a) Dinh b./ (Gauss hay Kronecker - Cape 11i)
He phudng trinh Za ii x j = h i (i= 1,..., m) ( 1 ) i=1
ce nghiem khi va chi khi rangA = rangA hs. b) Phining phap khet nem Gauss
Cho he pinking trinh (1), n6u dung cac pilau Men dpi sau day thi La van nhan dude met he phudng trinh thong doing \TM. he (1), nghla la he cO ding tap hop nghiem nhit he (1).
+ :Than hai ye cUa met phudng trinh nao do cem hee, vat s6 k #0. + Geng vao met phudng trinh cua he sau khi da nhan met s6 bat ky vao hai ye cem phudng trinh khac. -
Cap Oen bi6n ling during cl6i vOi he phuong trinh chinh la cac Olen bin d6i so cap thy(' hien Den cac clang dm ma trail 1)6 sung Abs cUa he.
Dung phuong phap khil Gauss la Dille hien cac phen bign ckii Liking during de chia he phuong trinh (1) v6 he phuong trinh ma ma trail 06 dung: P 0 1 I 131 h ip p+i m-p 0 0 b'm p n-13
phan t>i a phan goch (*) co the khac 0. Khi rid n6u 13,;+1 + + > 0 thi he ye nghiem,
n6u = b'„,= 0 thi he có nghiem phn thuOc n-p tham s6. §4. CAU TRUC CUA T11 DoNG CAU
1. Khong gian rieng - da thitc dac thing
Cho V la khong gian vec to tren truong K (1K bang K hoac C). Wit anh xa tuy6n tfnh to V d6n chinh no dude goi la mat to dOng calu tuy6n tinh. Tap ode tv ding cOu tuyeyn tinh eau V kY hieu la Hom K(V, V) hay End K(V). Gia f e End K(V), W la khOng gian con ena V; W chick goi la khOng gian con bkt bi6n elia V n6u f(W) c W.
Vec to a # 0 thuec V dude goi la vec to rieng cim f e EndK(V) ling vat gia tri rieng X ngu f(a) = Ira, A e K. Khi do khong gian met chigu sinh bai vec to a Et met kh8ng gian vec con bgt bign cim f.
Val A E 1K, tap ker(f - AId) khi no khac {(5} la khong gian con cim V, gam vec to khong va tat ca cac vec to rieng caa f ling vdi gia tri rieng X. Kitting gian nay goi la khong gian rieng cim f
v6i gia tri rieng A, ki hieu
Gia sit ta cl6ng eau f e End(V) trong met co sa Mao do cim V
co ma tran A, thi det(A - XI n) IA met da fink bac n dovi v6i bign X, khong phu thuec vao vice, e chon co sa, va &lac goi la da thitc (lac trang caa ta (tang eau f (ta cling nOi do IA da that da, e trang caa ma tran A), ki hieu M.(x) = det I A - X I n I . Nha vay A la met gia tri rieng cim f khi va chi khi X11 nghiem &la da that dac trang eim f.
GM sit , Ak la cac gia tri rieng cigi met phan biet caa I Px Pik la cac khong gian rieng Wang ling via ale gia tri rieng do, thi t6ng Pr + Px 2 Pxk la tong true tigp.
2 -Ong gian rieng suy Ong
GM sit V la khong gian vec to tren (Wong K, f e End K(V). Vdi mei A e K, xot tap {a e VI co s6 nguyen m > 0 de1 (f-XIcl,)m(a) = 6). DO la met khong gian con caa V, khi khac {0} no &arc goi IA khong gian Hong suy rang cua f ling vdi A va ki hieu IA
+ Vdi moi khong gian rieng suy rang 3 k 7r IA met gia tri rieng cim f va Y, c Rs.
+ V6i X IA gia tri rieng cua f, dim2h x bAng bei cern nghiem 7.
cua da thitc dac trung /. f(X).
+ Mot g?",, la met khring gian con cila V bait bi6n qua anh xa f. + Gia sf1 , "f.k IA cac gia tri rieng phan biet tiing cap va u, e \ {0} = 1, 2, ..., k) thi he cac vec td {u„ u k} doc lap tuy6n tfnh.
3 - Tit citing clu luy linh
a) Ta not ring f e End K(V) IA tu ding caiu luy linh nau tan tai s6nguy'en k > 0 de'f = f o .. of= 0, hdn nua, nau # 0 va
k hin
0' = 0 thi k goi la bac luy linh cua E
Tu dang udu f e End K(V) ma co cd sa te„ , e n} sao cho Re) = (i = 1, , n-1) va Re p) = 0, thi hay linh bOg n va ed sa {e i, ..., en} dude goi 11 cd so xyclic d6i v6i f. Trong cd SO xyclic ma trail cua f co clang:
0 0 0' 1 0 0 1 0 1 0 0 .. 1 0
b) f e EndK(V). U la khong gian the to eon cem V, U chide gui IA khong gian vec to con xyclic chid vdi f nEu U IA f- bas biEn va trong co mot ca so xyclic dee vdi f/U: U -> U.
c) NEu f e End K(V), dimV = n, thi V phAn tich dude thanh tang trip tiEp cua cac khong gian vec to con xyclic doi vdi f. Vdi mdi so nguyen s 2 1, sE cac khong gian vac to con s chiEu xyclic doi vdi f trong moi each phan tich dEu bdng nhau va bAng: (adsbygoogle = window.adsbygoogle || []).push({});
ran g(f 8') - 2rang(f s) rang(f
d) N'Eu 9 ) IA khong gian rung suy rang cua f dng vdi A, thi
(f - A.Ick) la td d6ng eau Iuy linh cern V.
4 - Ma tran clang chutha lac Jordan caa tat d6ng eau GiA sei V IA khong gian vec td hisiu han chigu tren truang K,