Vec to 13= k,a, + + k„,a„„ (a, c V, k e K)
goi Ht met to' hop tuyo'n tinh cua cac vec to 4 0 = 1, Ta ding not p bieu thi tuy6n tinh qua cac vac to a t , ..., an„
MOt he vac to la,, ..•, caa V dude goi la he phu thmic tuyeIn tinh nen c6 cac s6 = 1, m) kh8ng deng thoi bang kheng cna ]K sad cho = 0 . Met Oat hkeu along &king:
1=1
he (st„ am) ka he pha thuec tuy6n tinh nen c6 met vec Lo nac de cim he bleu thi tuyen tinh qua cac vec to can lai oda he.
Met he cac vec to kh8ng phu thueic tuy6n tinh doge goi he doe lap tuyen tinh. Nhu \ray, he {a„ am} la he dOc lar tuyen tinh ned.1 moi t6 hop tuy6n tinh k i a i =0 to suy ra k, = = k m = 0.
Cho he 14„ aid cac vec to doe lap tuygn tinh caa kh8ng gian vec to V ma m6i vec to caa no la t6 hop tuyeIn tinh caa ca( vec td cim he ([3„ thi k < 1.
. Cho he vec to {di } ; e trong kheng gian vac to V, I la tap chi so gfim huu han phan tit He con (a3) jedcI goi la he con de(
lap tuyen tinh t6i dai cim he da cho neu n6 la he dec lap tuyen
tinh, va nen them bet kST vec td ak nao (k E I \J) thi ta dude met he phu thuec tuyen tinh.
Cho he hau han vec td {a, , , a m} trong khong gian vec td V, thi s6 phain tit cim moi he con dee lap tuyen tinh t6i dal ciia he tren deal bang nhau, so/ do duoc goi la hang cim he vec td {ao ...,
3 - Cd sd va so clueu cua khong gian vec td
Met he {e„ , en} the vec td doe lap tuyen tinh am killing gian vec td V duos goi la met co sa oda V nen mm vec to cim V deli la t8 hop tuygn tinh cim vec to {c„ , e n}. Khi V c6 ed sa g6m n vec td thi moi co sa cim V deu c6 dung n vac td. S6 n goi
la so" chigu cim V, Id hieu dimV. Neu kleing tan tai mat cd sa
Om him han vec to, thi V goi la khong gian vec td v8 han chieu.
Cho co sa e = le i , , ej, yen vec tO bet kS7 a e V, ta co a , , x„) dupe goi la cac toa doo cim vec td a del vOi ed sa {e„ , e n}, ; la toa de thu i cim a doa vdi cd 56 do.
Gia sit co ed sa khac c = {c,, , E„} ciaa V, ma si = ]=1 = 1, n). Nen vec td a có toa di) (x i) trong cd s6 va c6 toe de' (x1 1) trong ed sa thi ta c6:
= Ec oxl i , , n.
Neu ta ki hieu ma tran C = va ma trail X = (x), X' = (x') la cac ma tran cot, thi X = C X'.
4 - Klaang gian vec tei con va klaing gian vec td thacing Tap con khong r6ng W ciaa khong gian vec to V duoc goi IA khong gian vec to con ciaa V nen W la khOng gian vec to, voi cac phep toan ciaa V han dig tren W.
Tap con khong rang W dia. V la khong gian vec to con ciaa V khi va chi khi W on climb dna vdi hai phep toan dm V, nghia la vdi a, 13 E W va k E K, thi +p E W va k a E W.
Cho W,.... W EI la cac khong gian vec to con cria khong gian
11
vec to V, khi do nW, IA mOt, khong gian vec td con Gem V, IA khong gian con ldn nhat nam trong moi W i , i = 1, 2, ... , n,,.
Cho tap hop X c V, khong gian vec la con be nhat cna V chga X duck goi 1a bao tuygn tinh cila tap hop X, ki MO <X> hay Vect(X). Neu X = , bao tuygn tinh ciaa X thick ki hieu la <a t , , a„,>.
Cho W„ , ho nhUng khong gian con ciaa V. Khi de bao tuygn tinh ciao, tap hop W,U... UW E, dude goi la tang cim cac khong gian con W, ,W„, ki hien Ia W, + + W„ hay ZW; .
Ta thay rang a e EW; khi va chi khi a = Za i , a, e i=1
Neu moi e /113/4 , a vigt chicle mat each duy nhat a i=1
dang a = a, + + a„ , a, e thi t8ng W, ch.toe goi la tang I=t
true tigp cua n khong gian vec td con (i = 1, 2, ..., n) va ki hieu la W, e W2 ED ... ®W„ hay S W,.
Gia sit V la klafing gian hitu han chikt,W, va W2 la hai khfing gian vec to con dm V, khi do: (adsbygoogle = window.adsbygoogle || []).push({});
dim(W,+ W2) = dimW, + dimW2 - dim(W, fl W7).
Cho W la khong gian vac to con cim khong gian vec to V. Ket quan he Wong throng tran V: a-Paa-Pe W. Lop Wong :lining cam vec to a dirge ki hiau la [a].
Tap thudng V/W vdi hai [top toan:
[a] + [in = [a + [I] va k[cx]= [kal
vdi moi a, 13 E V va k e K, lam thanh mat klihng gian vac td,
durie got la kitting gian vec to thtfong (ciia V chia cho W):
dimV = n, dimW = in (0 m n), thi dim V/W = n - m. Anh xa it : V -> V/W ma a(a) = [al (Inge goi la phdp chi6u
tdc.
§2. ANH XA TUYEN TINH
1. Dinh nghia. Cho V va W lh cac khong gian vec to tram twang K; Anh xa f: V -> W duo° goi la anh xa tuye"n tinh (hay itIng ca'u tuy6n tinh, hay toan tit tuyan tinh) n6u no bao
phop toan cua khong gian vec to, cu the' la: veil moi a. p e V. 1119i k e IK, to ce:
P) = f(a) f(P) f(k a) = k . f(a).
Anh xa tuy6n tinh dude goi la don din nett ne la don anh, :oan can n6u n6 la than anh, va dang 6.'11 n6u n6 la song anh.
Hai khong gian vac td V va W chive goi la clang cdu vol nhau nAu ce met ding cau f tla V len W.
2 - Cac phep than tren cac anh xa tuy6n tinh
Ta ki hieu tap cac anh xa tuy6n tinh tit khong gian vac to V dAn khong gian vac to W la Hom(V, W) hay Hom K(V,W) de chi rd K la tniang co sir.
HomK(V,W) la met khong gian vec to tit truong K vbi hai phep town nhtt sau:
Vai f, g e Hom K(V,W); anh xa f + g e Hom K(V,W) the dinh ben (f + g) (a) = f(a) + g(a) vdi moi a e V.
Vol k e K, f e Hom K(V,W), thi kf: V —> W xac dinh boi (kf)(a) = kf(a) vdi moi a e V.
Anh xa f + g dupe goi la tong Kin hai anh xa f va g. Anh xa k. f ddoc goi la tich eda anh xa f vdi vo hdong k. 3 - Di6u ki6n xac dinh anh xa tuy6n tinh
Anh xa tuydn tinh f: V —> W hohn toan chidc :Mc dinh khi Mei anh cOm met co so.
NMI le„ , e„) la co se, cOta khong gian vac to V va a,, a„ la n vec to cent MI:Ong gian vec to W (V, W la cac khong gian vec to tren clang rant trnong K), thi Kin tai duy nhdt met anh xa f e flom K(V, W) de f(e i) = a, yea j = I, ... f la don cdu khi va chi khi he , a„} doe lap tuyetn tinh, f lA clang cdu khi va chi khi he M I , , a„) la co sa oda. W. Gia = EJ la cd
in
tan dm anh xa tuy6n tinh f dOi vOi co sa {e,) va {EJ. Nhg vay 16u cho cd sa E cua W va co sa e cua V, thi dinh 19 tren chring to :Ang co mOt song anh girla tap Hom K(V, W) va tap 1),E#t(m, n, K).
FlOp thanh cua hai anh xa tuy6n tinh la mot anh xa tuy6n rhh, nghia la nOu f: V —> W va g: W —> Z la the anh 'ea tuygn
thi g f: V --> Z cling la mot anh xa tuygn tinh. 4 - Anh ca hat nhan cua anh xa tuygn tinh
Cho f: V —> W la thing thu tuy6n tinh gifla cac khong gian 7ec to, neh X la khong gian vec to con cua V, thi f(X) = { f(a) I a E X) a khong gian con cua W, va n6u Y c W, Y la khong gian eon
W thi f-'(Y) = e VI f(a) e Y} la mot kWh:1g gian con cua V. Ta goi Kerf = f-1 101 la hat nhan cda anh xa f va Imf = f(V) la inh cua anh xa f. S6 dim Imf doge goi la hang cua anh xa f, kf lieu rang E
Gia sit f e Hom K(V, W),
la don eau khi va chi khi Kerf = {0}. Nth dimV Fa hfiu han thi IimV = dim Kerf + dim Imf. (adsbygoogle = window.adsbygoogle || []).push({});
Cho ma Wan A e Mat(m. n, 1K), xem A nInt ma tran cua inh xa tuy6n Unh f: K n —> Km trong thc od sei chinh tdc. Khi do
tang cua ma trail A (da dude dinh nghia trong chudng I) hang
rang cua f va chinh la hang cua he vec td cot cua ma tran A. 5 - Ma tran aim t‘i thing cgu trong the cd ad khac nhau Cho f E Hom K (V, W), trong co se: e = (e„ , e n) f cO ma Wan
= (ad), nghia la f(e1) = Za ij e i . i=1
Gie sii & = (6 o••ogs) le mot cd sa &bac, ma E, = /Cijej , tron€ i=1
cd s6 6, f co ma tren B = (b ii) ughia = Eb ii Ma trar i=1
C = (co) chide goi la ma tren chuyen co sa. Ta ca: B=C' AC
Hai ma tran A ya B dude goi la deng deng neiu có ma trey khong suy bign C de B = C -' A. C. Nhu yey hai ma Win cue ding mot phep bign (lei tuygn tinh trong hai cd sd khic nhau &dug clang.
Ta goi vet min ma tren vueng A la t6ng cac phen to trer • &tang Oleo chinh. Hai ma tran tieing deng co vet bang nhau
Vet cua mot to deng cdu tuygn tinh la vet cila ma trail cfla ni trong mot cd sd nal] do. Vet clad ma tran A dude ki hieu la trace A hay trA. V6t cua td d6ng cliu f thick ki hieu la tracef hay trf.
§ 3 HE PHUtiNd TRINH TUYEN TINH 1 - He pinking trinh
He Ea ki xi = b k (k = 1: 2, ... , i=1
dO aki , b k E K cho trUoc, k = 1, , m; it= 1, , n.
xi la cac en, dude goi la h0 phudng trinh tuyen tinh (hay 14( phudng trinh dai s6 tuygn tinh) g6m m phudng trinh, n an so Khi K la truang s6 (nhu R hoac C), thi cac a to goi la the he 66,
hQ se" to do.
all a 12 a ln a21 a22 a 2n
b
b 2
Ma trail Abs = goi la ma tran
b6 sung, no có ducic tii ma tran A bang each them cot cac he so' tti do vac, cat thu (n + 1).
b
Neu ki hieu X = va B = la ma tran cot,
xni bini
thi he phtiOng trinh (1) co the vieit duoi clung: AX =B.
2 - HQ Cramer
He n phudng trinh tuyein tfnh n an s6, ma ma Iran cac he
s6khong suy bi6n goi la he Cramer.
HO Cramer c6 nghiem duy nhdt. each tam nghiem nhu sau:
Cdch 1: Xet phasing trinh ma trdn AX = B, vi detA # 0 nen tan tai ma trdn nghich dao Al', va ta co:
X = B (adsbygoogle = window.adsbygoogle || []).push({});
Cdch 2: Xet he vec td cat an}, ma a i = (aii ,a si ,...,a mj )
va b(b„ , b 0,) la vac to cat td do, the thi he viat dude dU6i clang =b. N6u ta goi D i la dinh thiic cda ma trail nhain dude bang each thay cat thit i cda ma trdn A bat cot cac he si6 tp do, thi x
I = D ado D la Binh thiic cim ma tran A.