VI ma Han A=(a n) khong suy men, nen co ma tran nghich dge 11 = A ' = (bd.
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> + µ <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: 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
(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 + 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 sä 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 = (µX 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
A. (X + Y) = (7. + ip) (X + iY). 6 day i la don vi ao.
So sanh cac phan Uwe va pfin no a ca h ai v6, ta ducre
AX = XX - (1)
AY = XY + p X. (2)
Tit (1) va (2) ta co:
VA X = Y`X - p Y'.Y X`AY=i.- p Nhung Yt. A . X= (XL. A Y)L nen ta có:
AY`.X-pYt.Y=XYLX+gr.X
p (XL . X + . Y) = 0. VI X, Y khong dang that bang khong, nen r. X + Y.Y>Oviy*yn=0,docloA,+ip=X e R.
3) Gia sit A la nghiem (lac trung cith A, theo tren X e Khi do co vec td cot X = 0 de' AX = XX, suy ra Xt.A.X = Vi X = 0 nen Xt.X > 0. TU do, ndu A the dinh (lacing thi X`AX > 0 nen X > 0. Ndu A the dinh am thi XtAX < 0 nen X < 0.
Vi du 3.6. Cho A la ma tran thong tithe ddi ximg cap u e End (R") có ma trgn A trong co sa tti ten. Chung minh rang ndu X lit vec td khac khong tha R", thi ton tai vac to thong Y cua u thuOc khong gian con sinh bai X, AX, A 2X, A" - ')C
LaPi
AX, A"X phu thutic tuydn tinh, nen co mot vec to bik thi tuyin tinh qua cac the to dung bathe no. Gia sit k ad
h6 nhat de AkX bleu thi tuygn tinh qua X, AX, Ak 'X. Khi
0 tan tai the s6 thuc b k., ..., bo dg
AkX + b k, Akd + + b iAX +13 0 X = O. Gia sit (x - p 1 ) (x - i.tk) la &tan tich cim da tilde
xk + +...+bo thanh the nhan tit tuy5n tinh, vdi g 1 , gk E C. Khi
0 = (Ak + bk., Ak- ' + + boI) X = (A - ... (A - ukI) X hay (A - = 0 vol Y = (A -1.1 2I)... (A - tikI)X x0.
Nhtt vay AY = suy ra g, la nghigm dac trung cim A. 'heo vi du 3.5, to co p, e R. 'Nang to go , ..., gk dgu thuc. Do do lh vec to rieng cUa u va Y thnOc khong gian sinh bat X, AX,
Ak- `X.
Vi du 3.7: Gia su ,\ is ma trail vuong thuc dal xiing cal-) n, u End (Y") có ma tran A trong co so to nhign. Chung mink ang tan tai ma trap trot giao P sao cho ma tran Pt . A . P = :1 ma tran cheo. Cite phan to tren during cheo chinh cim hinh la cac nghigm ciao trung cim A (kg ca bOi). Tii do suy ra ang A xac dinh throng (Wong Ung xac dinh am) khi va chi khi
nghiem dac trung cim A dgu dvong (Wang ung dgu am). Chu y: del chigu kect qua trong vi du 3.5).
Gia X,, s < n la mot hg trot giao gam nhung vec td
Ta cheing to rang có vec td rieng X„, trip giao vet cac i = 1, 2, s. That tray, gia sat X x 0 la mOt vec td true giao Xi (i = 1, s). Vei mai i= 1, s va r > 0, ta co:
X` . A' . X, = . . = (WIC) = 0.
(X, 11 gia tri rieng eng vdi vec to rieng X 1) to do (Alot X, = 0, s ra AIX true giao yea Xi. Theo kat qua trong vi du 3.6, co vec rieng Xs,1 cern thueic khong gian con sinh bei X, AX, A"- Nhung cac tree to AIX true giao \el cac X, (i = 1, s) vi tray 3
clang true giao yea Nhtt tray trong co X„ X„ trip g
gem toan vec to tieing ciaa u.
Dat - 1 X• thi he Y1 , Ya la he cac vec to rie .X,
sao choY,` . Y, = 8,, ki hiOu Kronecker). Vi tray ngu dal IA ma tran ma cam cot la Y„ thi P IA ma trait trvc giao.
Mat khac, AX ; = 3,X ; nen AY, = X ; lc ti/ do . A. Y; V, = X, va vie j thi Y,' A Y, = X; Yit Y; = O.
Do do PAP=A trong do A la ma tren chdo yea cac Olen eh& a ..., X. Vi P = nen A thing dang vdi A, suy
I A - XIn I = - I = (X, - X) •-• -
Pu do suy ra 3.„ chinh la tat ca Cite nghiem (lac tn./ cUa A, Ire ca bOi. ,
Cugi clang, do p AP = A suy ra A the Binh decing (am) va chi khi A xac dinh duong (am) nglâa. IA khi va chi khi 3 1 ,
Vi du 3.8: Cho A, B la hai ma tran that d6i xiing cap n, hen
nem B the dinh during. Chung mink rang ten 41 ma teen khong suy Bien C Ct. A. C=A va C'. B. C= In, 6 de A la ma trait chef), va In la ma trail den vi cap n.
Lai gicii:
Do B xac (huh &king nen t6n tai ma lit' khong suy bin T de 'FL . B . T = In. Ma tren Tt A. T doi thing nen theo kgt qua trong vi du 3.7, c6 ma trail tnic giao P de 13` (Tt A T). P = A la