tuygn tinh qua x,, xj+ : x. = a„ x i + + cc„,
Khi do aji = u(x r, = a, a lj + a2 aji + + nghia la thing thir r am ma trap (ad k tau thi tuythn tinh qua r-1 dOng dau. Ta do suy ra Gr (x l , xk) = det (ad k = 0.
Gia sr) x„ xk d'Oc lap tuyen anh. The thi fx,, la cd sa cua khong gian con F via E sinh bai he {x 1 , xk} va ma trail (adk la ma trail clic( (p i = (p1 F doi vdi cd sọ do. Vi xac dinh
during nen theo Binh lY Sylvester, to co det (a, i)k > 0 nghia la Gr{x l , > 0.
Vi du 3.5: Cho A la ma tran yang del xiing dip n tren R.
( xi
Xet khong gian R" cac vac to cot X = , trong do x i e It; u
n
la phep bign del tuygn t;nh trong co ma tran trong co sa' to nhien la A. Chung minh rang:
1) Neu X, Y la nhang vac to rieng cern u iing voi nhitng gia tri rieng khae nhau, thi X, Y true giao theo nghia V X = 0.
2) Moi nghiem day trung cim A deli la s6 that.
3) Nefu A xac climb duong (vac Binh am), thi mot nghigm da'e trung eim A dgu dutong (tttemg Ung: dgu am).
Lai gicii:
1) Gia sit X, Y la hat vec to rieng cua u Ung voi hai gia tri rieng X, n; n. Ta co AX = XX, AY = HY. Tii de
Yt.AX = VAX = X.Y.X;
Xt.AY = Xt.p.Y = g.Xt.Y;
NhUng Yt.A.X = (r.A.Y) t do At = A, nen X Yt. X = (àX t . Y)` =N. Y` . X.
to do (X - . YtX = 0 YL . X = 0
2) Gia sit X + in la nghtem dac trung cim A. The thi ten tai vee td cot (phitc) X + iY trong do X. Y la nhUng vac to cot that khong dong that bang khong