Embed Size (px)
Citation preview
PHAN HUY PHIJ • NGUYEN DOAN TUAN
BAI TAP DAI SO TUYEN TINH
NHA XUAT BAN HAI HOC QUOC GIA HA NOI
Chin trach nhiem xual bcin
Gicim doe: NGUYEN VAN THOA
Tong bien Op: NGUYEN THIEF N GIAP
Bien tap:
HUY CHU
DOAN 'MAN
NGOC QUYEN
Trinh bay Ilia:
NGOC ANH
BAI TAP DAI sq TUYEN TINH Ma s6: 01.249.0K.2002 In I .501) cudn, tai Xtiiing in NXI3 Giao thong van tai S6 xuat ban: 49/ 171/CXS. S6 Inch ngang 39 KH/XB In xong va Opt [Yu chi& CM/ I narn 2002.
Lai NOI DAU
Mon Dai s$ tuygn tinh dude dua vao giang day a hau hat
cac trUnng dai hoc va cao dang nhtt 1a mot mon hoc cd se; can thigt d@ tigp thu nhUng mon hoc khan. Nham cung cap them mot tai lieu tham khao phut vu cho sinh vien nganh Toan vi cac nganh Ki thuat, chting Col Bien soan cugn "BM tap Dai so tuygn tinh". Cugn each dude chia lam ba chudng bao g6m
nhUng van d6 Cd ban cna Dal so tuygn tinh: Dinh thfic va ma trail - Khong gian tuygn tinh, anh xa tuygn tinh, he phticing trinh tuygn tinh - Dang than phttdng.
Trong mOi chudng chung toi trinh bay phan torn tat lY thuyat, cac vi du, cac hal tap W giai va cugi mOi chudng c6 phan hudng dan (HD) hoac dap s6 (DS). Cac vi du va bai tap &roc
chon be a mac an to trung binh den kh6, c6 nhUng bai tap mang tinh 1± thuygt va nhUng bai tap ran luyen ki nang nham gain sinh vien higu sau them mon lice.
Chung toi xin cam on Ban bien tap nha xugt ban Dai hoc Qugc gia Ha Nei da Lao digt, kien de cugn sach som dude ra mat
ban doe.
Mac du chting tea da sa dung 'Lai lieu nay nhigu narn cho sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang
khi bier, soon, nhUng chat than con có khigm khuygt. Cluing toi rat mong nhan dude nhUng y kin clang gap cna dee gia.
Ha N0i, thcing 3 !Lam 2001
NhOni bien soan
3
rvikic LUC
Chubhg .1: DINH THOC - MA TRA:N 7
A - Tom tat ly thuyeet 7
§1. Phep th6 7
§ 2. Dinh thitc
§ 3. Ma tram 10
B - Vi dn 12
C - Bei tap 35
D HtiOng dein hoac clap so 43
Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH
• PHUGNG TRINH TUYEN TINH 57
A - TOrn tat ly thuyeet 57
§1. Kh8ng gian vec to 57
§2. Anh xa tuyeen tinh 61
§ 3. He phydng trinh tuy6n tinh 64
§4. Can true caa tai ding cku 67
B Vi dtt 71
C - Biti tap 96
§1. 'thong gian vec to va anh xa tuyeen tinh 96
§2. He pinking trinh tuy6n tinh 104
§3. Cau tit cna melt tu thing calu 106
D. Illidng sign ho(tc clap s6 110
5
§1. Khong gian vec td va anh xn tuyin tinh 11(
§ 2. He phudng trinh tuyeit tinh 12';
§3. Cau trite dm mot tg ang cau 12Z
Chtedng DANG TOAN PHUONG - KHONG GIAN VEC TO
OCLIT VA KHONG GIAN VEC TO UNITA 134
A. Tom Vitt 1t thuyeet 134
§1. Dang song tuy6n tinh aol xUng va dang town phuong 139
§ 2. Killing gian vec to gent 135
§3. Khong gian vec to Unita 142
B. Vi du 14E
C - Bai DM 174
D. Hitting dan hotic ditp so 179
Tai lieu them khan 192
6
Chuang 1
DINH THUG - MA TRAN
A - TOM TAT Lt THUYET
§1. PHEP THE
Met song anh o tit tap 11, 2, met phep the bac n, ki hieu la
'1 2 3
n} len chinh no duet goi la
\G I a 2 G3
15 del a, = a(1), 02 = a(2),..., a„ = a(n).
Tap cac phep the bac n yeti phep nhan anh xa lap thanh met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen
t3 cua nhom S„ bang n! = 1, 2... n.
Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich
the cem a n6u s6 - j) (a, - a) am. Phep the a &foe goi la than
ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6
nghich the ciaa a le.
1 neM s la phep the chan Ki hieu sgna =
-1 net} a la phep th6 le
va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the
cling bac, thi sgn(a = sgn(a) . sgn( ).
Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„
• - • , i k doi mot khac nhau dr coo = 1 2, coo = i3, a(ic) = i1
7
va a(i) = i vdi moi i x i„ i k . Vong )(felt do dttoc ki hieu IA ik). M9i phep th6 dau &tan tfch the thanh tfch nhung
yang xfch doe lap.
Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong xfch ••• , i k ) phan tfch chive thanh tfch 0 1 ,
§ 2. DINH THUG
I. Gia sit K IA met trueng (trong cuan sich nay to din yau xet K la &Ong s6thvc K hoac truang s6 phitc C). Ma tran kidu
(m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim
m hang, n cet cac phan tit K, i = 1,m, j = 1,n. Tap cac ma tran kidu (m, n) chive kf hieu M(m, n, R). Ma trail vuong cap n
IA ma tran co n dong, n cot. Tap cac ma trail vu8ng cap n vdi cac phan tit thuoc truong K ki hiOu IA Mat(n, K).
2. Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, ..., n. Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K dude xac dinh nhu sau:
detA = zsgn(a)amo) E Sn
3. Tinh eh& ceta Binh that
a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram A, thi dinh auk cim no ddi da:u.
b) N6u them veo met dong (hoac met cot) cim ma tran A met to hdp tuygn tinh cim nhUng thing (hoac nhung khac, thi dinh auk khong thay ddi.
8
fan ail
de a21 +alci
‘a n „ a,,, + an i a ll a l; ...a 1 ,„ all
= det a 21 21 +det a21
—a 1111/
an, a2„
...a2 n
" S ' Ill "S IM /
c) Ngu mot Bong (hay mot phan tfch thanh tong, thi
dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6:
d) Cho A = (It o) E Mat(n, K), thi = b) a do = a ij &toe
goi la ma tran chuy6n vi cim A.
Ta co detA = detA t .
4. Cdch tinh dinh that
a) Cho ma tran A E Mat(n, K). Kf hi'911 Mi; la dinh that cua
ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot
thu j cut ma tram A vb. Aij = (-1)H M u clucic g9i la pha'n phu dai
s6cUa phgn to a ii cna ma trait A. Ta có CAC tong thtic:
O ngu i k
det A ngu i = k
O ngu i x k
det A ngu i = k
Nhu fly detA = EamAki (k = 1, 2, ... n) 1=1
heat detA = Z
•
a ikAik /=1
9
CUT thac tit throe goi la cang thdc khai trim dinh tilde theo (long hay theo cot.
b) Dinh 1ST Laplace
Cho ma Iran A = (a, J ) c Mat(n, K). Vo; rn6i bQ
;2.••, ix), 1 s i, <1 2 < . <
ik va Oh ik), 1 =11 < j2 < ••• <jk n, 1 s k< n, dat A. la
11-11,
dinh thac caa ma tran vuong cap k nam d cae &nig i„ i k va cac
cot j,... j k cim ma tran A; M. la dinh thdc cita ma tran vuong h
cap (n - k) c6 dude bang each gach the clang thu ik va cac cot thu j 1 , , jk caa ma trail A. Ta c6 kgt qua:
detA = ZED' Si k+31+
A. ik
do j1.-- jk la k cot cgdinh. Tgng &toe lgy theo tat ca cac bQ ik) sao cho < i 2 < < i k Cong thdc troll dude goi la c8ng thdc khai trim dinh thae theo k cot ji, 'along tV, to có gong thdc khai trim theo k clang. Khi k = 1, to &roc gong tilde da not trong muc a.
§ 3. MA TRANI
1. Ma trgn kigu (m, n) vgi cac phan tU tr8n trang K da dude
gidi thigu trong §2. Tap the ma tran kigu (m, n) vdi cac phan ti tren tragng K dude ki hiQu la Mat(m, n, K) A E Mat(m, n K)
(Woe vigt A = (a ii) i = 1, m ; j = 1, 2... n hay ro rang hon:
10
811 a19 aln
A= a, 1 a99 aon
amt amt arnn,
2. Cac phep todn tren Mat(m, n,
Cho A = (a y ), B= (b,j ) thuOc Mat(m. n, K)
Ta có:
a) Ma tran C = (cg) a do cy = a ii +
&toe goi la tong cua hai ma tran A va B va ki hien la A +B.
Ma tran D= (d,,) a do di; = a ij -
dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B.
b) Vdi k E At, ma trail kA c8 cac phAn to la (ka ii ) duoc goi
la tick cua ma tran A vdi ph&n td k cua trudng K.
c) Neu A = (a ii ) c Mat(m, n, K) va
B = (bp() e Mat(n, p, K) thi
ma tran A . B E Mat(m, p, K) ma cac phAn tit &tele xac dinh INN
AB = (c, k), a do
e ik = Zaijbjk 5=1
&toe goi litich caa hai ma tram B ye. A.
Vol A, B e Mat(n, K), to có det(AB) = detA. detB.
11
d) Tap Mat(n, K) con ma tran yang cap n vdi phep toan
cOng lap thanh mot nhom giao hoan, con vdi phep Wan rang ma tran va phep nhan ma trail lap thanh mat vanh khong giao hodn, co don vi.
3. Hang ctia ma tran; Ma trim nghich ddo
Gig A E Mat(m, n, K), ta dinh nghia hang ciat ma tran A
la cap cao nhgt cua dinh thric con khgc khong rut ra W ma tran A. KM A E Mat(n, K) va hang A = n (ta cling dung ki hi3u hang
A la rang A) thi ma tran A goi la khong suy bign, khi do
detA * 0 va ton tai duy nhgt ma tran B thuOc M(n, K)
A.B = B.A = I„; d do I. lit ma trail don vi. Ma tran B &roc goi 11
ma tran nghich dgo cna ma tran A va ki hi3u la A'.
Gig su A= (Au )la ma trail plw hpp cim ma tran A = (ad,
Ab la Olga Ow dal see mitt phgn ht a ii ; A t la ma tran chuya'n vi cua A . Khi do:
At . detA
B- VI DTI
Vi cla 1.1. Xac dinh clgu rim cac phep th6 saw
a) a 11 2 3 4 5 2 3 5 4 1
b) 5= r1 2 3 n n+1 n+2 211 2n+1 Ill 4 7 ai-2 2 5
31
12
Lai gidi
a) Phan tick) a thanh tich cac chuydn tri:
= (1 3 5 4 1
2 3 4 5) = (1 2 3 5) = (1, 5) (1, 3) (1, 2)
2
(chi) 9 la pile)) nhtin cae chuydn tri dude thuc hign tii phai sang trai nhu hap thanh cua the Anh xa).
Vay sgna = (-1) s = -1
Co the lam each khan: Cam nghich the cua a la (1, 5), (2, 5), (3, 5), (4, 5), (3, 4).
Vay a có 5 nghich the nen sgna = -1.
b) Ta hay tinh sS nghich the cua boa)) vi (1, 4, 7... 3n-2, 2,
1 khong tham gia vao nghich the
4 tham gia yen 2 nghich the voi the s6 thing sau no.
7 tham gia van 4 nghich the.
3n - 2 tham gia vao 2(n - 1) nghich the voi the se dung sau no.
2 khong tham gia vao nghich the nao vdi the se dung sau no.
5 tham gia yen 1 nghich the voi the se dung sau n6.
S tham gia vao 2 nghich the vdi cae s6 dUng sau n6.
3n - 1 tham gia vim (n - 1) nghich the voi the s6 thing sau no.
Cae s6 3, 6, 9..., 3n khong tham gia vao nghich the nao voi the s6 (hang sau
13
Vay co tat ca 2 + 4 + 2(n-1) + 1 + 2... + (n 1) - 3(n -1)n 2
(n-1 )n
nghich the trong hoot vi da neu va do d6 sgn S = (-1) 2
Khi n = 4k hoac n = 4k + 1 thi sgn 5 = 1
con neu n = 4k + 2 ho4 n = 4k + 3 thi sgn = -1.
Vi du 1.2
_ 1 2 3 Cho phep th'e' f - en dgu la (-1) 1
3 12 fn
Hay gag dinh da"u dm:
a) 1-1
b) g = (In
2
fn-) • n )
Lift( gidi:
a) Vi sgn f. sgn = sgn (f.
= sgn(Id) = 1 non
sgn (e 1)= sgn (f) = (-1) k
b) X4t phep the"a = 1 2 nj
n -1 . 1
thi g = f. a
Do gay sgn g = sgnf . sgn a.
n(n-1) Nhung sgn a = (-1) 2 nen sgn g = (_])k+C;',
14
Vida 1.3
ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf
j) v6 ben trai Wring throng v6i viac dai cha car s6 i , j a clang
drah cna phep the.. Cling nhu vay, nhan mat phop th6 Nth ehuynn
trf (i, j) v6 been phai tunng during voi del eh?, I, j a dong tit 66a
phep th6.
Lo gidi
Gia sif a la
truong hop nhan
Gitisi2 a=
Theo d nh nghia
phop th6 cho j) la phep
ben trai tile la f = (i, j). a.
(3 n
1 2
(i, j) = ( 9 2
_ (1 2a
ai
nj
n
)
chuy6n tri. Xet
Trunng hop nhan ben phai dude ant Wring fib
Vi dy 1.4. Cho f va g la hal phop th6cua n strtn nhien clAu tien.
a) Chung minh rang có the' cilia f va g bang khong qua (n - 1)
phop chuyan trf (nghia la ton tai k phop chuyan trf a l , cr2 , ak ,
k 5.11 - 1 g = ak ak_,... a,. f).
b) Chling minh rang khOng tha giam bat s6 chuOn trf rah
trong cau a) titc la en the' chon f va g sao cho khong the dua f vd
g bang ft han n - 1 phop chuyan trf.
15
La gicii
a) Xet phep the g o f', phan tich g o e' thanh tfch cac vang
xich dOc lap T 1 , Tp .
g o e' = ... T2 .Ti
Neu kf hiOu m i nt do, dai cart yang 'dell T i thi
± m2 + =
rang mOt vong xfch (a l , a 2 , am) la mOt plop the
a cac s6 tv nhien Ui 1 den n sao cho a(a) = a 1+1 (i = m-1) va.
a(a„,) = a l , con a(l) = 1 nen 1 yen moi i = 1, .,., m. Vong xfch
(a l , a 2 , u„) goi la ce do, dai m.
Ta da hiet rad yang xfch do, dai m deu phan tich duo
thanh m - 1 chuyen trf. Vi vay g o e' phan tfch duo. thanh 'Lich
caa i(m i -1)= n -p = k phep chuyen trf. i=1
Nhungpa. 1 n-1:115n-1
Nhu vay g o f -1 = ak (s, - chuyen tri)
TV do g = ak 0 ak.,, ... 0 0 f, k n - 1
va cac a, la cac phep chuyen trf.
2 ... nj ri
b) Cho g = la phep the ddng nhA 0.
( 1 2 3 n Oa 1 2... n-1)
f ve g duo Wang it hdn n - 1 phep chuyen trf.
on f = . Ta se chUng to rang killing dua
16
' WA met phep the" h = 1
1 2
ta not rang la
2 n
phan tit chinh quy ngu i. De" rang neat nhgn vao ben trai
cua h met chuygn Lri thi A:C . 1)111in tit chinh quy tang cling lAm la
met don vi. That way, ngu ngudc kg, cheng hen i, j la hai phan
tit kheng chinh quy cim h ma nob d6i dig h i voi hj ta Jai &tog
hai phign tu chinh guy (cum phep th6 m6i) th6 thi: h, < i. h, < j
nhang hj 2 i, h; j vti 1Y.
Do f chi co met phAn tei chinh quy. ye g co n phan tV chinh
quy, vi vgy khong thg clua f vg g bring it hon n - 1 phep chuygn tri.
Vi du 1.5
Chung minh rang vdi mei so k (0 <k < C;) t6n tai met
phep thg a e S„ co dung k nghich thg.
1231. giai
Ditch 1: Ta hay cluing minh met It& gug manh
Nigu a = (a,, la met hogn vi cda 1, 2, ... n va ace 1:
nghich thg, 0 < k < , thi có thg ct6i che hai phfin tri nao
do de thu ridge hog') vi c-3 co k + 1 nghich thg. That Nay, int&
hgt ta nhan thy rang negi > vdi moi i = 1, 2, ..., n-1 thi
coa nghich the'. Vi vay, do so' nghich th6 caa a la k < ,
nen ten tai i o dg oc ii) < a i0+1 ;
Xet hoan vi p = 0„) trong do 111 i = a , ngu i # i„, i„ + 1,
con p io = a ;0+1 , p i+, =a,„ thi HI rang 13 co nhigu hon a met
nghich the". Nghia la s6 nghich th6 ciga p la k + 1.
17
Ta tha'y A= B. ca do
1 1 0
1 B= vi C=
11
t1
Nhan cot thu nhal ciia ma tran A vdi -k rdi cOng vac) cot this k, ta dude:
1 -1 -2 ... -(n-1)
1 0 -1 - (n - 2)
1 0 0 .. - (n - 3)
1 0 0 0
det A =
Khai trio'n the() dung Ulu n, ta ea:
-1 -2 ... -(n -1)
-1 = (-1)"+1. (-1)"-'=1
Cdch 2.
ma detB =1, detC = 1 nen detA = detB. detC = 1.
Vi du 1.9. Hay tinh
cosa 1 0 0 0
1 2cosa 1
= 0 1 2cosa 0 0
2cosa 1
0 0 0 1 2cosa
21
Lai gidi:
Khai trim dinh thac Lheo cot cub" to co
D„ = 2cosa . - 17,1 _2
De thay D, = cosa.
cosa 1
2cosa, = 2coi2a - 1 = cos2a.
Gia sa D1 = cosia \TM moi = 1, . k.
Ta có
Dk,I = 2cosa . Dk - Dk_ i
= 2.cosa . coska - cos(k -Da.
= (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a.
Nhn vay D„ = cosna
Vi do 1.10
Hay Dull
1
1 eP + e -(1)
a do the phan to tren &tang choo chinh bang nhau va band
e q) +e -9 ; the phan tit tren hai &tong xien Win nhat \TM (Mang
the() chinh bang 1, con the phAn ta khac bang 0.
+ a-9 1 0
1 e" +e 1 A„ = 1
0
0
N x0
22
Khai trin theo cot tht nhEt, to c6:
An = (e P +e-P)A n _i:
e 21' - e-2(P Nlinn xet rang 4 1 = 6 9 +Cc =
A 2 = ((i e ro e 6 36 - -39
e (P -
e(1+1)6 - e -(lrv1),p
Girt sit AR - e (-0
, k - 1, 2, , n - 1. -e (P
Ta c6 An = (ec e-P)A n _ i -A n_,
=(eP +e w)enip - e npe( n-1) n4) - e e
(:+1)o - e -(n+1)4'
ew e q) - —
e (n+1)p - e-(n+1),p
Nhu v .6}.: A n =
Vi du 1.11
Tinh:
l l a 1 a, a n
1 a l: +h i a, a n
D = dot 1 a1 a, +b., a n
a 1 a.9 ... a n +b n
-
e 1 -e
23
Lai gidi:
LAY ciOng dAu nhan vOi -1 r6i Ong vao cac thing con 1ai , to có ngay D = b, 1) 2 ...b„.
Vi du 1.12
Cho da thric P(x)=x(x+1)...(x+n)
Hay tinh Binh thdc:
P(x)
P(x)
P(x +1)
P(x +1) ...
P(x + n)
P(x +n) d =
P(n-1) (x) P th-1) (x+1) P th-1) (x + n)
P thl (x) Pthl (x +1) P (n) (x +n)
gidi:
Ta b6 sung de' dude ma Han dip (n+2):
P(x) P(x +1) P(x +n) 0 P(x) P4x+1) P4x+n) 0
D = P( n ) (x) Pthd(x +1) P0P(x + n) 0
P(„+l) (x) 1301+1) (x +1) pg+0(x +n ) 1
RO rang det D = d
Nhan dOng 1111 k cua ma trail D vdi (x +n
dc-ix( 1) k-1 r6i (k-1)!
Ong vao clang Hirt nhgt vgi tat ca k=2, .. n+2).
Khi do, phAn tii dung dau có clang:
poc + 0 + P(k) (x +0.(x +11.) k ok = n). k! k=1
24
Pkx+1.) P(x+n)
PTUx +1) . P (n)(x+n)
PT+I kx+1) PT+Ikx+n)
d = del D - (x+n)°+i (n+1)!
(-1) 1T1 (x +n) n+i
n; con phan tit a cot cuoi bang
nghia 13 clang thg nhaa c6 dang: (n+1)!
(0, 0, , (+1)"+1(x+n)ni ). (n+1)!
Do do
Ta ki hieu dinh thge a v6 phai bai C va ma tr5n Wong ring
hai (6-1 Vi da thge P(x) = n(x + i) Ken P'"0(x) (n+1) !, vi vay i=0
the s6 hang a dOng cu6i &au bang (n+1) !. dgn gian ki higu
va each viek ta dirt xk= x+ k, k = 0, 1, n. d dOng thg hai tit
&leg len ciaa ( ta co:
(Pw(x0), P ("ax,), PT)(x.,)) ((n+1) hco + a l , , (n+1) tx„+
a do a l la hang s6 &do do. Khi nhan (long cue"' ding voi
r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang
P(x 0 ) P(x] ) P (x i) )
P (11-1) (x 0 ) Pth-e (x l ) PT-1) (x n )
(n+1)!x0 (n+1)!x 1 (n+1)!x n
(n+1)! (n+1)! (n+1)!
a l (n+1)!
(*)
25
Deng this ba tit dUdi len caa ma Dan (*) co dang
(n +1)! 2 xo +a i x o +a,,..„
(n+1)! x n2 +a i x n +a., 2
2
Cong vac) (long nay hai clang °Ma sau khi nhan voi cac s a, va a 1 ta nhan dude clang (n +1)! (n+1)!
(n+1)! 2 (n+1)! 2 (n +1) 6 9 xo ,
2 2 2
Bang each bidn ct6i nhu vay vdi cac dong con lai, ta clan m Dan ( '61) ve clang sau ma khang thay d6i clinh thfic caa no.
c = det = det
x on X n
n-1 x n1 X0
(n+1)! = (n+vn
k=1 IC !
X0
n(n+1)
(-1) 2 .((n +1)!6'1 .D n .
J k! k=1
26
6 do D„ la dinh thiic Vandermonde cua clic so" x„, x„
n-1
D6' thay D„= n(x k - x i )= n(c_i) = 11(n-o!= 1. k>i>0 k>1 1=0 k=1
Vay
d = detD = (-1) 2 [(n +1)! i n .(x +n)n+1 .
Vi du 1.13
Gia sU A e Mat(n, K), A= (au ) a do ali = 0 vdi moil= 1, 2...,
con aij bang 1 hoac 2001 \Ted i t j. Chung to rang ne"u n chan,
thi det A # 0.
Lbi gidi:
Nhan xet rang neh to them vim mit phan to a jj nao do cha
ma tran vuemg A mot s6 than, thi dinh thfic cha ma tran nhan
dticic se sai khac vat dinh thiic cha ma tran A mot s6 than. Vi
the ne'u to hat di 2000 don vi a nhUng phan t,i bang 2001 cha A, thi tinh than le cha dinh thfic cila A khong thay d6i, nghia la:
detA = detB (mod 2) a do B = (14
h i; = 0 n6u i= j va = 1 ngu it j.
Ta có:
0 1
1 0 det B =
1 1 0
nhan Bong ddu veli -1 fee cling vao cac dOng con lei ta dttuc:
detB =
1 0 0 -1
Deng vao cot thu nhat ca car cot con lai ta cO:
detB = . (n-1).
Vey khi n than thi detB la sidle, do vey detA # 0.
Vi du 1.14
Tinh Binh theft:
' 1 1 1 1
0 C 3 C I C i
2 n+1
D = det q C., C 241 2 Cn+2 ...
n-1 n-1 c c C2-11 C.!;1-1 1- .)
n n+1 2n-1 >
LtlL gill
Vi CI, +Clr = C74, nen hieu cUa moi phan tit vol phfin I
dUng b8n trai no thi bang phen td thing ngay tren n6. De tir
D, ta ldy cot thit n tr3 di del n-1, r6i ldy cot n-1 tie( di cot
2,..., ldy cot the] 2 trU di cot thU nhat, ta co:
28
1 0 0 0 0
C12 1 1 1 1
D = det Cy c?c, C 1 C 1„1
" •
011 C112 cn-2
C 1,1,12 2n-2
lai lam nhtt tren, to co
1 0 0 0
C l2- 1 0 0
D = det C3 C13 1 1
c^-3 n-3 2n-4
sau n -1 budc nhit v'ay, to
1 1 0 0 0'
1 0 0
D = det
cn-1
CI3
Lin-2
1
cn-3 C I
0
1
= 1.
VEty D = 1.
1.71 du 1.15
Cho A= cos9 - sin9
sin9 cow , fray tinh An .
Lai gidi:
cosy -saw cos9 -sine cos2p -sin29 Ta co A2
sine cost° sing cosy, sin2c cos29
29
Gia A k = cosky -sinkp vOi k = 1,
n-1. sink cosk(p
Ta ttnh
cos(n - lap - sin(n cosy - situp A" = A n-1 .A =
sin(n -1)9 cos(n -1)9 2 \ sing cosy ,
cos (n -1)9 cosy - sin(n -1)9 since -cos(n - lap since- sin(n -1 9 cosy
sin(n - lap cosy + cos(n - lap situp cos(n - lap cosy - sin(n -1)9 sirup
cosmp - sinmp
sinmp cosny
Vay = cosny - sinny
vdi 11191. n E N. sinny cosny ,
Vi (11..t 1.16
0 1 Cho A = hay tinh A 200"
0
Lbi gidi:
-1 0 0 -1 (1 0 Ta c6 A 2 =
0 -1 1 0 0 1
ma 2000 chia het cho 4
\ray A200° =(A)500 =I, (I, la ma tran thin vi cep hai).
Vi du 1.17
Ma Iran vuong A e Mat(n, K), A = (ad throe goi la Ina tran
phan del xang netu a ii + = 0 vdi moi i, j = 1, n.
30
Hay chang minh: 'rich caa hai ma tran phan doi xang A va la mat ma tran phan doi 'ding khi va chi khi AB = -BA.
Lai gidi:
Gia sid A = (ad, B = (b id d do ait + ai, = 0, b y = 0
di mm L j = 1, ..., n.
Dat C = A. B = (c,); = =1
D = B. A = (c1,0 = Ib ija jk id]
Ta ca.: c, k = Za ji b ik - Eb kia ji - d ki
Nha vay AB phan del xang <=> c, = V i , k.
tt=> = vet mot i, k c> AB = -BA. Vi chi 1.18 Cho A, B la hai ma tran vuong c9p n. Hay minh
let(A.B)= detA. detB. gidi:
G1a. sit A = (a,,) , B = (It o ) yea i, j = 1.
Ta lap ma tram an 0 0 a1.,al n
a91 a22 a 2n 0 0
ant a
n11 0
- 1 0 .. bth
0 b21 b2n
C=
0 0
31
Khai tri6n theo n clang dal) (thee dinh 12) Laplace), ta co detC = detA. detB. ( 1 )
Mat khac, bi6n &it ma Iran C bai phep bi6n ct6i sd ca) sau: Nhan cot thu nhal vat b 1 , cot thil hat Null cot thin n vet btl; r6i Ong vac) Ot thu n + j (j = 1, 2, n), ta dude ma tran D Bang sau ma dinh thiic cua D va cua C bang nhau:
a l a a d ta d
a21
a m am ... ann dm d , , ... d nn
-1 0 0
0 -1 0
0
a do = riaik b ki , nghia la (dij) = A. B. k=1
Khai tri6n theo n cot cuoi (theo dinh 157 Laplace) ta có
detD = det(dij) = det(A. B) (2)
Tit (1) va (2) va do detC = detD nen ta co:
det(A0 B) = detA . detB.
Vi rtu 1.19
Cho X la ma tran vueng cap n. Chung minh rang X giao
hean vOi moi ma tran vueng Ong ca) = X co clang XI,,, a do I„ la ma trail dun vi cap n.
D=
32
Lari gidi:
Ni111u X = 1 thi re rang X giao ho an vdi moi ma Han \along cling cap.
Ngnoc lai, gia sit X = (X„) giao haul vdi moi ma tran yang cdp n.
VOi i„ # j 0, to chiing minh x inio = 0. Mudn stay chon A = (ad
trong do a joio =1 con one phan tit khac ddu bAng Thong. Phan to
ding i o cot j„ cim ma tran XA Wing x. •' can phan tit a thing i o
cot j, cim AX la 0. Tir di6u kien AX = XA suy ra =O. Nhu
y X co dang: k r 0
„
0 k„
Cho ma tran A = (a) a do a ] , = 1 vii moi i, j. Khi do phan tit
o dung i cat j cim ma Han XA la X , con phan to 0 ding i cot j
dia ma tran AX la 2. nen k , =
Vi vay: X= 7. I,,.
Vi du 1.20
Cho ma Iran cap n:
a b
b a A=
b
a) Chung minh detA = (a-b)° (a + (n-1)b).
b) Trong trudng hOp detA s 0. Hay Huh ma trdn nghich dao
A+ oda ma tran A.
X=
33
Lai giai:
a) Gang the clang vao dOng this nhat ro'i nit
a + (n-1)b a clang dau, ta (Woe
1 1 ... 1
b a ... b detA = (a + (n-1)b) x
b b ... a
Lay dOng chin aria (Anil th.fie tit nhan voi -b r6i Ong vao the dong sau, ta eó:
1 0 ... 0
0 a - b ... 0 detA = (a + (n-1)b) x
0 0 ... a - b detA = (a + (n-1)b) . (a-b)" -i .
h) A kha nghieh <=> detA 0 .(=> a b va a + (n-Gb O.
Goi B = (b ii) la ma tran nghich (lac) cila A = (a„).
1 Ta Wet rang b id = detA . A„ a do A. ; la phAn Phu dai so
etia phan trong ma tran A.
a b b
a Vol mdi i thi = la nh thiic cap (n-1).
b b a
Theo phan a) thi A i = (a-b)" -2 . (a + (n-2)b).
34
A a+(n-2)1( Vay b„ -
det A (a 1- (n -1)1).(a - b)
Vol i # j = (-1)H , d do kl„ la chub thue e6p n-1,
co dupe ba' lig each x6a &Ong thu i va cat tha j eila ma tran A. Do
A dovi ming nen = Gia sU rang i < j, khi do cot thfi i va
dung thu j-1 cua Mo gain town nhang phan t& b. N6u d6i eh .6
Bong j - 1 len tren Ming dau (Mu nguyen cac dung khac), rei lai MS( nit i len 6;4 thu nhai (va van gib nguyen the cat khac), thi to (Moe:
m = = (-1)'34 b.(a-b)" -2 .
Nha v(6y b dot A (a +(n -1)b)(a - b)
Do do
a +(n -2)b -b 1
(a +(n -1)b)(a -b) -11 a +(n -2)b
C BAI TAP
1.1. Chung to rang mai phep chuyan tri dm (n 2 2) lh
mot phep the le.
1.2. Hay phan Lich eau pile') the! sau thanh tieh ene phep
chuy6n tri
-b
35
1 2 3 4 5 6 7 8 ' a)
8 1 3 6 7 5 4 2 )
( 1 2 3 4 5 6 b)
6 5 1 2 4 3
1.3 Tim s6 tat ca the phep the a e S„ sao oho a(i) x i vol mm i = 1, 2, ...n. Cluing t8 rang khi n chan, so" one phep th6 clang tren la m54 3616.
1.4. Ki hi8u (n, k) la secac hoan vj rim 1, 2 n ce dung k
nghich the.. Chiing minh cong thile truy 146i sau:
(n+1, k) = (n, k) + (n. k-1) + + (n, k-n).
vdi guy vac (n, j) = 0 nevu j < 0 hoac j > 9
1.5. Ta goi 45 giam ena phdp the f Ia hi5u cna s6 the phan
tic khting bat dog (nghia la s6 the pilau tit i ma f(i) i) va s6
clic Yong xich <IQ dai ion him 1 trong phan tich cua f thanh tich
car yang xich 458 lap.
a) Chung minh f cotang tinh chat chan la vOi do giam cua no.
h) Chling minh rang s6 161 thidu one nhan tic trong phan
tich cna f thanh tich the chuydri tri bgng 45 giAm cua f.
1.6. DM \h hai s6 x va n nguyen, n x 0, to ki hi5u r(x, n) IA
s6 chi khi chia x cho n : 0 r(x, n) < n. Chiang minh rang nal
n > 2 va a la so? nguyen, nguytin to' d61 vdi n, thi tthing Ung
ki—>r(ak, n) la mot phan tit cua S„,, k e 11, , n-1}.
1.7. Cluing minh rang moi phep th6 cap k > 1, dou phan
tich dupe thanh tich nhiing chuydn tri dang (1, i) vdi i = 1, k.
36
1.8. Xac dinh davu cua cac phop the' sau:
a) I 1 2 n n +1 n + 2 ... 2n 2n +1 3n
43 6 ... 3n 2 5 1 3n-2
b) 1 2 n+1 n+2 2n
2 1 ... 2n 1 2n-1,
1.9. Chgng mink rang:
a) Dinh thiec cap ha ma cac Phan to Wang +1 hoac -1 le mot
s6
b) Dinh attic ay Ion nhig la 1.
c) Dinh thew c)41) ba ma cac pga'n tit la 1 hoac 0 dot gia tri
IOn nheit biing 2.
1.10. Tinh dinh thew cap 2n D = det(cM),
a vai i= j
dO d i; = b yen i + j= 2n +1
0 i=j vet i+j 42n+1
1<i, j S 2n.
1.11. Cho A = (an) la ma trn cap n, a1 e R. Chung minh
rang detA khong thay dei, neit cac phein tei a i; ma i +j 15 dooc
thay bai so' dal cga no.
1.12. Chung to cac dinh thew sau day bang khong:
1 cosa costa cos3a
cosa costa cos3a cos4a a)
costa cos3a cos4a cos5a
costa cos4a cos5a cos6a
a)
c+-1 b+-1
c b
2 a + —1 a
2
1 c+-
c
b + 1 a+ 1 2
b a
(abc -1)
b) =
1 1 1
-1 0 1 a) Dn = -1 -1 0
-1 -1 -1
0 -1 -1
1 0
-1
1 1 0
1
-b1 a1 - b2 a l -b n
a, -b1 a2-b2 a, -b a
h) ; n > 2.
a n -b 1 an -b.,-b n n
1.13. Cho hai ma tran A. B sao cho:
5 11 1 14 A B = I, B. A -
J1 25 ) 4 y
Hay tam x, y va A, B.
1.14. Hay khai tridn dinh thiic va chilng minh:
b)
(a+b)2 b 2 a2
b 2 (b+c)29 c - a = c = (a + c) 2
= 2(ab be + ca)2 .
1.15. Way tinh cox: dinh thric cap n saw
38
1.16. Hay tinh dinh thdc sau:
1+x'2 x 0 0
x 1+x 2 x 0
a) D„= 0 x 1+x 2
0 0 1+x 2
nghla Dn la dinh thiic cap n ma cac phan tit tran during cheo
chinh bang 1 + x2, the plidn tit thuOc hai [Wang chat) grin during
cheo chinh bang x, the phOn tit con lai bang 0.
2 1 0 .. 0
1 2 1 .. 0
0 1 2 0
1
0 0 0 1 2
1.17. Cho da thite P x) = (x - al)(x - (x - a„).
a d6 a, la cac so thvc dot mot phan biet. Hay tinh climb thde sau:
P(x) P(x) P(x)
x-a 1 x-a, x-a n
1 1 1 a l a, a n
a;
a an-2
n-2 an -2
1.18. Xet hai ma tram ph& cap ha:
'a b c ' 1 1 1
A= c a b va J= 1 j
b c a, v 1 j-
2n . 1 dO j = e = cos— +. sm-
2n = .
3 3 2 2
b)
39
Way chang minh det J # 0. Hay tinh ma tr'nn AJ. tit do suv ra gin tri detA. Hay nen hal Loan Wong to kin A ya J cite ma 1.rn cap n.
1.19. a) Hay tinh dinh thing cap n san:
a+13 (A.13 0 0 0
1 a+11 a.{3 0 ... 0
14„ = 0 1 a+ ^ 0
0 0 0 0 1 a+ 0
1)) Chung to rang dinh tithe sal' khong plat thuhe vao y,, )
••• Ynt
1
g1+ Yl
(x 1 1111 1)(x1 11 37 9)
1
x 2 + y1
(x9 + N.1)(x9 + y2)
(x n +y 1 )
(x„ +y 1 )(x 1 +y2) D„=
n-1
11 (X1 + Yi )
1.20. Cho ma 14.'0
Hay tinh A u1° .
n-I Fl(x.) +y i ) n(xn +)'i) 1=1 i=1
( 1 -2 1 \
A= -1 1 0
-2 0 1
40
1.21. (ha si:t ma trgn A e Mat(m, n, va rang A = 1.
Jhisng minh rang cac ma trgn B e M(m, 1, K) va C e Mat(1,
1g A= B. C.
1.22. Chung minh rang ma trap A = a b ) d thOa man (
pIntong trinh: X2 - (a + d)X + (ad - bc) = 0,
0 do 1 2 la ma Iran don vi cap hai.
1.23. Cho ma trgn
9. 1 0 v 4 1
A=
90
vOi 4 x 0. Hay tinh A- '.
1.24. Cho ma tran vuong c5p 4.
cosa sincx cosa sina
cns2a :;in2a 2cos2a 2sin2a cos3o. sin3a 3cos3a 3sin3a cos4a sin1a 4cos4a 4sin4a
Chung minh rang A khg nghich khi va chi khi a ♦ kg (Ice Z).
1.25. Cho ma trgn A = (a) e Mat (n, 1H) ma the phan tit
doge cho bai tong that: (-1)H- CV yin
vOi i = j
0 vai i>j
6 do Chi.,' kHrn-k)!
- . Chung minh rang A2 =1,1 .
(9, la ma Iran chin vi).
91
1.26. Gth sa X = ())) e Mat(n, R),
1-1 a do x ii 4- (-1)")! Chitng minh X' = I.
01- 4,
Chit Se voi aeR k e N, ki hiqu
)a, a(a - 1)... (a - k +1)
- a
())) )0 )- 1
0 k! ;
1.27. Gia = yXx„ x.„„ x„) vdi (i = 1, 2, ..., n) la ham
[ cua cac bi8n doe lap x 1 , x 2, .. x„. Ma tran J = J(Y, X) = aYi dx• 1 i
dliciC goi ]a ma tran Jacobi cua phdp bi6n din, con Binh thud dm no duck goi la Jacobien dm phep biers den do.
Bay gin /cot m6i quan hq gilla n 2 ham vii va n 2 bien xii dune cho Isdi ding thiic:
Y= A.X.B, a do Y= (y i ) , X = (xi), A . B e Mat(n, R) la hai ma tran the trn6c.
Chung minh rAng det(J(Y, X)) = (detA)" . (detB)".
1.28. Cho X = (x„) e Mat(n, R) la ma tran tam gide dudi; va Y = X. X.
Chung minh rAng det(J(Y, X)) =
1.29. Cho Z lA tap cac sqinguyen; A, S hai ma tran vu8ng
cap n, cac phAn t> la nhilng s6nguyen (ta vie) A, S e Mat(n, Z)). Hun nua detA = 1, det S x 0.
Dal B = . A. S; Chung minh rAng cOs6m nguyen dding de Bm e Mat(n, Z).
42
1.30. GM sit trong ma tran A = (a u ) c Mat(n, R) da cho
rude tat ea the phan ti a d (i # j). Chang minh rang có thk dien Mo &rang char) chinh the s60 hoc 1 de ma tran A kheng suy Bien.
1.31. Tim tat ca cac ma tran A e Mat(n, K), A= (dj), 0
na tan toi ma tran nghich dao A-' cling có the phiin t5 khOng am.
1.32. Cho ma tran vuong A co the phan to la s6 nguyen. rim diet' kien can va du de ma train nghich dao cung c6 the
-Men to la s6 nguyen.
D - HDONG DAN HOAC DAP S6
Xet T e S„ (n 2 2) gie sii i < j va T = j, T (j) =
T (k) = k vdi moi k s i, j. Khi do the nglach the eaa
ji, k} Nob < k < j
j/, j} Nob / i + 1, j - 1
With vey co tat ek la U - i) + (j - i - 1) = 2U - I) - 1 nghich the Vi so nghich the le nen t la phop the le.
1.2. a) Phop th6 da cho phan Deb thanh hai yang xich chic lap (1 8 2) (4 6 5 7) = (1, 2) . (1, 8) (4, 7) (4, 5) (4, 6).
b) (1, 6, 3) (2 5 4) -= (1, 3) (1, 6) (2, 4) (2, 5).
1.3. DS. S6 Dat ca cac phep the a e a(i) # i vai moi
f n (- 0
n +1 . NMI yky A, (i =1, 2,..., n-F1)
i =1,n la n! E k0 kl =
1.4. Xet tap A gam at ce eke Minn vj (Ma 1, 2, ..., 14, n+1 co
thing k nghich the. Ki hieu A i la b° phkn eaa A gdm cac imam NO
a n , a n+1 ) ai =ma
la nhang tap con rot nhau caa A va A = A, U A2 U U... U AnA.
43
Xet Anki = {a = (a p a 2 ,..., an+1)1 = n +1}
NMI 0y so cac nghich the cna a bAng se cac nghich the cue
1 9 nghia la bling 1+ Dieu do cheng to so ea(
U Ch 2
Jhfin to cria A„,, bang (n, k).
Xet tap Ai (i = 1, ..., n). cis su a e Ai, a = (a, a j , a„ 4 ,) Theo dinh nghia a ; = n+1. Nhti vay (aj , khong la nghich th( vdi j < i va (x„ ai) la nghich the vdi j > i. Do do a, tham gia vac n+1 -i nghich the.. Xet hoan vi a' = (a l .— aa,_„ (>61, cCia S2 S5 nghich the maa a' bang sernghich the cna a trii NM; vay ta có met song anh tit A ; len tap cac hofin vi cua S„ co thing k-n-l+i nghich the., do do so' phan t& cera A ; IA (n, k-n-1+i). Td do, sephan to cUa A la:
(n + 1, k) = (n, k) + (n, k-1) + (n, k-2)+... + (n, k-n).
1.5. a) Xet phan tich f = a l o a, o ... a,„ thanh tich cac veng
xich dec lap di) dai > 1. Gia sa do dai a l, la d k; ta they f(i) # i khi
va chi khi i thuoc met trong cac yang xich do. Vay do giam coal
IA d,+€1.2 M6i vOng xich dai dk phan tich dude thanh d k -1 chuydn trf. Do vay f phan tick deck thhnh d 1 +... + (.16-1n chuyen trf. Do do kha'ng dinh a) driec chting mink.
b) Goi / a di) giam can. f. Theo a) f phan tich (kw thanh / chuydn trf. N5u có phfin tich f thanh h chuyen tri nao do, ta phai chring minh h 2 1. Ta có bd de sau: Neu a, 13 tham gia van met yang ;rich cas phep the f, thi khi nhAn f vdi chuydn trf (a, f3) (ve ben trai hay ben phai) yang xich dji not phan thimh hai \Tong xich d'ec lap. Con neu a, tham gia vao hai vOng xich cua
44
Thep the f, thi khi nhan viii chuyen tri (a, (i), hai yang xich se
Thep lai lam met (136 de nay a thing chfing minh). Tr/ do neu g
a phop the. va T la met chuyen tri tin dp Mem cua g o T khong
vire); qua do giam cfia g ceng them 1. Vi the nen f phAn tich
luec thanh h chuyen tri thi do giam cUa f khong \wet qua h.
1.6. H.D. Do a va n nguyen t6 vfii nhau, nen a k khong chic
het cho n vdi mm k = 1, 2.... n-1. do de r(ak, n) la nhfing so
phan hied.
1.7. Vi moi phep the phan tich due() thanh tich cat ghee
chuyOn tri, nen chi can chfing minh bai town cho phep chuyen
tri(1,j)e Sk vei 1, j # 1. Ta ce (i. j) = (1, i) . (1, ) . (1, i).
1.8. Xem vi du 1.1
- 1-1(n+D
a) DS: (-1) 2 .
r(1121)
h) DS: (-1) 2
1.10. Iasi trien cot dau, to ca. D ykk =(a 2 _ " "--1 2n-2 •
Do D2 = -13 .2 nen D 2n = (a 2
1.11. A = (a id , detA = E sgna ai,(1) ana(N E S..
Trong moi tich a ko(k) a ngfrik , tong E(k +cr(k) )= n(n +1) k=1
la sir than; nen re met se cliSn the thfia so ak ., 0.;) ma tong k+G(k)
le. VI Vey khi thay ha; nen i + j 1e, thi cfic tich
a nc(k) khong dei, do do detA khOng del
45
1.12. a) COng dOng thu nhat vdi (long the./ ba, ta dude don tY le vdi (long thu hai.
b) Nhan dong thd nhat vdi (-1), rdi cUng vao die (long th hai va Ulf( ba, ta dupe hai dOng CY 15.
1.13. Ma tran A, B e Mat(2, K). Ta có hai bat bian detAB
detBA va tr(AB) = tr(BA). Vi vay:
fx+y=30 {x = 20 {x =10 hoar
x.y = 200 {y=10 {y=20
20 14 \ a) x = 20 va y = 10 o BA=
14 10
20 14' 5 11 Ta co ABA = A A.
14 10 , 11 25 2
Tit do ta tinh dude A = 5k -14X
, A E R -11k 0
va B = A - 5 11
11 25,
10 14 b) x = 10 va y = 20 BA=
:14 20,
10 14 5 11 -3 5 l ABA = A X A= P.
14 20 4 11 25, X 55 32
1.15. a) D i, = 1.
b) A n = -A„. 1 + 4,1 = 1.
Tit do suy ra A 2p = 1, A 2p+ , = 0
46
1.16. a) Khai tri4n D„ theo cot thu nhat, to cee
D T, = (1+ x 2 )D n_i -,4 211)„,, , D, = 1 + x 2 ,
D 2 = (1 + x') 2 - x' = 1 + x 1 + x 4 .
Taco: Da -D„-1= x 2 (D„_i -Dn _ 2 )= x 1 (13 1-2 - D n _3 )
x 201-2) (D9 _ D1). x 21
Tct do Dn = Dn _ 1 + x 2”
Do vay D r, =1+ x 2 + x 3 +...+ x 2"
b) Ap clung eau a) vdi x = 1. DS (n+1).
1.17. Khai tri .on A(x) then (long tha nhat, to co:
P(x) P(x) A(x) = D(a 9 ,... a n )
x - a l -
+ ( 1)"7 I3(x)
; a 46 D(a 2 , x -a n
la dInh Dane Vandermoncle eaa cac s6 a 2 , , an;
Cho x = a,, to co A(a l ) = (a, - a 2)... (a, - a„) n(a j>i22
A(a,) = (-1)" n(a i -a 1 ). j>141
Thong ta A(a 2) = = A(a„) = A(a 1 ). Da tithe A(x) bac nhO
hOn hoar bang (n-1), co the gia Da hang nhau tai n diem phan
biat, vi vay A(x) la hIing.
Tit do: A(x) = (-1) 11 ' Fr (a 1 -a.). x ISIS“
47
1.18. J In ma tran Vandormonde vat (Me s6
biet nen dot J * 0.
Ta
a+b+c a+ bj+ cj2 a + bj 2 + ej
A . J = a+b+c c+aj+bj 2 c+aj2 +bj •
a+b+c b+cj+aj2 b+cj2 +ttj
dotAJ = (a + b + c) (a + bj + cj 2) (a + bj 2+ cj)det
1.19. a) Khai trien theo cot the] ula. La
D„ = (a + a0D«-s.
Ta co: D, = a + 13, 1/2 = az + an + 13 2 ,
k
Gia sit D k = Ea i f3 k-i (k = 1, 2, n-1). =o
n-1
Ta co = (a +p) a ir -1-i - an Ia'0 2-2-I i=o i=o
n-1 n-1 n-2 = a i+lpn-i- ork ira .2 a al n n-i-1
1.0 i=0 i=0
n-1 n-1 = r3 n ct i v _i-
i=1 ]=1
= E =o
Mtn tray D„ = Ear . i=o
4 8
b) Cheng minh qui nap thee n.
1 1 I, n = 2 o D2 = = x2 - x, kheng phi thuec y,.
+ y i x, +yi
n = 3 D„= (x, - x,)(x,, - xi)( 119 - x1)-
Gia stir Dk =
11(X j -x i ) vdi moi k = 2, n-1. Sii <MX
Ta chting minh ding thim tren dung voi
Khai triL 1) 1, theo clang flu./ nhal, r6i rut cac thira soi chung 6 die cot ye sit (King gia thiel qui nap, La cc):
D i, = (-1)" n( Yi) n(X j - X1) (1); k=1 /=k 151<“n
i,j #1:
Neu ee i x j ma x, = xp thID„ = 0; ve ming theft quy nap Van dung.
Neu x ; x x vol moi i #1; thi to Ming thee (1), to co:
D T, = 1 i<g nI
(4) k-1 n (xi +
44 11(x, -x k ) Fl(x k -x j )
j>k
Net 14(y 1 ) • (-D kjjl 11(x 2 +y 1 )
lxk
k=1 n _ x, n (x k k=1
(x / +y 1 ) ixk
(x j -x k ) jxk
P(y,) 13 da thvc bUc < n - 1 doi vdi y,. Vbi y, = -x ; win moi i = 1, 2, Ta co P(-x,) = 1, do de P(y,) = 1. Tiido ding thirc quy nap three chiing minh .
49
1.20. agA=B+I3 L la ma tran don vi,
0 - 2 1' 0 0 0 \ B= -1 0 0 , t3do B 2 = 0 2 -1 ; W = 0
-2 0 0 \ 0 4 -2
Ding khai trie'n nhi thric Niu-ton to co:
A100 = (0 , 13 ) .100 13 +100.B+
100;99u 1- 2
1 -200 100
Suy ra A 100 = -100 9901 -4950
-200 19800 -9899
1.21. Vi rang A = 1, nen tat ca car (long tS , le vb7 mot dong.
Wiy ma tran C E Mat(1, n, K) la ma trqn dong do. Giq sv clang
thd icim ma trqn A bang b,. C, lay B = (b) la ma Lean cqt. Ta co A = BC.
1.23.
A -1 =
1.24. Tinh detA = Tit db suy ra ket qua.
1.25. Bat A2 = (b (), khi b i; = iaik a k; k=1
+ Vdi i > j, 13 ;; = 0 vi = 0 ne'u k j con k > j thi ak; = O.
50
+ VOi i <j bli = Za ik a ki =
= E( _ 1) k_i cry-1)Hc i ( k=i k=i
. • 0-0! _ z ( 1) )+1; (J -1 )•
k=i (i -1)!(j - i)! (k - i)!(j- k)!
(j-1)!
(j - k)!(i -1)!(k - i)!
= (-1)1+1 c 1 - 11 J-.(-1)kIck• 1 : fiat k - = / J
= (- 1) 1+3 cl-1• Z_.r J- (-1/ c i 3 = 0 , vi
1 -0
(- 0 1 ,c = 0 - =-0 . 1-0
+ Vert = j, ta co b ii = Ea ik =1 ki • k=i
NhLt vi,ty A7 = I„.
1.26. Triloc h6t to có nhan xet sau: Cho a, b, C e N, c 2 b
Khi do: b 7 (2.-1,'
E(-1) k
kzo - ‘a., - a,
Cong thtic tam co the thong minh guy nal theo a = 0, 1, 2, ...
va hiu y la = 0 voi k > a.
flat X- =(3.7 .5 ) th y ] , = [_,x lk x kj k=1
•• =
•
= k=r n-k, 3 - i ,
xot tren).
(do dp dung nhan
51
Gia sit = (20, to ed za = Zy ik x kj kst
kbl in-i‘ - Ora
k=1 Tr -1, „n - j,
k „ iEnn+1+1+k n- i
ki=1 „k -1 n
Neui<j zu =0
Neu i= j = 1
Ngu i > j z,i = E Enh-Ft
aXt<l) t b
day t = k -1. a = n - b = n - j).
Nha vSy zij = D_o b+t t! b!
a<t<b a!(t-a)! t!(b-t)!
b
b ! x ( _ i ) 1)-0
a 1)+1 t-a Cb 1) C b_ ik
a<t<b (b - t)! (t -a)! t=a
Ixa = (-1)am
s=0
Nha vky = 8,j nen Xx = I„.
1.27. Trade het to Chang mink rang ne!u Z = AX,
a (Id Z, A, X e Mat(n, R.) thi det(J(Z, X)) = (detAr
That vay, gia sit Z= (20), 21; = a ik x kj . k=1
52
Tif do a ik n6u / = j
0 nau / j • (1)
Ta coi ma tran J(Z, X) E Mat(n 2 . 1K) vdi can dOng va can cot
duct danh s6bai can cap c6 thit Lu (i, j) , 1 < i , j n. Thii ti cac thing clang nhu cac ctit la (1, 1), (2, 1)... , (n, 1), (1, 2), (2, 2),...
az i (n, 2), (1, 3)... (n, n). Phgn t& a clang (i, j), cot (k, /) la 3 . Tn
ax k/
do, do (1) ta en: A
A J(Z, X) =
0
va nhn vay det(J(Z, X)) = (detA)".
Turing to n6u Y = Z. B thi ta clang ce):
' det J(Y, Z) = (detB) ° . Xet Y = A .X. B = Z. B.
Ta co: J(Y, X) = J(Y, Z) . J(Z, X). Mut vay
det(J(Y, X)) = det(J(Y, Z)) . det (J(Z, X))
= (detB)n . (det(A))n .
1.28. Gin s>1 X = (xi) e Mat(n, K); x,, = 0
vdi i < j. Ta có Y = X. Xt = (Y,)
Y;i = xi (x k Exik jk
k=1
Nhv vay Yii =
53
OY • Xetax 13 . Do (*) to có
,j aX
2X 11 vat i = j
X JJ j
x is ngu i=j,r=j
Ta có ay 'J ngu r = = j, j (*)
42/c i„, neat r = i = j
0 trong the twang hop con loi.
Ma trOn (1(Y, X) e Mat(n 2, R), the (Ping nä cac cot dttoc
darth s6 nhu trong bai 1.27. Phan tn a hang (i, j), cot (r, s) Pt 13 o rs
Ta nhan tha'y vdi (i, j) < (r, s) (nghia Itt i < r ngu j = s hotic dy
j < s) thi - 0. Mt n;
Nhu vay J(Y, X) la ma tran tam gitic &RR
Nhu \ray det(J(Y, X)) = aXij _ -
'J=1 jX j1
=1l fl - n(2xn)=- JJ j1 J
1.29. Goi A = (AO la ma tram phu hOp cat) A.
Ta co A41 = 1 A t - A l e Mat(n, Z). det A
Ta co B = SI 1 . A. S B rn = S4 . S
VI ma tran S g6m cac so nguyen nen k = Ida S e N* (do gia thigt dotS = 0). Vol m6i se; nguyen dming p, )(et ma tran AP e Mat(n, Z), gia sit A P (ap ; goi R la set du khi chia a!; cho
54
k = Idet S. Idat = 0'0 vi 0 < r i; < k, nguyen nen so' dile ma
tran dang Rp la huu hen. VI Oy t&n Lai p, q c N*, p > q sao cho
the ph&n tii taring ung cad AP va A P bang nhau thee modk.
Do do AP = +(detS).0 , C e Mat(n, Z).
Tit do: = +(detS).C.(Aql . that m = p - q.
Rhi do: I - 1 A"'S=1„ +detS.S -1 .C.A -4 S;
Vi S e Mat(n, Z), (detS). Sl' = S i la ma Iran phu hop cua S,
nen S e Mat(n, Z) va vi vity: B" c Mat(n, Z).
1.30. Ta chfing minh quy nap then n.
n = 1 : hien nhion,
a ll al p
a21 22
Neh cho track a 2 , . a 1 0 t 0, to chon a, 1 = a22 = 0.
Con nein a21 . a u, = 0, La chon a„ = a 22 = 1.
Nha vay n = 1, 2 thi bad Wan dung.
Chi sit bai toan dung laid moi dinh tithc cap < n - 1.
Ta chiing minh no dung ved dinh at& cap n.
Gia silt A = (ap) e Mat(n Xet A„ la phan phu dai so' caa
ph .dn tit an . Theo gia thi6t quy nap, to da Hein dude a22, a„„ de?
A, 1 0.
n=2, A= =an a 22 -a 21 a t2 .
55
DAt a ll = x, khai trien theo clang thu nhAt ta co:
detA = x. A, 1 + Zali A li
Ta co I:A uA li la hang set NAu hang so nay kluic khang, ta J=2
chon x = a„ = 0; nAu hang s6 do bang killing. ta chon x = a t , = 1 se. clime detA 0.
1.31. Trade he'd ta chting t3 rang nen ma Iran A va A-' E Mat(n, co the phAn t& kheing Am. Chi a m61 cot cila A co
dung met plan
ThAt vAy, gia sa a cot thu j cua A co 2 secdim/1g, a clang i, vA dOng i 2 . Chon clang k vdi k # j cila ma tran A'. VI A -1A = I„ non tich cua dOng k cua A' vdi cot j CEla A bang 0. Nhu va t), cac phzin t0 cfm cot thil i, va i 2 nam tren clang k deli bang ki:Mg (k x j). Do do hai cot i 2 va i2 cua A-1 tY le vdi nhau. Diu do clan deIn mau thuan.
NMI vAy m61 cot cim ma tram A co dung met s6 clueing (con lai dgu bang 0). Dicing te, m6i deng cila ma tran A clang có dung met s6 throng. Giao hoan cac demg (hoac cac get) ta duoc ma tran chg. °.
Nguoc lai, tat ca the ma tran 'Then dude tt ma tran cheo (a li > 0) bang each chuygn clang host cot deu kha nghich. Ma trAn A-I nhAn dude to ma tran A bang each Iay nghich dao cac phan to khac kitting rdi chuyAn vi.
1.32. DS det(A) = ± 1.
56
Chuang 2
KHONG GIAN VECTO - ANH XA TUYE-N TINH - HE PHUONG TRINH TUYEN TINH
A - TOM TAT Lt THUlt
§1 KHONG WAN VECTCS
1. Dinh nghin: Gin sit K la mot traong. Mt tap hop V
kirk rung cimg vdi hai phep town "+" : V x V —> V
(a , p) 1—* +
va phep than " •" : K x V —> V (k, a) 1--> k. a
dime goi la mot kheng gian vec to tren twang K n6u no thOa
man cac tinh chgt sau. mot k, /, c K va moi a, p, y, c V,
to CO:
a) a +p=p+ a
b) (a+ p)+ y= a+(p+ y)
c) CO phlin tit 0 e V sao cho:
a + 0 = 0 + a = a vin moi a c V.
d) Vdi a e V, ton tai (-a) e V sao cho:
a + (-a) = 0
e) k(a + is) = ka +103.
g) (k + /)a = ka + la.
h) (k. / )a = It (/ a)
k) 1. a = a, 1 la phan ta don vi cita truang K.
57
Mot kh8ng gian vec CO tren throng 1K con &foe goi K -
kKong gian \ecto.
Khi 1K = R, kh8ng gian vac to &loc goi la kh8ng gian vec to
thee. Khi K = C, kh8ng gian vac to dude goi la kh8ng gian vecto
phiie.
2 - Sit crOc 15p tuygn tinh vit phu thqe tuye'n tinh
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 , ..., a n „
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, = = km = 0.
Cho he 14„ a id 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(
58
lap tuyen tinh t6i dai cim he da cho neu n6 la he dec lap tuyen
tinh, va nen them bet kST vec td a k nao (k E I \J) thi ta dude met
he phu thuec tuyen tinh.
Cho he hau han vec td {a, , , am} 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„ , en}, ; 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 o xl 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'.
59
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,. =1
60
Gia sit V la klafing gian hitu han chikt,W, va W2 la hai khfing gian vec to con dm V, khi do:
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.
61
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
flomK(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
ciaa W, thi f(e,) = E, = vaaha tran A = (ad goi la ma
62
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.
Cho ma Wan A e Mat(m. n, 1K), xem A nInt ma tran cua
inh xa tuy6n Unh f: K n —> K m 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 HomK(V, W), trong co se: e = (e„ , e n) f cO ma Wan
= (ad), nghia la f(e1) = Za ij e i . i=1
63
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 x i = b k (k = 1: 2, ... , i=1
dO a ki , 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.
Ma tram A = (ak;) goi la ma tram cac he sg.
64
a ll a 12 a ln a 21 a 22 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
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 xI =
Dado D la Binh thiic cim ma tran A.
65
3 - He phtiong trinh tuygn tinh thuAn nhiIt
HO phudng trinh tuy6n tinh thuan nhal ce clang:
AX = 0
(2)
Xet ma tran A = (a o) nhu ma Wm cua anh xa tuyen tinh f:
K" —>. Km trong cAc co s6 chinh tac cua 1K'l va Km, thi tap hop
nghiem cilia (2) chinh la Kerf. Mei cci sa caa Kerf goi la met hee, nghiem co ban cilia (2). HO (2) ludo có nghiem x, = = x„ = 0, nghiem nay goi la nghiem tam thuong. KM rang A = n, thi he chi co nghiem tam thuang. Khi rang A = p < n, thi tap hop nghiem la kh8ng gian vec to n - p chieu (n la s6 an ciia phudng trinh).
4 - He phudng trinh tuye'n tinh tong gnat
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. -
+ DAi the to cua phudng trinh cem he.
66
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 EndK(V). Gia f e EndK(V), W la
khOng gian con ena V; W chick goi la khOng gian con bkt bi6n
elia V n6u f(W) c W.
67
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 , A k 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 Ta thgy rang:
68
+ 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 EndK(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 EndK(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 , ..., e n } 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
69
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:
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,
f e Endx(V) ma da that ddc trung 94(X) cd dang:
i(X) = (>1 (X ,-X)`2 ... -X)s k
(cac &Ai mat phAn biet, i = 1, 2, ..., k), khi do V la King true tiEp
tha cac khong gian riAng suy rang V = x G3.? X 2 e X k va trong V co mot co sa (ei) = 1, n; (n = dimV) dA trong co ad do ma trdn din f tao bdi the khung Jordan
0
70
nam doe (Wang cheo chfnh so khung jordan cAp s vai phan
eh& X, bang:
rang(f - - 2rang(f - kildv)s + rang(f -
Ma tram clang tren cua f xac dinh duy nhAt sai khae each sap xap cac khung Jordan. Ma tran do dude goi la ma trail dang
chuan tac Jordan cila tp (tang eau f.
B- VI DV
Vi d4 2.1: Xet R", R!" la the khOng gian vec td tren R. Cho
x„ x 2 , , x„, la m vec td thuee R". ]E la met khong gian vac td
con eila ChUng minh rang tap IF the vec td ena K" dang
Zt i x i (t 1, , e E Fa khong gian vec to con ena K', dimF =
dimE - dim(E nN), trong de N la khbng gian cac nghiem tha
phtlong trinh:
E t i ., =0 (a do t i , tn, la cac An). i=1
Lai gidi :
Xet anh xa tuyan tinh u c Hom(Rm, R") a do u(t,, t„,) =
EtiXj . Khi do F = u(E), vi vay F la khong gian vec to con cna
R'. dimF = dimE - dim(Keru'), u' = uI E.
Kern' = (KerU) fl E = E (1 N.
Nhtt vat dimF = dimE - dim(E n N).
71
Vi du 2.2. Cho x i , , xi, la nhUng vec to khac khong trong khong gian tuy6n tinh V. Gia sit c6 Oleg bi6n d6i f e End V sao cho f(x 1) = x 1 , f(xk) = xk + vdi moi k = 2, ... , n.
Chung L6 rang he {x 1, , xj la doe lap tuy6n tinh.
Lb gidi:
Ta chiing minh quy nap theo n.
Via n = 1, thi {x 1} dee lap tuy6n tinh do gia thiat x 1 # 0.
Gia sit menh de dung vdi moi he fx„ xki; k < n - 1. Ta chimg minh menh d6 dung vdi k = n.
Xet t6 hqp tuyen tinh
ECiX i = 0 . (1) i=1
Khi do:
) = CIX ] Zcif(xi) = c,x, + Eci (x i +xi_ 1 ) = i=1 i=2 1=2
= ZeiXi . (2)
i=1 i=2
n-1 Tit (1) va (2) to suy ra: Ec i x,„ = = 0.
i=2 i=1
Do gin thik quy nap hee, fx,, , doe lap tuyen tinh nen
C2 =e2 = = cn = 0.
Tit do suy ra c,x, = 0 c i = = = c„ = 0 Nhu vSy he {x 1 , , xj doe lap tuy6n tinh.
72
Vi du 2.3. Trong khong gian vec to V cho he ak} cac
vec to doe lap tuygn tinh ma m8i vec to dm no la t6 hop tuyen
tinh cua cac vec to cim he 113„ , Hay ehung minh k < 1.
Ta eó the' gia thigt ••• • la doe lap tuy6n tinh, neu
khong to se lay he con doe lap tuyen tinh t6i dal eim {p,, , pi}
g6m m vec to va se elnYng minh k s m < 1.
Theo gia thigt a„ , a k bieu thi tuyern tinh qua D i , p, nen
ten tai cac vS hudng a.„ e K sao cho:
a ; ; 0 =1 k) (1)
Ta gia sit k > 1. Ta se cluing minh
tuygn tinh. Xet t6 hop euyen tinh
k 1
ak phty thuec
/Xiai
1=1
I
j=1
<=> i=1
( k
.i=1
j=1
pi =o
=0
<=> EauXi =0 j=1
(2)
vdi j = 1 do he pi leriic lap tuyen tinh.
Vi he phuong trinh (2) la he phuong trinh thuan nhat, có s$ an nhigu hon sg phuong trinh, nen no co ve s6 nghiem, nhu vay
73
ton tai nghigm (x„..., x k) khac khong. Tn d6 suy ra he la,, ...,a k phu thuOc tuydn tinh. Dieu nay trai vai gia thi6t. Do do k < /.
Vi 4 2.4. Xet hai khong gian con E I va E2 cua khong giar vac to E. Gia e>i E/E 2 IA khong gian thacing va h: E, —> E/E 2 li han chd caa anh xa chiou chInh tde troll E,.
1) Tim didu kien can va chi dd
a) h 1a toan anh
b) h la don anh
2) Chung tO khong gian (E, + E,)/E 2 (fang eau v6i kholu gian E i /E i nE2 .
L of Rich:
1) a) X& E —> E/E 2 la anh xa chidu chinh cac, h =
h toan anh a ME,/ =11(E).
E, + Kern = E + Ker n = E
E, + E2 = E (vi kerb = E2)
b) Ta en Kerh = E, r1 Kern = E, n E2
Nhtt vey h don anh a Kerh = 0 a E, E, = {O}
2) GiA. se F = E, + E2;
Xet F F/E2 va k = 1-1/E, •
Do phAn 1) k la toan anh va kerk = E l C Kern = E l 11 E2.
Do do to co E l /E l fl E2 ding can not E 1 2>E 2/E2 .
74
Vi du 2.5. GM. sit V la khong gian vec td tren truing K. Ty
Bong can p: V -+ V ducic goi la met phep chien ngu p 2 = p.
1) Chung to rang ngu p, q la hai phep chigu, thi p + q la
phep chigu khi va chi khi pq + qp = 0.
2) Chiang t6 rang p.q IA phep chi gu khi va chi khi [p,qI = qp
- pq la anh xa tuyern tinh chuye'n Imq van Kerp.
3) Vol p,q Fa hai phep chigu sao cho p+q la phep chigu, hay
cluing to Im (p+q) = Imp + Imq va Ker(p+q) = Kerp n Kerq.
Lei gied:
1) p+q la phep chigu ra (p+q) 2 = p+q
<=> p 2 + p.q + q.p + = p+q <=> p.q + q.p = 0
2) p.q la phep chigu <=> (p.q) 2 = p.q.p.q = p.q = p 2 q2
Er> p (qp — pq) . q q (pq — pq) (Imq) = 0
.(=> (qp — pq) (Imq) c Kerp.
3) Vi mai phep chigu p co rang p = trace p,
va vet cim tong hai anh xa tuygn tinh bang tong cac vet cim no,
cho nen ngu p + q la Agri chigu thi:
trace (p + q) = trace p + trace q.
va rang (p + q) = rang p + rang q.
Tit do dim Im (p + q) = dim Imp + dim Imq, suy ra
Im (p + q) = Imp ED Imq.
D6 °hang minh Ker (p+q) = Kerp n Kerq
to nhan thay kerp nKerq c Ker(p+q).
75
Nguuc lai vdi x e Ker (p + q) thi p(x) + q(x) = 0.
Do Im (p.+ q) = Imp $ Imq nen MI p(x) + q(x) = 0
suy ra p(x) = q(x) = 0, hay x E Kerp n Kerq.
Vi du as. Gis sit E va Fla hai khong gian vec td tren
trulang K; Horn (E, F) IA tap cac anh xa tuyeal tinh tit F Mn ]F;
Ft la khong gian vec to con cna F.
a) Chiang to rang £ = e Horn (E, F) / Imf c F 1) la khong
gian con cria Hom (E, F).
b) Gia. s5 F= K la mot phAn tich cua F thanh t"o"ng tryc tipi=1
ciaa khong gian F. Vbi mdi F, xet 2, = if E Horn (E, F) I Imf c Fd.
Chung to rang Horn (E, F) = 2.
WL gia'i:
a) Do g ding kin vai phep town tong anh xa va nhan anh
xA vdi mat vet hriong thuac K, nen hieln nhien la khdng gian
vec to con crIm Horn (E, F).
b) Vdi h E Horn (E, F), vbi m6i x e E, to phan tich h(x) = y
theo eec thinh ph'An y = h(x) = E yi , yi e F1 id
Xet E -, K, e Horn (E, Fi) = 2 hi (x) = yi
76
Va vdi moi x e E, thi h(x) = Eh; (x) i=1
Do 05 h= , e
Gia six Eh ; = 0 nghia la vial moi x e E,
( ) (x) = h i (x) = 0, h,(x) e F, 1=1 1=1
to do suy ra 11,(x) = 0 voi moi i vi F = e K. I=1
Nhu \ray Horn (E, F) = e g. =1
Vi du a 7. GO Q la Huang s6 h> u ti va WI la mot tkp hap
khong rkng. Ki hiku E la tap cac anh xa tif c32 vao Q. E la
Q - khong gian vac to viii hai phep than: f, g e E, a e cc9, k E Q the
(f + g) (a) = f(a) + g (a)
K. f) (a) = k. f (a).
Gia sif V la khong gian vac to con ciaa E.
1) Chang to rang: nku f e V va co n dim x„ x„E Ca
sao cho
ingui=j . . — k(x,) = = ej=1,n Ongui#j
thi he 10, •••, f,,} dOe lkp tnyen tinh.
77
Goi W la khong gian con cea V sinh bed f„), khi do
m81 g E V 11611 viet dude met each duy nhat dudi clang: g = h, +112 ,
a do h, e W, h 2 E V va. h = 2(x)= 0 yea mai i = 1,n.
2) Chung CO rang hai tinh chat eau day la Wong clueing:
a) dim V n
b) Ten tai n ham g„ thuec V va n diem x„ x„ ciaa
W( sao cho g, (x,) = 8, ; vet moi i , j = I,n .
Lai gidi:
1) Xet to hop tuyen tfnh EX i f, = 0, c Q. 1-1
Khi do vdi moi x E cam , LW; (x) = 0;
Chon x= x„ thi (xi) = 7.j = 0 ( = 1,2, n)
suy ra he {f„ , f„) doe lap tuyen tinh.
Val g e V, ham h, e W cAn tim phai thaa man
g(x) = (h, + h 2) (xi) = h r (x,) vdi moi
Do If„ f„) la co sa cem W nen h, :11 1 (x, = E gfrA . lei 1=1
Dat ha =g - h, eV
thi h2 (xj) = g(xj)-11 2 (xj) = 0 vdi moi j = 1 n.
78
2) Theo phan 1) neu b) clime th6a man, thi he (g„ g„) doe lap
tuyen tinh, do \ay dim V 2 n; nglila la menh de b) a) thing.
Ta chung minh a) suy ra b) bang phydng phap quy nap theo n.
Vdi n = 1; n6u dim V 2 1, tan tai fi x 0 va vi vAy có x, e c14
de fi (x,) = A x 0. Chon g, = thi (x,) = 1.
Gia sii menh de dung vdi moi k = 1,2, n.
Ta chiing minh nd dung vdi n 1.
Theo gia thi6t quy nap dim V n + 1 dim V ?. n nen tan tai n ham g, E V va n diem e A de gi (x,) = 8,, (i, j = 1,2, ..., n).
Goi W la khong gian con sinh bid {g„ g n}, to cd. dim W< dim V.
Theo cau 1), vdi f e V \ W, to cd f = h, + h 2, vi h, e W nen h2 0; vi th6 c6 x,,, d'e' h1 0; (x,„., x x, vdi i = 1, 2, ..., n).
h 2 hi -
) t bin+, vdi MOt i = 1,2, ...,
n+1. Bay gia = g - = 1, ..., n),
thi vdi j = 1, n to co gi (x j )= (x J )= s i; va gi (xn+1 )
=g. (x„. 1 ) - x; = 0 n6u 2, 1 = g i (x„,)_
Nhu Nty co n4-1 ham {g- i } th6a man digu kien bai town.
Vi du 2.8. Gia sit 98 la ho dem doe cac anh xa tuy6n tinh
tit R" (16n R"I; vdi mai a e R" xet 0(a) = {f(a) / f e :43}. GM sit
g la Anh xa tuyein tinh to den âR" sao cho g(a) E :0304 vdi
moi a e K. Chung minh rang g c PC.
79
Lo gidi:
Ccieh 1. \TM m6i x e R, xet vec to dx (1, x, x2 ,
H9 eac vee to d x c6 tinh chat:
a) vdi x„ doi mat phan biet, thi ta xi dPe ld
tuy6n tinh, vi Binh
# 0 (Dinh thite Vandermonde)
b) V6i m6i d x e R', tan tai f E sao cho g(Ci;( ) = f( ).
Ta phai chiing minh g c 93, nghia la c6 f e d f
g@t„ i ) tren bb la co so cua R".
Gia sii node Lai, vdi m6i f E c6 khong qui (n - 1) sd the
phan biet x i cl6 g(d xi ) = *xi ).
Do hop &dm Mtge eac tap bop hfiu han phan to 1a mat to hap kh8ng qua deM dude, nen sod cle vee to d x e ma g(d x
e g3(a x ) le killing qua d6m dude. Digo nay trai vdi gia thie
Nhu vay phai co co sa Vt xi ,...,a x ) oda an vti 06 f E tgi d
= f(di xi ) vOi moi i= 1,n; nhu vay g=f e
80
(rich 2. (Mtn}, cho doe der hied mot sa sri kien cilia khong gnin metric).
Theo gia thiet vdi moi a e der f e de" f(a) = g(a). nhu
vay — g) rad= 0 => a e Ker (rig).
NInt vay U (Ker (f — g) f E gq} = R.
Hia sir g vSy vdi mai f E thi f — g x O. do vay
dim Key (1— g) n — i. Vi vay Ker (1 - g) lie met kliong gian con
thing, khbng dau tra mat cent R". Dieu nay gay nen man thuan
do RS la khong gian metric day vdi metric dieing Hwang vii
dinh 19 Haire aid rang: met khong gian metric day kheng the bang hop dem dude mita nheing rap hop khong dim Hu mat.
III. du 2.9. Cho hai so nguyen dining r, n ma r < n:
(it c Mat (Th R). rg." = ran k rang cl = r m& (CP =
Hay chdng minh vet ()f = a,, + a i„ + + a„,, = r. (Vet can
ma Man red thudng duo( ki Heti bdi trace cel).
Xet f e End (IR") co ma trhn trong cd ed tly &nen
e = le,, e„). i(e,)= apie
Theo gia Hired to cd f2 = f.
Bat Y = - f(a) I a e khi de Y c Kerf, nen Yea a e
thi x = a - 1(a) e Keil, do vay a = f(a) + x e IMF+ Karl
8 1
Ta co Imf n Kerf = {0), that Gay, gia sit p e Imf Kerf, thi co a e R" d f(a) = p va do 13 E Kerf nen f(0)= 0 suy ra:
f(p) = 12(a) = f(a) = p = 0, vay Imf n Kerf = {0). Nhu vay R"
Imf O Kern dim Imf = r va f I Imf = id, Chon ed sa s Rua R" sac)
cho {E,, £,.} c Imf, {E„,„ E„) e Kerf, thi trace f= trace al= r.
Cho Nhaat lai rang n6u f la anh toa tuyeat tinh cie'n
Jrco ma Ran oat = (ad trong cd sa e = (e,, nao do ciaa Tthi
so' a„ +,...,+ a„„ khong phu thuoc vita vice chop cd so oda °L va
clack goi la vat ciaa anh xa tuyan tinh f va chiac kf hieu la trace f.
Vi du 2d0. Gia sa °W la hai khbng gian vac td treat
twang K, f c HomK CP; °Tf).
Xet anh xa f: Kerr Gil"
[a]-a f [al = f (a),
Hay chttng to f la dun eau tuyen tinh va Imf = Imf , ta do
f la dang cau to /Kerf len Imf.
ianh xa f lh sac dinh, khong phu thuR vao dai dien. That way, vdi [al = thi a - a' e Kerf, W do f(a) = f(a") hay II-al = f
thd thay f la Maya tinh va Imf =Imf.Ta chung to f la ddn cau. That Ray RR [a] # [a'] thi a - a Kerf f(a - = f(a) - f(a) # 0 do do f(a) # f(a) suy ra f [a] # f Nhu way f don cau va do
do f la clang cau tit cliKerf len Imf.
82
Chu" y: neat so chieu Irau hen, thi tfx vi du Hen to
dim tillierf = dim Imf say ra dim "P= dim Kerf + dim Imf.
du 2.11. Giei he phudng trtnh
+2x 2 +3x 3 +4x 4 =30
-x i +2x., - 3x 3 + 4x 4 -10
x, - X3 +x 4 = 3
x i + +x3 + X 4 =10
1 2 3 4
-1 2 -3 4 D =
0 1 -1 1
1 1 1 1
Day 1a he phudng Huh Cramer.
gthi:
= -4
80 2 3 4
1 30 3 4
10 2 - 3 4 -1 10 -3 4
1), =
D,= = -8 3 1 -1 1
0 3 -1 1
10 1 1 I
1 10 1 1
1 2 30 4
1 2 3 30
- -1 2 10 4 -1 2 -3 10
D a = -12 D'= = -16 0 1 3 1
0 1 -1 3
1 1 10 1
1 1 1 10
NMI vey , = 1, x 2 = 2, x 3 = 3, x 4 = 4.
83
Vi dii 2.12. Gihi va bran luhn theo thaw s6
A.X1 x, + x• 1
Xx, + x. 2,
+ 4x 3
I) =
a) Veil X s 1 vic a x -2. Day la he Cramer.
it +1 1 va ta ce, - , x -
X +2 2, + 2 +2
b) vdi A = 1, to c6 he Wring during vdi:
x i + x2 + x,, =1 hay x 3 = I - x, -x.,
do x i , x, lay trtyl,
c) Vol X = -2. He co dang:
-2x 1 + x., + x. = 1
x i 2x, + x3 g - 2
x i + x, - 2x 3 1
GUng vg still ve curt ba planing trinh Ln c6 0 = 3, nhn vhy he
voi nghiam.
84
Vi du 2.13. Dung phitong phap khn. hay giai he phuang trinh:
xt + 3x, + x3 + 2x 1 )( 5 = 2
3x 1 + 10x 9 + 5x3 + 7)( 4 + 5)(5 = 6
2x 1 + 8x, + 6x 3 + 8x 1 + 10x5 = 6 (I)
2x 1 + 9x„ + Sx 3 + 8)41 + 10)( 5 = 2
2x 1 + 8x, + 6x 3 + 9x + 12x5 = 1
Lm giai:
Nhan hai ye aim phtiong trinh (tau vdi Inning so/ thich hop,
I.()) acing vac) eac phtaing trinh khae, ta clia; he pinning trinh
()rang throng vdi he (I):
x + :3x 2 + x,, + 2x + X5 - 2
X9 + 2X3 x 4 + 2x5 = 0
2x, + 4x 3 + 4x4 + 8x5 = -2 (II)
3x, + 6x3 + 4)( 1 + 8x 5 = -1
2x, + 4x 3 + 5x 1 + 10x5 = -3
Nhan pIntong trinh thit hai cna he (II) vdi gag s6 thich hop
roi tong viva du; phuong trinh kink cita he, ta &toe he Wong
dyeing -
x 1 + 3x2 + x3 + 2x 1 + x5 = 2 (I)
x, + 2x 1 + x, + 2x5 = 0 (2)
2x 4 + 4x5 =-2 (3) (Iii)
x 1 + 2x5 = -1 (4)
(
3x 1 + 6x4 -3 (5)
85
Car phtfting trinh thit (3), (4). (5) trong he (III) la tudng during. Vi 04 he (III) tudng during vol. he
Giai
x i + 3x 2
y 2 +
he (IV), to (little
xn = —1 —
X2 = 1 —
x i = 1 +
2x 3
2x 5
2x.
5x i
+
+
+
2x 4
x - I
x 4
3x 5
+ 2x 5 =
+ 2x 3 =
a do x4 . x
2
0
-1
Inv 31
(IV)
Vi dv 2.14. Cho hai ma lien A, B thuOr Mat (n, K),
A = (ad, B = 09.
Khi do A + B =(a i +1)0 (bloc goi la tang hai ma tren A va B.
Chang minh rang:
1) I rang A -rang B l < rang (A + B) a rang A + rang B.
2) rang A + rang B -n a rang (A B) a min (rang A, rang B)
3) Nan A' = E, tin rang (E + A) + rang (E - A) = n. (3 do E la ma tran don vi cap n).
1) Gia stir f, g la hai ph&n tit cilia End ( â "), co ma 'Iran A, B
Wong fing trong m(llt cc; se( s = (s 1 ) di( cho. Khi do f + g c6 ma
tren A + B. VI Im(f + g) c Imf Img, nen:
.dim(lm(f+g)) < dim(imf + Img) dimlmg.
86
Tit de suy ra:
rang(A + B) < rang A + rangB.
Mat khdc:
rangA = rang(A+B-B) < rang(A+B) + rang(-B)
suy ra: rangA < rang(A + B) + rangB
Tit do:
ningA - rangB < rang (A +
ta: rangB - rangA < rang(A + B). VI yay
rangA - rangB 15 rang (A + B)
2) Ta co f: K" , g: K" la hai anh xa tuy6n
tinh, Im(f o g) = Im(f img) c Imf nen rang(AB) < rangA.
Mat khan: rang(AB) = dim(lm fog) < dimlm f = rangB.
Do yay rang(A o B) < min(rangA, rangB).
Bay gio ta churig minh
rangA rangB - n < rang(AB).
Ta co dim lle = n = dimIm(f o g) + dimKer(f o g).
Mat khde 26t anh xa Kor(f g)/Kerg Kerf (1 Img
a do tI/4xl = g(2), yin X E Ker(f 0 g). De they (To la (Tang au tuyeal tinh. Vi
dimKer(f o g) = dimKerg + dim(Kerf fl Img).
Tit do dim Im (f 0 g) = n - dim Ker(f o g)
= dim Img - dim(Kerf Img).
87
Nhu vay:
rang(AB) = rangB - dim(Kerf (I Iing) je rangB - dimKerf
rang(AB) -e rangB+rangA- n.
3) Vi A 2 = E (E la ma (ran den vi), nen:
(A - E) (A + = 0
Vt vay, then phan 2), to co:
rangA -E) + rang(A + E) - n <0
hay rang(A - E) t- rang(A + E) n.
M3t khac , then phfin 1)
rang(A+E) + rang(A-E) 2 rang(A = rangA = n
Do vay rang(A + + rang(A - E) = n.
V( du 2.15
Cho ma tran A e Mat(n, R), cac phan to (ran duang cheo
chinh bang 0, con cac phan to khac hang 1 hoac bang p, d (16 p
la mot eel nguyen lan hdn 2. Chung to rang A e n-1.
142i
)(et ma tran II = (h ij ) e Mat(n, 1R),
.3 do b ij = -1 vol moi i, j. Khi do rangB =1.
va A + B = (CO, vdi C,, = -1, con thc phfin to khOng thuOc during
cheo chinh bang 0 hofic bang p - 1.
Nhu vay det(A+B) = (-1)" (mod(p-1)).
88
Do p-1 > 1 nen det(A +B) # 0 hay rang(A+13)= n.
NhUng Chao vi du 14, to co:
rang(A+B)< rangA + rangB = rangA + 1
Do vay rangA 2 n-1.
Vi du 2.16
Gui sit (p la 1 -1101, L1.1 long cau run khang gian vac td phis n chieh V. Ching (Dinh rang tan tai (ig khong gian vec td con 9 - hat bin k chieu Vk:
0= V„ V, c c V„ = V.
gidi:
9: V —> V la Di dOng cau clic( C - khon g gian vec (5 V, 11611
ce mot vec td Hong e t
Bat V, = Vect< e l >, thlkhong gian v@c t d con 1 chi6u
(p - brilt biOn.
Gia sv (ili xily dung duo() car khong gian con (4) - bat Heal V,.
V2, ... Vk ma V, = Vect <e l >,
V2 = VITA. <ei , >.... Vk = Vect e h 8: dimVk =k.
Tel get long cau Wk : y \wk. vk [U. I H> [ (P( )]
89
cam sinh bai clang cau p. Do tir k IA ta clang can caa khong gian
vac to V/Vk tren truang se" phtic C, nen Co vac td rieng rea +i
ling vdi gil tri rieng can chfing minh:
Ubl Vect le, ....,e k , e k*, I
khong gian con k+1 chieu dm V va la - bat Man.
kik Xet re hop tuyan tinh e i = 0 .
i=1
Neu ?L k+ , 0 thi ek,i e Vk [ ek+1 = [6], trai vei vice k
chon{ a k ,, I la vac td rieng, vay 2 k, l = 0 tit do =0 nen iat
= 12 0 do he {a ; ...6 k doe lap tuyan tinh. Nha way he
lei , eki doe lap tuyan tinh = k + 1.
Ta chUng minh Vkk i lap - bat Bien. Chi can chang minh:
{14 ) e Vik*I-
Ta cif) pk k im(rp )1 ) 1 k- k+lk = ` T‘ -k+1 kk = •`k-hlk k-kk = `- k+1 k
p (e +, ) - X e k+ , e Vk C Vk+l, tit do suy ra
Bang each do, ta say dung clam cat khbng gian:
Vo = 0 c V, c c Vn = V p - bat hien
Nhain xet: Ma tran tha ta clang cgu p trong co sa 1 , ... e „} )(ay clang a tran c6 clang tam giac tren. Dung ngan ngil ma [ran, ta c6 the phat hie: moi ma tran phew vuong cap n dgu d6ng Bang vOl mat ma tran tam gile tren.
6 k+1 ) e Vk-kik
90
Vi du 2.17. (Dinh It Hamilton - Cayley)
Cho V la mot K - khong gian vac to; cp e End(V); 9 c6 ma
tran A trong cd so e = (ed i = 1, ... n; 1,, - la ma trail dun vi, khi do det(A - xI„) IA ma da thdc bac n, doe gut la da thitc da, c
trung cUa tii (ItIng cau cp. D6- thay da thuc nay khong phu thuc
van ed so e = (u) da chon. Ta cling goi da tilde tron la da HI& dac trung dm ma trail A.
Gia sit det(A - xI„) = a ox" + a l x" - '+... +a„_,x +a„.
Khi do to co: a 0A" + a,A" - ' + +an_,A +a„ = 0
hay 42'1 + a,""-1 + + an .Id = 0.
Lai gidi:
Xef ma luau B la ma trail chuyi1n vi ci'm ma tran phil hop am A - xI„. Cac phan ;I'M ma Iran B IA nhfing da thilc cua x
v6i he tiI trong truong K, c6 hac khong qua n-1. Ta
B = B0 +13,x + .„ + x" -2 + Bn _ 1 . x"-1
trong do Bo , Bi , B, 1 la nhung ma trait vuong cap n, khong phu thuiic x. Theo tinh chat dm ma tran phu 110p, to co:
(B0 + 11,x + +B R,. x" 1 ') . (A - xI„) = det(A - x1,). =
= (aox" + + + an_ i x + a„). I„.
Ta co mot he cac clang thitc sau:
B0 A = an . I.
- Bo = and • In •
B2A - B, = an-2 • In.
9 1
B„-,A - 13,12, = a, 1„.
-B„., = a,,T„.
NhAn vg voi vacua the clang their trcn Bin Met vdi I n , A. X- A" rOi enng Jai, to (kith
a n A n + + a, ^ A+a n I,= 0.
Chu Dinh 19 Hamilton-Cayloy thadng dude ghat binu:
MO ma tran vuting dela la nghiem maa da tithe dar trnn, ua ne.
Vi rig 2.18. GiA s t B la ma trap ley linh, la ma tran giro halm vdi B. Hay chtIng mink
det(A + B) detA.
Leh gidi:
a) Xet twang hen detA = 0.
cia stY n la bac lily linh rem B, nghia la Bn = 0. Ta re:
= zc nk A k B n-k = Ecnk A k w_k =
(A + B)" A Ick A k-1 n--k
k=0 k=1 k=1
Vi detA = 0 nen det(A+B)n = 0. Vi way
det(A + B) = 0 = detA.
b) Ttheng Mk) detA = 0.
Do AB = BA non BA-' = A-1 B.
92
Theo gii thiat B lay linh, nghla Isl co se n de 13" = O. tit do
-1 B)" = 0 vi vay A -1 B lay linh. NM( vay tan tai ma tran kheng
ay Men C do C - 911 -1 B).0 la ma tran dime tan hal nhning ma An inking Bang:
' 0 o 1 0
1
0 1 0
am riot thco during cher) chinh. tr .( do ta co det(1 + A -1 B) = 1, a do la ma Ban don vi (tang dip vdi B va A.
Nhu vay det(A+ B) = det[A(I + A -1 13)I= detA. (-Iot(I +2.1 11 B)= detA.
Vi dy 2.19. GiA six V la met khong gian vec td tren truang
va f e End,(V). Chiang minh rang Kerf = Imf khi va chi khi
= 0 vh ten tai h e Enci,(V) do h o f+ foh= Hy.
Didu ki(es chi: Tu f = 0 say ra Imf c Kerr.
Bay gia ta chUng minh Kerf c Imf. VOi IV la Itheng gian ni bat 14 cna V, to co:
(1) 0 f + f 0 II) (NV) c hof (W) + f 0 h(W).
Do ta co of + f oh= Id, nen
W c ho 1(W) + fo Cuh(W) vdi mei W c V.
Lay W = Kerf, thi f(W) = 0 WI ta
Kerf c f o h(Kerf) c Imf.
Nhu vay Karl= Imf.
93
Digit ki•en din
Gia si7 Imf = Kerf = V,. Goi V2 la phein bet tuygn tinh ego
V, trong V nghia la V = V, $ V2; p l : V V I la phep chi&
chinh tac, nghia la ven x e V, x = x, + x 1 , x, e V i thi p,(x) = x l .
Xot anh xa ?: V2 -)
x2 1—› 1 (x2) = nx2).
De thgy If la Bang ego. Dat h = I -1 o P I , to en:
f 0 h(x)= f -'. pi(x) = pi(x) = xi.
Thong tki h f(x) = ho f(x, x2) = ho f(x2)
= -I 0 Pi 0 f(g2)= f((2)= -1 • ?(x) = x.2-
Do vgy (10 h + h (x) = x l + x2 = x.
Do do foh+hoN Id o .
Tu gia thing Imf = Kerf suy ra f 2 = O.
Vi du 2.2a GiA si V la khong gian vac to tit' truong 1K. v?
f e End K(V). Vol m61 da thfic P e IK[x], P = ±a i x i , dat P(f) =0
'a; f' thu6c Endk(V). De thyIKPc] EndK(V) (13(P) = P(f =o
la mat d6ng cau tuyen tinh.
Da thitc Pf &toe goi la da thew t6i tie'u cum to d6ng au f new
P 1(0 = 0 vix n6u Q(f) = 0 thi Pf la vac cua da thew Q, voi Qe IK[x].
94
1) Gia sia V I la kh8ng gian con oda V bar bin do=i vdi f va f, = f I V, la clYng au cam sinh. Chung to rang da thfic to) tiaai P, ena f, la Pdc cua da thfic to) tiPu P r .
2) Gia sfi rang V phfin tfch throe thanh tong cac kheng gian
con baat bign del vdi f: V = la dying (Au cam sinh trim =1
V, va P, la da thuc tot tiPu cPa f,. Chung LO rang P, la boi chung nhO nhavt cua cic da thfic P.
Liii gidi:
1)Vi f = f1 nen vdi moi Q E K[X], La co Q(f) = WO fit VI.
kreh Pf lA da thfic ten tiUu rim f, thi P r(f1 ) = 0. Do vhy Pf la bOi da thfic toi ti6u P, oda f,.
Ta cling nhyn flay rang P, khang nhal thi6t la uoc thpc sp Lim P,, khi V, la klieg-1g gian con thpc so cim V. Vi du ne).1 '= %H", thi La c6 Pf = P, = x - vdi moi V, c V.
2) V, la khong gian con f bi6n, nen do can 1), da thfic :61 thi6u Pr chia hect cho vdi moi i = 1, 2, p. Do vyy Pf chia Mich° bOi chung nhO nhiit P cua Pi .
Ta chUng minh P(f) = 0. That vyy, vdi moi i = 1. p to c6 I)
= Q . P, tit c16 vdi moi x c V, x = Ix ; , I'(f)(x) = LP(f)(x,) = 1 = 1 1 =1
P p P(f; )(x i ) = EQ ; (fi )13; (fi )(x ; ) = 0 do P r (f) = 0 vdi i.
r=-1 i=1
NMI vyy P(f) = 0, to do P chia 116t cho da thik toi tiPu Pf ty dang can f.
95
X = (a,), A 1 =
2.1. Cho ma tran X E in. ]K)
7 0,6
)1 9 - H la cot tha j ca» ma tran X, =1,2, 1)
C - BAT TAP
§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINH
a )
B,, B.,. , B„ la cite cot cUa ma tran X. X'
Chang minh rang cite khong gian con rim R" sinh boa
A l , A„ va sinh bai 13 1, , B„ trimg nhau, tit de suy ra
rangX = rang(X.
2.2. Chang minh rang mei ma ran hang r d ou via dude
duai clang tang cua r ma tran hang
2.3. Cho E 19 la hai K - khang gian vec to va u e
Hom w (E, F). Cac vec tel x 1 , , x , thuoc Kern; y l , y, la nhang
vec to brit k9 thuec E. Chang minh rang hai trong ha khang
dinh sau keo theo khang dinh MEI ha.
1) lx,, , la cd sa can Ecru.
2) B(y,), u(ydl la cd sa cUa Imu.
3) K r , ys} la ed so ciia E.
Tit do suy ra nen E hau han chieu thi
dimE = dim(Keru) + dim(tmu).
96
2.4. a) GM sit E va Fla hai kheng gian vec to him han chieu tren truring IK, dimE = dimF = n. u e Hom K(E, F). Chung
minh rang u don cgu khi va chi khi u town
b) Neu vi du chUng to rang neeu E va F có so" chie"u vo han, menh da tren kfrong dung MM.
2.5. Cho a la mat ph6p thgbac n va u E End(C) xac dinh bar
u(Z,, Za) = (Z.K0 , Zem •• • ;II) )-
a) Hay tam ma tran A cim u trong co sa to nhien te r , &la C.
It) Tim tat ca the ma trait B e Mat(n, giao hoan duple vdi A.
2.6. X6t kh8ng gian vec td E him han chigu tren truang K.
a) GM s>i fe„ , e„) la mot co so cim E, a„ , a,, la nhiing vo hudng doi mat klMc nhau, u e End(E) the dinh bai u(e i) = ° ; e ; = 1, , n). Chung minh rang nth v e End(E) ye u.v = v. d thi ton tai nhUng vo hudng p„...,j3„ sao cho v(e i ) = [3 ;e; .
b) ChUng minh rang nth to clang cgu tuygn tinh u giao hoan vdi moi to clang eau tuyeIn tinh tha E, thi t6n tai ve hitting
E K u(x) = 7rx vdi moi x e E.
..2.7. Cho A =(aIii)e Mat(n, K); det(A) x 0; V la mat killing *4,
vec to tren truang K va uj e End(V), j = 1, ..., n. Chting minh
rang nth cac to deng cgu tuygn tinh vi = (i = 1, 2, n)
giao hoan vat nhau, thi cac u 3 cling giao hoan vdi nhau.
97
2.8. Gia sil A e Mat(n, R), detA # 0 va trong moi (long efn
A co dung mot s6 khac khong, bang ± 1. Chung minh rang:
a) Al = A-1
b) Co seta nhien m de Arl = A'.
2.9. Cho V la khong gian vac to Hen truing K va u
End(V), x la vec to cria V them man IOW = 0 va u° 4 (x) # 0 v6 mat se' nguyen throng q nao do. Chimg minh rang he {x = u°(x)
u(x), , 10-1 (x)} lap thanh mat he [lac lap tuygn tinh.
2.10. Gin sir) V la lit - khong gian vec to n chialt; f, g
End(V), Id la anh xa clang nhdt cim V. Chung minh rang nei Id - g o f la clang cdu dm V thi fog - Id cling lA (tang cdu ciaa V.
2.11. a) Hai tu citing cdu u, v E EndK(V) duac goi la Worn
during ngu c6 die dang cdu p, q cem V sao cho uep=q v Chung CO rang u va v taking during khi va chi khi chung NS cum hang.
b) Tit a) suy ra rang nalu X E Mat(n, K), hang X = r, thi tar
tai cac ma trnn khong suy bP6n P, Q e Mat(n, K) sao cho:
'Ir 0 I, 0 X = Q . . P a do la ma tran vuong cap n
0 0 0 0 ,
al gee tren b8n trai la ma trnn don vi h cap r.
2.12. a) Cho V a khong gian vac to n chigu tren trUang
Ira u, v e End(V) sao cho u o v = 0.
Chung minh rang hang(u) + hang(v) < n.
98
(^ri±ng 11111111 rang vo. mai hi ci6ng eau u e End(V) deu
Ong cdu V e End(V) sao cho u 0 v = 0 va
hang(u)+ hang(v) = n.
2.11 Cho E, F, G, H la cac khong gian vec td hitu Mtn cht4u
men trodng K, u c Hom(F. 0), u ce hang r. Hay tim hang dm
cac anh xa tuyeho t(nh sau:
a) cp: Hom(E. F) —> Horn(E, G) u v
b) di; Hom(G, H) —> Bom(F, H)
F--> 0)01.1.
2.14. Cho E la khong gian vec to n chien, u, v e End(E) sao
cho rang (u - Id E) = r, rang(V - Id E) = s. Chiing minh rang
range o v - id s) r + s.
2.15. Cho E la khOng gian vec to tren trtiang K Nth u e
End(E), rang u = 1. ChUng minh rang t6n tai 1. e 1K a u2 =
Min ntm, neAl X # 1 thi Id E - u la clang mill.
2.16. Gia s>i V la khong gian vac td tren trtfong so thuc R;
dim V = n; f e End (V).
Chdng minh rang en sdN nguyen ding sao cho rang (fk) =
rang (F`') vol moi k 2 N.
2.17. Cho ma tran A e Mat (n, K), A = (a id ma a ii = 0 vol
moi i = 1, 2, n. Chimg minh rang t6n tai ma trail B, C e Mat
(n, R) de A = BC — CB.
99
2.18. Cho hai ma tram A, B E Mat (n, K).
a) Chiang minh rang ma tren A d6ng clang vol X In thi A=X I„ X e ]K, In la ma tran ddn vi.
b) Neu A, B la hai ma train d6ng dang, thi A2 deing clang vet B 2 , At &Ong clang vdi 13' va ndu A, B kha nghich, thi A -1 &ling dang vdi B- '.
2.19. Chung minh rang m8i ma trail vuong cep hai tren truring K &et' d6ng clang vdi ma tran chuydn vi eila no.
2.20. a) Chung minh rang to d6ng ceu tuyen trnh u e End(V) thCla man didu kien
u2 (Id — u) = u (Id — u) 2 = 0 thi u lily clang, nghia la u 2 = u.
b) Flay tim phep bign del tuygn tinh kluing lily clang u thda man (Id — u) = 0.
c) Hay dm phep bign ddi tuygn tinh khong lily ding u th6a man u (Id — u)2 = 0.
2.21. Cho V la kh8ng gian tree td tren trtiang K.
a) Neu f, g la nhfing dang tuyen tinh tren klreng gian vac td V thOa man f(x) = 0 nett g(x)-e 0, thi co v8 Inking a e K M f=a. g.
b) Cho f, f„ f„ la nhung dang tuyen tinh trail khong gian •vdc td V sao cho f(x) = 0 neM fl(x) = 0, i = 1, 2, ..., n. Chung minh rang ton tai nhung vo hudng a l , c(2 , , an sao cho f = a 1 f, +a2f2
+ + a„fn .
2.22. rot khong gian vec td n chigu V veil cd sa le l , e„) vk V* la khong gian lien hop ens, V, nghia la khong gian cac dang
100
tuyen Utah tren V. Vdi moi = 1, 2, n xec dinh clang Wyk
tinh e V* bait
Qej) = Sij = j
1 i = j
Chung minh rang (£ 1 , ...,1„1 la cd sa cua V* va do do.dim V*
= dim V = n. Cd sa fb, f„} ciudc goi la cd sa lien hop hay ed sa
d6i ngau N:i ed sa ••.,
2.23. Chang minh rang nen fil, la nhfing dang tuy6n
tinh tren khong gian vec to n chieu V, vdi m < n thi ton tai
phiin ti x t 0 trong V sao cho fi (x) = 0 vdi moi i = 1, 2, ..., m. TU
do hay suy ra met k6t qua v6 nghiem cua he phudng trinh
thuan nht.
2.24. Cho V* Fa khong gian lien hdp cna khong gian vec td
V; u e End(V). Xet u*: V* —> V* the dinh bat u* (y) = you.
a) Chling minh rang u la phep Men dot tuyen tinh cua V*,
u* chive goi la phep Bien d6i tuyen tinh lien hop vdi u.
b) co sa{e..., en} (1) cila V va cd sa{f i , •••, in} (2) cua V*
lien hdp vdi ed sa (1). Chung minh rang ma tran (l u), cua u*
trong cd sa (2) la ma trait chuye'n vi cua (ct ii)„ cua u trong Cd sa (1), nghia la Po = a3; vdi moi 1 5 i, j n.
2.25. Cho V* la khong gian lien hop vdi khong gian vac td
V, S la be phan dm V. RI hie, u 8° la tap tal ca the phan to f E
V* sao cho f(x) = 0 vdi moi x e S.
a) ChUng minh S° la kitting gian con cua V*
b) ChUng minh rang nett dimV = n va S la kitting gian con
m chieu cua V thi dim S ° = n-m.
101
2.26. GiA s& V* IA khong gian lien hop vdi V, T la met be phan ciia V. Ki hieu T ° IA tap tat ca cat x e V sao cho f(x) = vdi moi f E T.
a) Ch'ing minh I"' la khong gian con ciaa V.
h) Xet S c V va S' ) c V* duct dinh nghia a bai 2.25. Chung minh rang (St la khong gian con oils. V shill bal. S. Dao biet.
S la khOng gian con thi (S")" = S.
c) Gia st V hfiu han chi6u. Chung minh (T") ) la Xining gian con cfm V* sinh boi T.
D4c biet, neal T la khong gian con cUa V* thi (T") ) = T.
d) Chting t6 rang Meal V ve han °hien thi trong Wee khong gian con T sao cho (11° ) ° # T.
2.27. Cho V* la khong gian lien hop ciia khong gian vet td V. IV,I, €1 la ho nhfing khong gian con oda V. Chiang minh:
a) n VID = ier Biel
(
.0 1, fly; =IVi° neat V Mitt han chi6u.
IET / ki
c) Khang dinh b) van ching n6u I Mill han, con V ce the vo han chigu.
d) Khang dinh b) khong con dting neal I ve han va V ve hat. chi6u.
2.28. Cho (1',), e i IA ho the khong gian con ciaa khong gian V* lien hop vdi khong gian vec to V.
102
Chung minh:
a) ETi I = nrE \ iel iE=.1
n = n6u V heti hen chiau. ieL
c) NC, u N.2 va han, thi trong V* cia hai khong gian
can T, (T,11T,r, Ti r12.
2.29. Cho E la te'ng true tipcim hai khong gian con V va
W; x e E viet dupe duy nhat dUdi clang x= y+ z; y c V va z e
W. Ta goi phep chigu cea E len V song song vdi W la clang eau
tuyen tinh p: E —> E xAc dinh bai p(x) = y.
Ching minh rang cl8ng cad tuyan tinh p: E E la phep
chi6u khi ve. chi khi p= = p.
2.30. Cho E la khong gian vec td va p e End (E). Chiing
minh rang p la phep chi6u khi va chi khi Id — p la phep chi6u.
Khi do Imp = Ker (1d-p); Keep = Im (Id-p).
2.31. Cho V Ea khong gian vec td tren tnidng K va f E End(V).
Chung minh rang P = 0 khi va chi khi Imf c Kerf.
ChUng minh rang trong trUCing hap nay g = hi + f la melt tkt
deng ceu dm V.
2.32. Cho V la khong gian vec td hilu han chiau va u e End(V).
b)
103
a) Cluing minh rang co hai co sa {e„ e„) va {E D ..., e n} cna V via soi nguyen s (0 s n) sao cho u(e,) = 6, vdi i i< s va u(e) = 0 vii s < i <
b) Tit do suy ra bin tai to clang caiu tuygn tinh v mart V sao cho you la phep chigu.
2.33. a) Cho p la met phep chigu caa kheng gian huu han chigu V. Chung minh rang trong V co cd sa trong do ma Wm A = (a3) ena p co clang dac biet: a,4 = 0 ngu i # j;
a il = 1 neu 1<i< s va a;; = 0 ngu i> s.
b) Tu (Icing Pau tuygn tinh u duos got la del hop ngu u 2 = id. ChUng minh rang clang thuc u = 2p - id thigt lap mat song finh
tit tap tail ca cac phep chigu p trong V len tap Cal ca cac doi hop
u. 6 day V la khong gian vec to tren truing K voi dac s6 khac
hai.
c) Tit a) va b) co thd not gi lid ma tran cua phdp del hop teen mat khang gian huu han chidu.
§2 HE PHUCNG TRINH TUYEN TiNH
2.34. Tim met he nghiem cd ban caa he phuong trinh sau:
x1 - 7x 2 + 4x3 + 2x 4 -0
2x 1 -5x.2 +3x3 +2x4 + X 5 =0
5x 1 -8x.2 + 5x 3 + 4x 4 +3x5 = 0 4x 1 -x2 + X 3 + 2x 4 + 3X5 = 0.
•
2.35. Cho hee, phudng trinh thuan nhat:
2x 1 5x, + x3 + 2x4 = 0
5x 1 - 9x2 2x3 7x4 = 0
-3x1 7X9 X3 + 4x 4 = 0
4x1 6X2 + X3 - kX4 = 0
Hay giai va bien Man then A he phudng trinh Wen.
2.36. ret he phudng trinh:
1 a(b -c)x + b(c -4 + c(a. -13) z = 0
(b-c)x 2 + (c -4 2 + (a. -b)z 2 = 0
GM sit a, b, c cloi met phan Wet. Chung to rang he tren co
ghiem hoac x = y = z hoac
ab -be + ea bc -ca +ab
-
ca -ab +bc
2.37 Gigi he, plutAng trinh:
X -X1 + 2 4- 1 X3 x 4 a - x2
▪
X3 + X
xl + X9 + X3 + x4
X1 + X2 + X3 + x4
2.38. Giai va bien Man he phudng trinh tuyen tinh sau:
2x + y z = a
X + my z
3x + y mz = c
105
2.39. Tim digu kien din va du a 4 dim A,(x l , VI), Ai(xi, 31 2), A3(x3, 310, Ai(xi. y1 ) khong ce 3 digm nao thang hang ngm tren met &tang trOn.
2.40. Tim da the bac 3 vdi he so thoc f(x) sao cho £(-1) = 0, f(1) = 4. f(2) = 3 va f(3) = 16.
2.41, Ch9 A = (a ig e Mat (n. K) sao cho dot A = 0. Goi la phan phu dal ce cUa phgn to a,,, gia thigt A 11 # 0.
Hay tam mat he nghiem co ban cila he phtiong trinh thuan nhal sau:
L Auxi 1=1
§3. CAU TRUC CUA MOT TV DONG CAU
2.42. Ching minh rang m6i gig tri rieng ciga phep chiegi dgu bang 0 hoar 1 va m6i gia tri rieng oda phep dM hop dgu bang 1 hogc bang —1. (xem bai 2.29 va 2.33).
2.43. Cho A, B la hai ma trail vuong gang
a) ChUng minh rang neu A, B deng Bang thi cluing ca clang met da fink dac trting.
b) Chung minh rang n6u A, B ce cling da thug dac thi det A = det B.
c) Cac khang dish dao dm va cua b) có dung khong?
2.44. Gia sit ip la met to d6ng ca/u cua khong gian vec to thkic n chigu V, trong mot ca sa nao do co ma Han A. Chung minh rang da tilde dac trong oua pee dang:
106
PTO,) = (-2,)" + c,(-2,)"1 + + + 'Prong do ck
Yang cua tat ca the dinh thew con chin)) cap k cua ma Han A,
)inh thew eon chinh la dinh thfic con lap nen tit the (long va
re c.0t vdi chi s6giang nhau).
2.45. Gia sa ( la to deng Ha.' cua khong gian yen to V n
nen trru trtfong K.
Gni la mot nF,Thi .em cua da ',hue dac Hang cua 9, 2,„ co
hieu r= rang (9-2, 0M).
Hay cluang minh I < n — r 5 p.
Y.46. Havain) nghiem dac trung cua ma tran AeMat (n, C).
0 -1 1 0 -1
1
0
1
2.47. Tim da the dac bung cua ma tran vuong cap n.
0 1
0 1
0
a va a
1
an
Tit do suy ra m6; da thin; bac n yea he t>i cao nhat (-1)" den
a da their dac Hung cua met ma trdn valuing cap n nao do.
A=
107
2.48. GM V lk khong gian vec W n chieu troll trUang K,
cp E End (V), X° la mOt gia tri rieng ena gyp, 700 la khong gian
con rieng cna V, ling vdi gia tri rieng A 0 . ChUng minh rang so
chigu cim g1 /.0 khOng vuyt qua sei bOi p cem nghiem 700 cua da
thdc (lac trung dm. T. Hay chi ra VI du trong do dim "91 A, < p.
2.49. Cho o la mot phop thg bac n va u e End (0 1) xac dinh
ban
u (Z„ Z„) = (74,„„ ) , ...,
Hay tim da thfic dac trung va the gia tri rieng ctia tu clemg tau u.
2.50. Cho f e End (R"), trong co se, to nhien {o„ ..., e n} co
ma tran A=(a d),tldoa1 = a vOi moi i va a d = b yen ix j; a # b.
Hay tim mOt ccI so cem R" a ma tran cua f trong cd sa do co
Bang
2.51. Cho A, B e Mat (n, C).
a) Chung minh rang the ma train AB va BA co cling tap hdp the gia tri rieng
b) HOi AB va BA co clang da thne dac trung hay killing?
2.52. Gia sU V la khOng gian vec td phito him han chieu, f e
End (V) ma co se N, nguyen during dg fN = Id. Chtng minh rang trong V co co sä gom nhfing vec td rieng cua f.
2.53. GM sif ma tran S e Mat (n, K) co anh chgt: Kai moi
ma tran A e Mat (n, 1K), to Mon viet duye mot each duy nhfit
108
dual. dang A = A, + A2, a do A, giao haw vdi S va A2 phan giao hoan vdi S, nghia la A,S = SA„ A 2S = -SA2 .
Chung minh rang 52 = X. I„ la ma tran dun vi.
2.54. Cho E la khOng gian vec td huu han chieu va f e End(E). Vai m8i i = 1, 2, ... dat V; = kern va W, = Im(P).
a) Chung minh V, c Vm , voi moi i = 1, va t6n tai s6 m
V„, = v„,+, va khi do V. = V,„„ vdi moi j x 1.
b) Chung minh E = V„, W„, vdi m tim chide trong can a).
c) Vm va tim duo trong a) la nhUng khong gian con bOt Bien doi vdi f. Hon nua, f Iny linh tren va kha nghich tren W„,.
2.55. Cho V la khong gian vec to tren truOng 1K (1K bang R
hoac C), 9 e End (V) va p lily linh. Chting minh rang:
a) Moi gia tri rieng cua cp deli bang 0.
b) Neu dim V = n thi da thlic dac trung cem 9 la oon.
2.56. Chiang minh rang neu W la tv d6ng ca'u cua khong gian vec td phac han han chieu ma moi gia tri rieng deli bang
khong thi linh.
2.57. Gia s>t V la Haling gian vec td tren &Jiang ]K va dim V
= n. f e End (V) la tv clang cali lily linh dm V. ChUng minh rang PI = O.
2.58. Gin sO V lh khong gian vec td tren trUang K va f e
End (V), rang f = 1.
109
Chung minh rang ton tai met co so dm V de trong do r tran cern f c6 dang:
hoac
0 0 0 \ 0 0 ... 0
Neu f co ma Den a dang thu hai, thi f lay linh va c2 = O.
2.59. Chung to rang neu hai tp deing cau L g gem kh6) gian V Dacia man he thew f.g—g f = Id (0 thi vai moi k e ta — &c.f. = k.g" (2).
Tit d6 suy ra rang khong the t6n tai the tp dgng cit'u tiv man he theic (1).
2.60. Chung minh rang, d6i yeti ho bat kY CAC torn to tuy tinh giao hoan vdi nhau Ding doi mgt trong khong gian vac hem han chigu- tren trang so" Ode, tan tai yen to rieng chui cho tat cA the Loan to tuyen tinh cria ho do.
D. HU6NG DAN HOAC DAP S6
§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINI
2.1. GM sU V, W Ian hurt 1a cac kheng gian sinh bai A„ ka B 1 , ..., Dr Veli mei = 1, 2, n thi B, = a,,A, + a 2A2 +
u„,A„.. Do do W c V. De chUng mini) W = V, ta chi can char minh dim W dim V.
Ta c6 dim W = rang (X . X') = n — d, trong do d bang chigu cua khong gian H the nghiem ena phudng trinh:
'a 0 . 0 '
0 0 .
to p 0
0 0 0
110
t„). (X . Xt) = (0, 0, .., 0).
Neat Y = (t,, t) E H Y. XX' = 0
X.)4' . = 0o Y . X. X' . = (Y.X) . (Y.X)' = 0
Y.X = 0. Nhtt vay Y e M la khOng gian nghiem cim phuong trinh t„) . X = 0 to do dim W=n—cln— dim M = rang X = dim V.
Do do W = V.
2.2. Gia su A„ ..., la the vec to cot. (-Ala ma tran X hang r. C6 th6 gia thi6t A„ (lac lap tuy6n tinh. Khi do cac Ai ; r <
5 n bie'u thi tuyett tinh qua A„ ..., A,.
Al = a,,A, + cii2A2 + + a„ A,
N611 dung ki hieu (A„ A2, ..., A ,) (16 chi ma tran co cac vec to cot la A l , A2, A„ thi
aciAl)
Mei ma tran 6 v6phili c6 hang 1. 'Pa co digu phai cheing minh.
2.4. b) Xet R [x] la khong gian vec td the da thew sr& he s6 thitc.
Xet u: R [x] —> R[x]
f(x) --) x .
116 rang u la chin caM, nhting khong than eau, vi Imu khong chiia nhiing da theic bac khOng.
2.5. a) Theo each the dinh caa u thi trong ea so tu nhien
(e l , ..., en} cern C", to ca u(e) = e,„„ Vay A = (ad la ma trait cua u
trong co so {e„ thi
111
{1 i (j) auo =
0 voi # a (j)
b) Gia sii B = e Mat (n, C)
Xem B IA ma tram cila phep bien den tuygn tinh trong cd
to nhien cna en : v(e i )= Eb na . Khi do AB = BA a uV = vu <
uv(ej) = vu(a) j = 1, 2, in n Eb u(a)= v(e a0)), j = 1, n <
n n n Ebije cr o) = Ebi,wei = <=> b ;,= b c,(;) „„,
i=1
1, ..., n, nhu vay, the ma tran giao hoan dime vo tran B = (130 sao cho b u = bum on) voi myi i, j = 1, 2, .
2.6. a) De dang.
11) Gia stl u e End s (E) giao hoan voi
EndK (E). Theo a) u (e) = p ie, voi {e„ cd so Gia sV i # j, 1 g 3 , j < n to hay chfing minh 13, = EndK (E) sao cho v(e k) = Vk 1, 2, ..., n. Tit vou
vou(e) = uov(e,). Nhu vay j3, e3 = u (a) = rija a (Di P: -pj=A. Tit dou(x)=Axvoimoixe E.
2.7. Ta co the bieu din n clang (laic Ea nu jet
n) &MI dang clang (Mk ma tran sant
an1 ant ann
( ail 312 aIn ‘ Ill 1
[a21 a22 v2
/ n
, . . v
Vn
voi mci i , j
i A la cic m n.
d6ng eau v
nao do dm I
p i . Clam v
= uov say
-13i) = 0 =
i=
112
VI ma Han A = (an) khong suy men, nen co ma tran nghich dge 11 = A- ' = (bd.
Do b(au) = (ba) u voi nun vo Imang a, b va voi moi phep bi6n
den tuyen tinh u nen La co:
u l (u i
B. =[B .A]. •
Aun j
Tie do to co:
1
.. 0
0 0
l 0 1
u l u2
Lon)
=
1 bin
621 b2n
••-
bn bnn
v2
vni
Hay u ; = ybi,v ;
Theo gig thigt, eac vi giao &ban voi nhau, nen suy ra the u, thing giao honn voi nhau.
2.8. a) Vi A khOng suy bign nen mei cot ena no cling co
dung met phAn to klthe khong.
Gig set A = (a n), A t = fa t) a t _ a ,, ij
Hat A.AL = B = (b 0). Ta c6
bii = Laik -a ki = L aikaik=sij k=1
Vi vgy A.AL = In = A-1 .
113
b) Cacti 1: Goi G la tap cac ma tran clang troll
Ta thdy A e G, B E G thi A. B e G, A-' =A° e G. Do do C lam thanh nhgm con cim nhian GL (n, R) cite ma Dan vuong
cdp n khong suy bign. Vi s6 cac phial to °Pa nhOm G la lulu han, nen vdi A e G, t6n tai k e N k 21, dd = Ak- ' = = At, nhtt vay A"' = At vdi m = k — 1 e N.
Cad' 2: Gia sit (e ) , ..., e n) la co sa W nhien cim f e End(R")
c6 ma trait A trong co sa {e l , e nL Ta e6 f(e,) = ± no) 6 d6 a la phep thg bac n. Vdi m61 i = 1, 2, n xet a(i), a 2 (i), tai k,deg ak'(i)=i, khi do f(e,) = eap o = e, PLi(6) = e,
Dat k = 2 . k, kn thi gc(e) = e, V ; = 1, n hay f" = Id Ak = In, ldy m = k — 1 to c6 = AL -1 = At.
2.10. GM sit f.g — Id khong la clang cdu claa V, khi do t6n tai x E V, x # 0 a (fog — id) (x) = 0. Hay fog (x) = x. Dant g(x) = y e V, to co f(y) = xz 0= yx 0.
Xet (Id — gof) (y) = y — g (fly)) = y — g(x) = y — y = 0. Digu nay
chfing to rang c6 y x 0 dO (Id — gof) y = 0, trai vdi gia thief Id — gof la (fang cdu.
2.11. a) D6 thdy u, v Wong ding thi co gang hang. Ngudc lai, n6u hang u = hang v = r, theo bai 2.3, trong V c6 cac cd sd {6, ..., en}, (e ) ,, e'„1 t(ei) = e' t = 1, r), u(u) = 0 > r), va c6 cac cc' sa e„}, e'„} v(e) = e,' vdi i = 1, r, v(el) = 0, j > r. Xgt p E End (V) p (e,) = e; voi moi i= 1, n; Xet q e End (V) sao cho q(s) = e', (i = 1, n) thi uop = qov.
b) Suy ra tit a)
114
2.12. a) De dang do Tiny c Keru va dim Imu + dim keru =
dim V.
b) Xet co so ..., en/ va {c 1 , c„} rim V sao cho u(e,) = e, (i
= 1, 2, r) va u(e,) = 0 (j > r). Xet v E End (V): v (c,) = 0 (i 1,
r); v(c,) = e i vai r + 1 5 j < n. Khi (id vou = 0 va hang u + hang
2.13. a) rang rp= dim E x r
b) rang ty = dim FI x r.
2.14. WE( u, = u — Id E , v, = v —
Tit do u.v —Id E = + IdE) (v, + IdE) — Id E =
= u, . v 1 + u, + v, = u, (v i-lId E) + v, nhung rang u, (v, + IdE) ic rang
u, va rang fu l (v, + IdE) + rang u, (v, +14) + rang v 1 .
Do vdy: rang (uov — H E) < rang u, + rang v, = r + s. (xem vi
du 2.14).
2.15. Vi rang u = 1 dim (Imu) = 1. Gia sii {x,4 la ed sd GM
Imu. Do u(x 0) e Imu nen co 2, e 1K: u(x o) = Ax„.
Ta hay thing minh u = = Au. Vol x e E u(x) = a . x o
u2(x) = a u (x0) = a . Xxo = X a. xo = Au(x) vdi moi x e E u2 = Au.
Gia # 1, tim 71 a (Id — u) (Id - nu) = Id.
Di5u do tudng &tong vdi (fik 1
x_1 .14.y
Id-u Icha nghich va nghich dao mid no la: Id + 1 u.
1-X
115
2.17. HD: Chon B la ma Dan cheo vdi cac phan to tren
during cheo chinh khac nhau doi mgt: b,, Chon C = (au)
a dsao Cho C v itt va Cli tuy y, tin A = BC - CB.
b i - bi
a b' 2.19. GM X =
d 1
E Mat (2, K). Hay chUng to c6 P =
x y e Mat (2, K), det P t 0 de XP = P.Xt.
z t
bz = by
(a-d)y= cx - bt
-d)z = cx - bt
cy= cz , xt zy
(1)
Xet cac trudng hop sau:
+) b = c = 0, chon y = z = 0, x t = 1
+) a = d, chon y = z = 1 x = t = 0
+) b 2 +cz> 0 va a t d: chAng ban b s 0,
chon x = 0, t = 1,y= b -z .
a -d
Nhtt gay trong moi trUdng hop, hP (1) dgu c6 nghigm
2.20. a) Tit u2 (Id - u) = 0 suy ra 112 =
Tit u (Id - u)2 = 0 suy ra u - u 2 - u2 + = 0
W vgy uz = u.
=
116
b) Vi du V = K3 , u (x i , x 2 , x3) = (x„ x 3, 0)
Ta thSy u2 * u nhung u 2 =u3 .
e) V = K3, v(x„ x2, x3) = (0 , x2 - x3, xa)
Ta có v 2 * v nhting v (Id - v) 2 = 0.
Chu Y: Co the neu nhieu vi du khac nfia.
2.21. a) Ngu g(x) = 0 voi moi x c V thi f(x) = 0 \raj moi x e V
va f = a . g voi a e K Neu có xo e V ma g(x0)* 0.
Data- f(x)
. g(x {,)
Ta chUng minh f(x) = a g (x) voi moi x c V.
Vi g(xo)* 0 nen g (x) = g(x), khi do:
g(x -;xo) = 0 f (x - )( 0) = f(x) - 4f(x0 ) = 0
f(x) = 4 . 1(x0) = g(x) f(x ) = a.g(x). g(x 0 )
b) HD: chtlng minh quy nap theo n.
n = 1 dilng vi day la trUang help a).
Gia eu bai town dung voi k = 1, 2, .. n - 1. Ta chUng minh
dung voi n.
Trubng help 1: Neu co i de fl Kerf c Ker f, thi fl Kerfj = joi j#i
fl Kerf, c Kerf. Nhu vay theo gia thief quy nap, to co i=1
f= ifJ=Eaifi (ai = 0) i=1
117
Truang kip 2: Vol mdi i, n Ker fi ¢ Ker j#i
Khi do vdi mdi i (i = 1, 2, ..., n) co xi e V de? f(xi) # Ova fi (x,)
= 0 ( # i). D'.1t a , -
f(x ) f
Ta chting minh dude f = + + a„ f„.
2.23. Ta chUng minh fl Kerf, 101.
Hay thong minh quy nap rang dim ( n Redd n - 1 vat i=1
1 < t < n.
TU do suy ra ket qua: Alai he phudng trinh tuyen tinh thuan nhdt vdi se; pinmng trinh ft hon s5 do dgu co nghiem khan khong.
2.25. a) De' clang
b) HD: xet 1x 1 , x„) la cd sa cua S, {x„ x„„ x,„„, x n} la co se) clia V. Voi moi i = m+1, n, set E V* : ii(Xi) = — ki hieu Kronecker). Ta thdy f, E S° .
Hay chUng minh {f„,,,, f„} is co so cfm
TU do dim S ° = n - m.
2.26. a) Hidn nhien
b) Theo bai 2.25, S° IA khOng gian con cent V*, (S°)° la khong gian con dm V. Vdi x e S, fin, F e S° , to co f(x) = 0 x e De cluing minh sinh hal S. Ta gia su W la khong gian con cna V, S c W. Ta ehling to (S°)° c W. Vdi x EW3f c V* de f(w) = 0 va f(x) x 0; Khi do vi S c W nen f(S) = 0 f c S°.
118
Nhung f(x) ± 0 nen x e (5 °) ° . Nhit vay vdi x e W x (S°)° tit
do (S°) ° c W va nbut vay (S°)° la khong gian sinh bai S.
c) Ta cd (TY la khOng gian con cim V* va T c (T°)°.
Gia sa W* la khong gian con cim V* va W* = T.
Ta °hang to (r) ° c W.
Gia sit (fi , f„,} la co sa cim W. Neu f e V* \ W* thi f
hong la t8 hop tuyen tinh cim cac LI. Do do, theo hal
21, cO x € V (x) = 0(i= 1, ..., m) nhung f(x)t O. Vi {f} la co
V1?'" (x) = 0 vdi moi i = 1, 2, ... m nen g(x) = 0 vdi moi
g e via do do, vdi moi g c T (vi T c W*). Tit do x e T°,
Nhung f(x)# 0 f e (T° ) ° .
Chiang minh tren chiing to (T° ) ° c W*, nghia la (T°)') la
khong gian con cim V*, sinh bai T.
d) HD. Gia sit (x i ) c i la co sa cim khong gian V, I — tap vo
ban; vdi i e I, Kat f, e V*: fi(xj) = S i (Su — ki hieu Kronecker).
X6t T la khong gian con cim V*, sinh bai (fJ ; c 1.
Hay chUng minh (T° ) ° = V* nhung T x V. (vi (In f E V* ma
f(x) = 1 Niel moi i f 0 T).
2.27. a) va b) (16. clang
c) chi can thing minh netu V„ V2 la hai khong gian con cim
V, thi (V 1 n vo° = vi° + v2"
Ta cO V, n V2 C VI, V2 VI°
v,0 c (NT, n v2)°.
D6 chiing minh (V 1 fl V2 ) ° c V 14) , lay f e (V 1 n V2) °
c or, n v21,°, v20 c (v, n v2)"
119
f(x) = 0 vol mot x E v, n V2 . Ta cluing minh f e VI ° + V2 0 . )(et WI la khong gian con bit maa V, id V2 trong V, o V, = WI ED (VI n V2); Wong ta vet W2 sao cho: V2 = W2 $ (V, n V2) via V = (V, n V2) e W. Khi do V la teng trip tip gun. V, n V2, WI, W2, W.
Chon g, h e V* the dinh bat:
lb nab x E Vt n V2 hoitc X E WI
.if(x)ngu x e W2 boas x e W
t0 ngu x e VI I)V., hoac x e W2 boas x e W.
f(x)ngu x e WI
WO rang g e V1 0, he V2° va f = h + g.
Nhu vgy (VI n V2)" t-_- W 1 ° + V2°. Tit dO (V, n V2) (' = Vi ± 1.72°- d) Vi du cluing to khAng dinh b) khong con dung ngu I v6
han va V co chigu ve han.
Gia sa (e) i e I la co so v6 han phis to caa V, mOt phAln to caa V co dang Ecc,e, trong do a i hau hgt bang khOng, tit mot
jel so hilt! han.
Vol mOi i e I, ky hiQu V, la khong gian COD cila V gam cac phan tit E cgs, vgi al = 0.
ieI
Taco n o n vir = v*. ieI
Trong khi do Vi° F {f E I f(e) = 0 V,
g(x).
h(x)c
120
Do do V ic' la khong gian con sinh bai f, (NhAc lai f, (e) = 6,j —
'hi 2.26) ma EV? x V* (bai 2.26). idl
2.28. c) Gia sit {6} i E I la ed so vo han dm V; f, e V* sao cho
, (e) = §i
T, la khong gian con sinh btli {f1}, T, V* (Kern bai 2.26).
f e V* sao cho Re) = 1 vOi moi i E I, xet T2 la khong gian
inh bai f T2 (1 T1=101 (Ty n T1P = V; nhung T 1 0 =401 cV.
T 2° t V vi T 2° chi chita cac MAI) ti) El:t i e, ma Ea ; =0. lei lei
Slut very (T2 (1 T1) ° a V+ T20 .
2.30. Ta c6 p la phep chigu .(r). p 2 = p
<=> Id — p — p + p' = Id — 2P + p 2 = Id — p
(Id — p) 2 = (Id — p).;=, Id — p la phep chigu.
Bay gib n6u y e Imp thi y= p(x) (Id — p) (y) = 0
y e Ker (Id — p).
Node lai, n6u y e Ker (Id — p) (Id — p) (y) = 0
p(y) = y= y e Imp.
Nhti very, neM p la phep chi6u thi Imp = Ker (Id — p).
D6 y rang p = Id - (Id — p), to co Kerp = Im (Id — p).
2.32. a) trang hip u = 0 la tam thtlang.
Gia sit u 0r dim (Im u) = s > 0.
121
GiO sit {e l , ..., e s} la cd sd cim Imu. Khi do en e, (i = 1, s) sao cho u (e,)= c i . Ta có dim (Keru) = n — s. LO {e sti , e„} la co sa dm V (xem bai tap 2.3). Chi can LIS sung de" co co sä ..., E s ,
£ 0 } Oil to dude hai cd sa can tim.
b) Xet v e End (V) ma v(c) = e„ i = n thi you la phop chi6u ciia V len khong gian sinh cj song song voi khong gian con sinh btli {e„„ e„}.
§ 2 HE PHUT$NG TRiNH TUYEN TiNH
1 -7 4 2 0'
2.34. A = 2 -5 3 2 1
rang A = 2. , 5 -8 5 4 3 4 -1 1 2 3,
Nhu vdy khong gian nghiem co so" chiOu 3.
FLO da cho tUdng dudng vOi
7x 2 + 4x3 + 2x4 = 2x 1 5x2 + 3x3 + 2x4 + x5 = 0
1 4 7 xi
9 — X 4 9 9 x5.
5 2 1
X 2 = 9 —
9 X5 + —
9 X4 — —
' Xr .
'
Ta có mot he nghiem cd bin, chAng han la:
Et i =(4, 2, 3, 3, 3)
a2 = (-7 , -1, 0, 0, 9)
a3 = (1, 5, 9, 0, 0)
122
2.35. HD. He c6 nghiem khong tam thang khi va chi khi
ih thiic cua he Wang khong a X = 35. Khi d6 hang Mitt ma
.n cite he s6 Wang 3 a khong gian nghiem 1 thigh.
2.36. HD. Coi he da cho la he phudng trinh thudn ntat viii
biers (It - c)x, (c - a)y, (a - b)z. Ta dan Mit he thuc:
(I) (c-a)y -b)z 1 bz -cy cz - ay ay -bx k
De he c6 nghiem khong tam thitang, to phai c6 k = ±1.
2.37. 1 ,
=4 a + b + c +
1 x2 =-
4(a-b+c+d)
x4 = 4 —1
+ b + c -d)
2.38. A=
2
1
3
1
m
I
-1
1
-m
, det A = 2m (2-m).
a) v6i m* 0 va m # 2 a He la he Cramer.
b) m = 0 rang A = 2 vi 2 1 # 0 . 1 0
123
=0.
1
1
1
1
Yt
y2
32 3 Y4
Digu kien có nghi5m la 2 1 a
1 0 b 3 1 c
=a+ b - c = 0 .
x=b-z Khi do
{,y ,=a-2b+3z, z my
c) m = 2 =4 rang A = 0.
Digu kien có nghiem: -5a + b + 3c = 0
Khi do
2a -b x -z +
3
y- z+ 2b
3
22 x 2 +y 2 x2 2 2
X 3 + Y3 2 2
x4 +Y4
2.39. PS: x3 x4
2.40, f(x) = ax3 + bx2 + cx + d
0-1) = -a+b-c+d = 0 1(1) = a+b+c+d = 4 f(2) = 8a+4b+2c+d = 3 03) = 27a+ 914+3c+d = 16
Tito: a = 2 b = -5, c = 0 , d = 7
f(x) 2x3 - 5x2 + 7
2.41. Ta có A = (a 4), nen ZadA ii = 0 (1) (i = 1, 2, ..., n) 1=1
124
Do A„ # 0, nen rang A = n-1 va khong gian nghiOni cna (1) co so. chigu 1. Hay rang (A11) = 1.
TV do hO EA ii x i =0 (i = 1, 2, n) Wong doing voi j=1
X2 X2 ± Al2X, = O.
Ta có mot he nghiOm co ban, riling ban
= (-Al2 , A 11 , 0, ...
n2= 0. An,
0)
0)
Et„_i =(-Alloo,o, A„)
§3. CA.0 TRUC CUA MOT TV DoNG CAU
2.42. Gia s& u la Mt chigu u2 = u. Ngu x la vec to rioting
ung voi gia tri rieng ?. u(x) = Ax.
u2(x) = u(x) = Xu(x) = X2x. Nhung u2 = u nen: A 2x = Xx
(1.2 - ),.)x = 0. Do x s 0 nen - X = 0 X= 0 hoar = 1.
Tudng to del vdi phep doi hop v2 = Id.
2.43. eau a), b) dO dang.
c) Dao ciia a) khong dung. Vi du:
Cho A = In (ma tran don vi), B = ma (On tam gihc (Judi ma
the phan t& dgu bang 1 Khi do det (A-AIn) = det (B-AIn) = (1 - Xr.
Nhung A AM B khong &Ong dang, vi A chi ding dung vii chinh no (xem bai 2.18).
125
Dao cua b) cling khong dung. Vi du: Chan Ala ma trar (cap n > 1) va B e Mat (n, It); B= (b id ma b 11 = 1, cent; = 0
mm (I, j) # (1, 1).
Khi do det A = det B = 0, nhung det (A - XIn) = (-k)° con I
(B - Mn) = (1 - A) (-X) 11-1 .
2.44.Vi N e End V, trong cd sa nao do co ma tram A nen thtic dac trung cUa N co
IA - XIn I = (-X)" + C1(-X) 111 + + C.
De thgy C„ = detA (cho X = 0). Ck 11 he so cim cac se; hang s6 mu (-X)^4, (Woe tao thAnh bai tich cua n-k phan to tr &fang cheo chinh dang: (ayi - X)... (a ind, , ,, -X) vdi cac phgn
trong khai trie'n dinh thtic IA - I, khOng chVa T. va ni trong cac dang, cot ba vdi cac dong, cOt c6 chi se; {i 1 Do dinh thtic con chinh cap k ciia A co cac phgn tv tren &tang ch chinh veil chi so" ba .•• VI 60 {II ..• Lk} la thy $, nen s6 cua (4)"4 chinh bang te'ng tat ca cac dinh thuic con chi dang tren.
2.45. Ccich 1. Gia su A la ma tran cila cp trong cd sa nao
dm V, thi = rang (9 - XDId) = rang (A- koI). CIA sii X la vac cOt, X thuoc khOng gian rieng P, ling vdi gia trt rieng X 0 khi chi khi ( A - A01)X = 0 > dim P kn = n-r. Goi R 1 la kitting git 0
rieng suy Ong ling vdi gia tri rieng Xo, thi dim E xci = p (130i cl
ko), ma Pxo c Rko nen n - r < p. Mat khac do det I A-X.011=0, ni
ran= 1 S. n-r. Vay 1 S n-r p.
126
Cdch 2. Dat B = A - kJ. Khi d6 = I A - (p+X, D)I I =
A - XI I , voi A = µ + X. Nhd vay A = Xo la nghiem bet p cua
IA - XI I = 0 thi µ= 0 la nghiem bei p cna. IB-µII = 0. Theo
bid 2.44, I B -µII = (-0" + C1(-0"1 + + Ck (-11) "a C, a do Ck 11 tong cga cac Binh tilde con chinh cap k dia. B. Do rang
B = r nen cac dinh there con chinh cap k > r dgu triet lieu Ck
= 0 voi moi k> r.
Vey I B - I = (-P)"+ +
Doclop=n-s?.n-r.
2 i 2.46. DS. Ak = 21 cos k
n +1
C. (-1,)" 5 , s < r.
HD. Xem bai 1.19, cam a) dat a + p = -A, a.p = -1.
2.47. Goi p(x) le da tilde dac tning cga ma trap A.
-A 1 G ' -A 1
an-1 an-2 thing coot to cg:
(1
P(A)=(-1)""a2,2det :v 1 +(-1)"*2 a,, 2 det
P(x) = det Khai trim theo
o'
1,
f- k 0
0'
1 )
(-1)2" (a. - A) det
127
Nhu v0y:
P(X) = (-1)' (a._, + a„. 2 X + + asXn- ') + (- 1)11 . l. n .
Ke't qua tiep theo suy ra tit ceng dr& nay.
2.49. HD: Phan Hell a thanh tich cac yang xfch doe 10 a = ... T2 . T I ; gia {e1 , ..., en} la cd s6; tn nhien cua Ca, tt u(e) =ea . Sap x6p lai co sa fe;} theo thft ty thich hdp ta dude c
sa mdi cua C". sap thanh P nhom, m6i nh6m g6m m a vet t (mk bang de dai Tk), k = 1, p. Chang han nhom thit nhat gen fe' l , e',„ 1 } thl u (e';) = e't+1, u(e'nil) = e' I . Tuong t>i cho ca nho khac. v0y. Da flak cl0c trung cua u co clang
I A - kin I= D I ... D„ a do Dk la da thile b•c m k dang:
- x 0 0 1 1 -X 0 0
Dk = 0 1 - X =(— OnIk+I-P (- Ark
0 0 0 1-1.
= E lrk ()Lin k Nhu vay
A-xin Hoy. (Am! -1). VP -1)
2.50. HD. Theo vi chi 1.20 (chttdng 1), ta có det (A - AI) = - b - X) 111 . (a + (n-1) b - X).
Com gia tr} rieng cua f la: a - b (bel n - 1) va a + (n-1) b. vó X = a - b, co (n-1) vet to rieng doc 10p tuyen tfnh thaa man
+ + to = O.
128
VOi it = a + (n-1)
VS'.y c6 the than Cd va xn (1, 1, ..., 1) gem
2.51. HD. Xet dang
(XIn - AB A
0 AIn i
b, co 1 vec
sox, = cac vec
-aide
'In 0'
2 B . In,
to rieng: t i
(1, -1, 0 ... 0), to rieng cita
ma tran:
In 0
[ B
In
= t2 = ...
..., x2. 1 1
( )Lin
O
= t„ = 1.
(1, 0, 0,...,
A
XIn - BA,
-1)
a) Ta có det (XIn - AB) . A n = A" det (Mn - BA).
+) N a3i X # 0, Ta có det (AB - Mn) = det (BA - Mn) nen AB va. BA c6 cling gin tri rieng khac khong.
+) Vol X = 0, do dot AB = det BA, nen no cling cO nghiem chung X = 0.
b) Do det (AB - Mn) = det (BA - Mn) vdi moi A, nen hai da thnc dac trung eim AB va cna BA trung nhau.
2.52. Xet cci so (6,, ..., e n} ciia V, trong do ma tran f duac tao heti m khung Giooc clang Ung vdi m gia tri rieng phan hi@ A l , ...Atm .
(2`k 0
1 Xk Ak = 1 ' , cap Gila Ak bang dim/Kt)
0 1 X i<
(Rtk -khong gian rieng suy rang Ung yea gil tri rieng VI la khung gian con bit bie-n cua f, nen theo gie thiet to co
Ak =I nk (I„k la ma trMa thin vi can dim Rkk = Ilk. Ta co Ak =
'0 0 ' (Xki n r r_- EN cN xika nNk i , a do J nk = 1 0
1 i=0
‘0 1
129
Nein nk > 1, thi c6 cac phAn tit Oleo Wang 0 vdi moi i, vi
\ay AI co phAn tit cheo Wang 0 vdi mm N, digu de trod gia
thigt. Vi \ray nk = 1 moi k. Nghia la ma Pearl cim f trong cd sd
(or e„.} co dang then, hay {e l, ..., en} la co sa g6m nhang \Tee to
rieng cim f.
2.53. Ta c6 bd de: Vdi S e Mat (n, K), det S = 0 thi co ma
tr6n M e Mat (n, K), M 0 a MS = 0. Trude het to sit dung be
de tren de chung minh bai than.
Gia sit S e Mat (n, K) co tinh chat: moi ma trail A e Mat
(n, K) den vigt dude mgt each duy nh6t 6 clang A = A, + A2, SA,
= A 1 S, SA2 = - A2 S.
Ta nhan thdy S phad khong suy bign, ngu ngdde lai S suy hien thi theo be de tit cO ma tran M # 0 de MS = SM = 0; Khi
do 0 = M + ( -M) = 0 + 0 la hai phan tich ma tran 0, th6a man digu kien bai town, trai vdi gra thiet va tinh duy nh6t.
Vdi mr6 A e Mat (n, K), to co the tim dttdc A l , A2 W di6U
(SA = SA1 + SA2 kien:
AS=A I S+ A 2S , SAk -SA 2
A l =j-k+S-1 .AS)
A2 = 2
(A -51 AS)
Vi A1 . S = S.A 1 nen to he thiic tren to co
AS + S'A. 52 = SA + AS to do AS 2 = S2A.
{
2S.A 1 = S.A + A.S
2S.A 2 =SA -A.S
130
VOi mm A e Mat (n, 111) S' in (xem vi du 1.19); A # 0
S khOng any bien.
(Meng minh bd de: xet P la da thiac tea tieu cith ma tr n S
(chi min xet S # 0). CAC, P = xk + + + ak 2 x + a k , k21, va p(s) = sk a isk- , = a k _,S + a k . In= O. Voi S # O. Ta nhgn
they k> 1, vi neu k = 1 thi P(S) = S + a, . I„ = 0
S = a, . I„; do S suy bien nen S = 0.
Ta lai thely ak = 0, vi neal ak # 0 thi S kha nghich, trai gia thiea. Nhu vay:
S (S 1' + S k22 + ± at.; - 1 0= 0
DM M = + ad1 1-1 + , I„ # 0, to co SM = MS = 0.
2.54. a) DO (ang
b) Nhan they V. n W m = {0}. That vay, gia se x e V„, n vv„, [moo = 0 va co y E E de fth(y) = x, tit do el(y) = 0 yeV2m= V„, l'"(y)= x = 0.
Nhu vay dim (V, 1V„) = dim V„, + dim W n, = dim E 'E_ V„, W.
c) Re rang V„, va W m tim clime trong a) la nhiing khOng gian
con bM bien dila \ril f. Vi f" (V„) = 0 nen f lhy link tren V„. De
chung minh f kha nghich tre.n W„„ gia su x e W m va. f(x) = 0, IV
do co y e E c/6 f"(y) = x elt(y) = 0o y e V„,, y e Vm
ray) = x = 0. nhu vay f/W,„ lk chin anh nen no IA song anh.
131
2.56.
Ccich 1: Da the da'c trting cua W co dang P,p (a.) = (-I)n . A".
Theo dinhb) Haminton — Cayley. Ta có p" = 0 4. 9 luy linh.
Ccich 2: Theo vi du 2.16, ta co the' du) mot co se trong do
ma tram A cna (;) cc) dang tam giac tren. Theo gia thiel cac phial
tU tren clang cheo chinh bang khong, d6 do An = 0 p" =0 cp
uy linh.
2.57. Cach 1. Gia q la se nguyen diking be nhal d5 f' = 0.
Vi f9-1 x 0 neu co vac to x e V (I5 f^ - '(x) m 0, r(x) = 0. Theo hal
tap 2.9, ta ce {x, f(x), 0-1 (x)} dOc lap tuyen tinh. TpY do q 5 n.
Vi = 0 = 0.
Cach 2: Dat W' = Ink(?). Ta co W' = W 2 = ... to do có m n
d5 W" = vi ne'u kheng ta co n = dim V > dim W' > dim W'
>...> dim W6-1- dim W"+' < 0, ye 1.3"7. Mut Sy W" = =
Wm+i suy ra W" = 0 hay P"= 0 f" = 0.
2.58. HD: rang f = I dim (Kerf) = n-1.
Chen e, e Kerf va e,J la co sa cua Kerf xet-f(e l ) = ale, + a2 e2 + ... + aue„, phan biet hai truong hop a, = 0 va a, x 0.
2.59. He tinic Igk — gk.f = k.g" dung vat k = 0,1.
Ta chUng minh rang nen no dung voi k-1 va k, thi h& thiw
dung vol k+I.
Gia sit co f.gk — gk.f = ta suy ra gk+1 gk.Eg = k.gk va
gigk — gickf.f = k.gk:
Tit do f.gk± 1 ,7, g'+' f + g (fgk - l- e- , . f) g = 2k.gk .
132
Do gia thiCi guy nap: gk-' — = (k-1) gk•2 nen: fel
gk+1 , fkti (k-i) g k = 2k.gk
hay (k+1).gk; nghia la he thtic (2) thing vdi
k+1. Vi \Tay, do he UMW (2) dung vdi k = 0,1, nen n6 dung vat
moi k e N.
Ta cluing td rang khAng ton tai cac ty dling lieu thaa man
he thin: (1).
Thai vz;)) , , )(et P(x) = xP + a p .,xP-1 + + a o la da thdc tot lieu
cua g; nghia la da thtic kluic khbng, cd bac nh6 nhet sao cho P(g) = 0. Khi viol he thug (2) vdi k = 0,1, P to nhan dude:
f 0 P(g) — P(g) f P' (g). d do 13' la dao ham dm. P. VI P(g)= 0
nen P'(g) = 0, trai vdi gia thiet P la da thne t6i tint' clan g.
Chi) y: CO the nhan xet rang khong the ton tai cac tq d8ng
ceu f, g thea man fog-g0f= id vi trace (f ug-go I) = 0, trong
khi trace(Id) = n = dimV # 0.
133
Ch uang III
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.
134
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
135
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 (y 1 , yd IA toa dO cam x va
y Wong N6u trong 100 cd sä 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
136
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
r rye giao: V = e V, e e 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, HI V_ 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.
137
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, >
138
Ta có halt clang theic sau, goi la bat clang these Cos Bunhiacopski:
<, >2 < N2 MYM 2 •
b) Hai vec td x, y &tee goi IA true giao vol nhau n6u <x, y> = 0. Khi do to co:
)1 2 = D1 M 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 , e 2 , ...,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
139
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 , a k } 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 y i e i thi <x,y > = Ex i y i . 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.
140
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.
141
+) 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.
142
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 .1131 2 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>
143
(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* 144
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>
145
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 , X 2 .
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.
146
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
147
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 , y o} 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 , x o , xozz }.
Dat ao = 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.
148
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 xo, x).
g(y, x) = a g(xo , x) = a g(x, xo) = 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 , x o) = g(x, Y xo)
Suy ra g (x - a xo, y + x o) = 0
g(y + xo , x - a xo) = 0
149
g (y + x0, x) = a g (y + xo, x 0)
= 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 a id = xi), 1 j s k.
Ta goi dinh Ulric det (a 1 ) la dinh thuc Cram (Ma cac vac td x,, x k va ki hiOu la Gr xk ).
Chang minh rang Gr (x i , xk) 0 va Gr (x 1 , . xk) = 0 khi va chi khi x l , 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 a ji + + nghia la thing thir r am ma trap (adk 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 , x k} va ma trail
(adk la ma trail clic( (p i = (p1 F doi vdi cd sä do. Vi xac dinh
150
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
151
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 2 X, 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
152
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
igng cua u. HO nhtt thg vdi s = 1 lam tan tai theo vi du 3.5.
153
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 X s, 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) ngl â a. IA khi va chi khi 3 1 , dtiOng (am).
154
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
ma trail cheo. Dat C = T.P, to chide Ct AC = A va Ct BC = In.
Vi du 3.9: Vei mbi ma tran (pik) A, ki hieu A* la ma Oran
lien help veil ma tran chuyen vi cim A, nghia la A* = A . Ma
trait vuong A throe goi la ma tran Unita n'etu A* . A = In. Cho A
FA ma tract Unita, eheing minh:
a) Neu k la nghiem (Mc trong cern A thi I XI = 1.
b) Ngu A. la nghiem dac trong eUa A thi 1 — cling la nghiem
dac thing cim A. c) Ngu A ante vi co cap le thi A co ft nha't met nghiem dac
trung bang ± 1.
Lel gidi:
a) GM sei X la nghiem dac trting &la A, khi do c6 ma tran
eet X # 0 AX = XX, tit de (AX)* = X*.A* = 7r X*. Nhan vg vat
v6 hai clang thtic tren, to co X*A* A X = X* X. Vi X x 0,
nen X* X x 0, vi A* A = In nen:
X*X = h.. X*X (i. . - 1) X*X = 0
.1.=1,dovay 114=1.
155
b) Gia sit X la mat nghiem dac trong cga A. det (A-X I) = det A* det (A - XI) = det (A*A - ) = det (I-A = 0
det (-1 1 - A*) = 0 det (-1
I - A*) = 0
det (A - —1 I) = 0 nhu vay X — la nghiem dac thing dm ma tran A.
c) N6u A la ma trail thge co cap le, thi det (A - XI) la d thitc bac le voi he s6 thgc, vi vAy co it nhat, met nghiem thg Theo phan a), nghiem time do phai bAng ± 1.
Vi du 3.10. Cho E IA kitting gian vec to n chieu tren tradn
so' thuc Tk va f IA met clang toan phudng iron E. We to x e
dude goi la clang huong nett f(x) = 0. Chung minh rang n6u f di data, nghia la t6n tai x,, x, sao cho f(x,) > 0 va f(x 2) < 0 thi tron E t6n tai co sa g6m nhung vec to dang huOng.
Ldi gidi:
Gia sii (e), 1 < i < n la co so chudn tAc cua E, nghia la ed s
ma trong do f c6 dang chudn tac. Hun nua, gia sit AO = 1 vdi i 1, 2, ..., r; f(ei) = -1 vdi j = r + 1, r + s va f(e,) = 0 voi t = r + + 1, n. Do f da't ddu nensz 1, r21.
Nhu vOy yen x = LU i e j E lE , ta co:
too ai 2 + arz _ cts r_fri _ _ coo_s.
Ta say dung cd se {v,} (i = 1, n) g6m nhung vec td clan Intang cim E nhu sau:
v, = e, + er,„ 1 < < r v = e - e r+1 r+s , , v,= r+s < t < n.
156
De thdy he {vi), 1 < i < n gem nhiing vec to clang hung cua
1. Ta chfing minh he NJ la cd se cita E. Xet EX,v, = 0, to
hung minh = 0 vdi moi i. rrs
Ta co ZA.,(e, ei.+1 ) + e j ), + Ek te t = 0 j=r+1 t=r+s+1
( r+s r
) + EX , EX e '.1 j ei + , , j=r+1 r 1=2 =1
yi X r+1 er+1
r+s
E A i e i Excet = 0 j=r+1 t>r+s
Do he ted, 1 <i < n doe lap tuydn tinh nen = 0 vdi moi i =
Nhu vay he (v,) la cd s i eida E, gem nhung vec td clang
tudng.
Vi du 3.11. Ta goi non ding hung cna clang Loan phudng f
a tap K tdt ca cac vac td dang hUdng. Chung minh rang non
Vang hudng cita clang than phudng f tren khong gian n chieu E
A khong gian con cud E khi va chi khi f khong ddi clgu, nghia
A f(x) 0 vdi mm x e E hoac f(x) < 0 vdi moi x e E.
1,64 gidi:
Dieu kien Gia sit 1K la khong gian con cim E. Neu f
thi then vi did 3.10, trong E có met cd sa g6m nhfing vac td
dang hang. NMI vay K = E, nghia la moi vec td cim E den
157
dang hudng. Digu do mau thuan vat anti Witi clAu cita E Do vay khong dau vi digu kiln can chn}c cluing minh.
Dieu kik gia si f kitting de). dAu, eking han f(x) 0 voi
moi x e E. Xet co sä {n) chudn tic cart f, to ed
f(Ici,e,)= 4 +. i=1
2 Nlw tray, non clang hudng la tap cac
vec to x = Ewe, me. a, = = a, = O. Do do tap K cac vec td
Bang hudng lk khong gian con dim E sinh bai {a„„
Vi du 3.12. Giit sit f tia g la hai dang Wan pinking tren
kitting gian tree W V, có bjeu auk toa da trong mOt cd {n}
= 1, 2, ...n) nao do In WS la: f = Za iix,xi , g = Ebo i x i .
Ta goi hop thanh cua hai dang toan Oath:Mg f, g la clang toan
phudng (h g) = bij x i x i .
Chung minh rang:
a) Neu cac dang f, g khong am, thi dang (f, g) cling khong am.
b) Neu cac dang f, g xac dinh during, thi dang (f, g) cling xac dinh chidng.
Ld gidi:
a) Hign nhien la (f, + f2 , g) = (f„ g) + (f2 , g).
Dua cac dang Wan phuang f, g ye dang clutAn tic:
158
EyF , g = , a do 37 1 , z3 la cite ham tuyen tinh dei vdi i=1
• Bien x 1 . Do la bieu tbiic toa do, cim NT* chuyen cd so. Ta co
= Xig? • Elf = EZ(37?' 4) • g)
i=lj=1
Xet mot s6 hang (y2, z2 trong do y= Ea k xk , z = Ebk xk k=1 k=1
• do y2 = Eak a / xk xis z2 Ebk b i xk xi , va (2z2) =
k,1=1
2
ai b k b / x k x / = Eakb kx k I 20. Nhu vay khang dinh i=1 k =1
dude chUng minh.
b) Gia sit f > 0 va g > 0. Ta dua g v2 clang chua'n tAc
= E2 y 3 , yi = Eq 33 x i (1 = 1, 2, ..., n), det (n o) # 0. KM do
i =1 i=1
g)=E (f, i=1
\ Nhung = Ecniq ik x 3 x k , nen (f,y 32 )= Eaik kq u x i lkk xk ),
do f = Ia ik xix k . Do f xac dinh dtidng nen (f, n2) 0. Neu X j,k=1
(xi) 0, thi co; a x; # 0, tea do do có i qi; # 0 n8n ci n xy x 0, va
o f xac dinh dudng nen (f, 37, 2) > 0. Do \Tay (f, g) > 0.
159
Vi du 3.13: Chung to rang trong kh6ng gian vec to Odclit chieu, c6 the tim dudc ho n vec to don vi u l , u2, sao cho di vai tat ca cap s6nguyen phan biet i, j, yen to - 11j cling la of
1 " to don vi. Ta {tat x= n +1 In, . Hay tim goe giva vec to x - 1 1
va x - U 2 .
Ldi
Gth six {e 1 } (i = I, n) la mot co sa bye chuSn °Cm khan gian vec to dclit E. Ta xay dung hen vec to {u„ u„) nhu sat he (ti t , u2) thuOc khong gian sinh bai {e„ e a} va {0, u,, u2} lar
thanh tam gine deu, nghia la ria l 11=1111 2 E1=1 va 14.u 2 = Ta b
sung tr, nhu sau:
u3 = + ).2u2 + X3e 3 , u3 u 1 = 11 3 . 1.12 = 2 —1 , u3 2 = I.
1 X? 1 Xi
+ 1 Tit do co —2 = X, +=
2 ' 2 2 A2 A1 X2 - —3 .
IF Va vi Al + x.22 + 4 + Xi X2 = 1 X 3 = + —2 3
Bulk xay dung vec to UM n to (n-1) vec to ducte xi{ Wong ty. Gia su to co (u1, u2,—,u.,-,) la he n-1 vec to nam trong klning gian con sinh bai {e„ ...e„. 1 { va th6a man (lieu kien bai than. Ta tim vec to
U.,= X, + + .;.
Sir dung dieu kiOn u n u; = 2 — vdt j = 1, n-1 va u,12 = 1, to
tim duac = 12 = = An_ = —1 va I n — n +1N hit vay to da
2n
160
xdy dung dude hee {m, u„} cac vec to don vi trong kitting gian dent E" they. man (lieu kiOn: goo gliva hai vec to bet kyt cim he
bting 60°, tit do I tt, -uli = 1 (i j).
1 Bat x -n
+1 + u2 +...+ ti n ), goi 01a gee gala x-u, va x-u 2 .
Ta ce 1 = (u 9 - 110 2 = [(x-u 1) - (x-u 2 )] 2 = (x-u 1 ) 2 + (x-u2 ) 2 -2(x-u2) (x-u 2);
va (x-1.1 1 ) 2 = (x-u 2)• =
Tit do cost/ =
1 tux + -
M +1r - 2M +1)
Vi du. 3.14: Gia sii V la met kh8ng gian vec to Unita, B: VxV->C
IA dang tuyen tinh rued tren V, nghia la B tuyen tinh dot vdi Men thil nhet, Ong tinh dei vdi Mon th>Y hai va B (x, ky) = (x, y) vdi moi x, y e V va e C. Khi do tan tai duy nhet met
phop del tuy6n tinh A cim V sao cho B(x, y) = <x, Ay> yin moi x, ye V.
gidi:
Truisc ta xet bit sau:
BS de: Gia sit f la dang tuygn tinh tren khong gian vec to
Unita V, khi do tan tai duy nhet met phdn tit h E V, sao cho vdi moi x e V, ta co f(x) = < x, h>.
Ghiing minh de: Net cd sa true chua'n te,, e 2, ..., ea} trong
V. ret phdn tit h e V, h = Ehk e k , a do hk = fk). Gia sii k=1
x= EXicek e V, Ta co f(x)= Exk k = xk hk = k =1 k=1 k=1
n n
< Ixke k , Eh k e k > = < x,h > .
X=1 k=1
161
Ta chfing minh phan tit h la duy nhat. Gia sii co 11 1 , h2 li hai vac [Al sao cho f(x) = <x, h,> = <x, h 2> vOi moi x c V. Tit d. <x, h, - h2> = 0 vdi moi x e V. Dac biet, lay x = h, - h2, ta ci 11h, - h211 2 = 0 h2 = h2.
Bay gia ta chung minh bli Loan: Gia sit y la phan 'eft c11 din} Mt Icy caa V. Khi do B(x, y) la clang tuyen tinh clSi vat than x Theo be de' co phan to h xac Binh duy nhat (phu thuOc y) sa( cho B (x, y) = <x, h>.
Xet anh xa A: V —> V
y —> A(y) = h.
Ta chitng 6> A la anh xa tuyen tinh. Ta co B (x, y, + y2) = B (x, y 1 ) + B (x, y2) <x, A(Y(+Y2)> = <x, A(y,)> + <x, A(Y2)>
<x, A(37 1+Y2) - A(Y1) - A(y 2)> = 0 vdi moi x c V. Do v'OY A(Y)+372) = A(37 1) + A(Y2)-
Tudng Lu B(x, 1y) = <x, A(Xy)> = y) = 7<x, A(y)> = <x, 1 A(y)> <x, A(1y) - X A(y)> = 0 vdi moi x e V.
Tii do suy ra A(1y) = X A(y).
Ta chiing minh tinh duy nhat maa A.
GM sii A, va A2 la hai phep Man de4 tuyen tinh thea man B(x,y) =<x, A 1 y> = <x, A2y> vdi moi x, y e V. Suy ra <x, (A, -A2) y> = 0 vat moi x c V, ye V, to do (A l -A2) y =0 vdi moi y E V, hay A l A2 .
Vi du 3.15: GM sit A NM B la hai ma trail thuc, xiing xac Binh during cap n.
Chung minh rang det (A + B)?, det A + det B.
Lai gicii:
Try& Mt ta giai bai toan trong twang hop rieng, A = In la ma tran ddn vi. Vi B la ma tran doi 'ding xac climb during nen
162
ton tai ma trail truc giao C sao cho C 4 B C c6 clang cheo ma cac phan tii tren duang cheo chinh la cac gia tri cna B.
Vi vay det (I + B) = det C - ' (I + B). C= det (I + C -1 B C)
= (1 + ... (1 + a 1 + X, ... X„ = det I + det B.
Ret truong hop t6ng gnat, vi ma tran A xac:dinh duong, nen c6 ma tran D sao cho A = D 2, a do D the dinhducmg. That
vay, có ma trail D, de2 D,' . A D, = A co clang cheo, D, la ma tit)
true giao (vi du 3.7) A la ma tran cheo ma cac pith) tit tit) cluong cheo chfnh la cac gia tri rieng cim A: p„ n„ > 0. Dat P la ma tran cheo mA cac phan to trot) during cheo chfnh la
P2 = A va A = D I A D l d = D,P1 D1 = D I PD1 1 .
D,PD 1 -1 = D2, vei D = D, . P . D, - ' = D,PD,t, D la ma tran d61 xang.
Td de A + B = D2 + B D (I + D - ' B D - ') D .
Vi vay det (A + B) = (det D) 2 . dot (I + B D-1 ).
Do D - ' B del sung , the dinh dming nen set dung k6t qua trot), to of):
Det (A + B) Y. (det D) 2 (1 + det D - ' B . D - ') =
= det D1 + det B = det A + det B.
NMI vay, bai town dude chting minh.
Vi du 3.16. Ta ky hieu M la khang gian cac ma Iran yang cap hai tren truong C.
a) Gia sit f: M C
X f(X) = det X.
Chung t6 rang f la met clang toan phudng. hay tim ma dean ciaa f trong cd so chinh tac caa M. Hang cna f bang bao nhieu?
163
j al as = a, X, + a2 X2 + a3 X3 + ai X4.
as • a) Gial sit X =
4
b) Ta nhac lai rang f(X Y) = f(X) . f(Y).
Gia su p la clang toan ph/dna tny y khac kh8ng tren 1M thOa man di6u ki6n p (X Y) = cp (X) . p (Y). Hay chfing t6 rang
= 1 va n6u X khong suy bieh, thi p (X) x 0. Hay tinh gin tri cim p tren cac ma tran lily tinh va cac ma Han suy bi6n not Chung.
c) Chung to rang p = f.
Lai
do {X I , X 4} la cd so to nhien dm M.
f(X) = a,a, - a2as . Day la mot dang than phudng tren M. Trong cd so tg nhien cem M, f có ma tran
0 0 0 1
2 0 0 0
2 0 - 1 0 0
2 1
0 0 0 2
1 det A = — 0, vay clang 'Loan phtidng f co hang 4.
24
b) Vi p khac khong nen co X e M de? p(X) x 0. Theo gia thiet to co p(X) = p(X . I) = y(X) . 9(I). Do 9(X) x 0 nen p (I) = 1. Neu det X x 0, thi co . = I. Vi \Tay
9(X . Xd ) = 9(X) . 9(X -1) = 9(I) = 1, cho nen p(X) + 0.
A =
164
Gia A E M, A x 0 va Ala ma tran lily Kith. Nhu vay A 2 = 0,
vay W (A') = MAW = 0 tii do (A) = 0.
Gia s5 X e M, det X = 0 th6 thi hang cem X bang 0 hoac
sang 1. Neu hang X = 0 thi X = 0 va vi bay p(X) = 0. N6u hang C = 1, thi eó ma trail khong suy Bien P, Q cle X = P . A . Q, a do
'0 1` = (xem bat tap 2.11).
,0
Tit de cp (X) = w(13) . W(A) . (p(Q) = 0,
vi cp(A) = 0, do A ley linh.
c) Vol X Lily Y thuee M, ta ehiing minh 9(X) = f(X). That bay
set ma trail X + AI e M, ta co
f(X + XI) = f (X) + 2 k F(X, I) + )L2
9 (X + XI) = (X) + 27r4 (X, 1) + x2
6 do F, (F la the dang cue wong fing cua f va cp. Khi X + i, I
suy bi6n nghla la khi k la gia tri Hong cim X, thi f(X + AI) =
(p(X+Xl) = 0. Til do suy ra hai da thfic tren bang nhau vat m9i X.
Dab biet, cho 7%= 0 ta co tp(X) = f(X). Vi X tay ST nen f=T.
Vi du 3.17. Khong tinh dinh thilc, hay giai thfch tai sao ma
tran A sau day kha nghich:
'1-i 2 3 4
2 3-i -2 0
3 -2 - i 1
4 0 1 2 - i
A=
165
Zd gidi:
Ta co A = A, - a do A l = 1 12
2 3 0
3 -2
4 '
3 -2 0 1 4 0 1 2
Ma trail A I la ma trAn thvc dai xting cap 4, nen tat ca 4 gin tri rieng dm A, deo thvc, (xem vi du 3.5), nghia IA da thfic POO = det (A, - ce, the nghiOm deu thuc, ter do P (i)* 0, nghia lA det Ax 0.
Vi du 3.18
a) cis. sit {e h e2, e3} IA mot co sa cim R3 . Cho clang Loan phttong Q tren Q (x) = x 1 2 + x2 2 + x3 2 - x1 x 2 - x2x3; x = (x 1 , x2 , x2) la toa dO cua x trong co sa {e 1 , e 2 , e3}.
Chang to (tang town phvong Q xac dinh mot can trtic delft fret) Re Hay tim moot cc( sa trvc chua'n cim R. 3 dot voi Q.
b) Gia sit f e End (R2), co ma trap A trong co sa 02, e3}:
'3 -1 1 '
A= 0 2 0
1 -1 3 Hay kie7m tra rang A la den xiing doi yea c‘u trot delft
trong phan a) (nghia la f la tv dang ca2u del xi:Mg)
Ldi gidi:
a) Ma train cua Q trong co sä {e t , e2 , e 3} co clang
166
Taco H, = 1, H
1 det
2 H = — > 0. Vi H la ma trail xac dinh during nen Q
la dang town phudng xac dinh during. Mut vay Q xac dinh mot
Lich vo hudng tren Ra (do chinh la clang cip tudng ting vdi Q).
Bay gia to tim mot cd so tn.tc chuAn del vo . Q. Neu x = (x,,
12, x3) thi Q(x)=(x, --1 x2 )2
+ (1 x3 - X 3 )2 + 2
1
x l = '1-
2
2 2
1
x2 = x3
1
x3 = --)r2
Hat
X =
x 2
+ x3 2
+ —x3 2
KM do
Tit do, xet cd sa {e' 1 , 0' 2 , e' a} ma
167
e el
ez = e3
e3 2 —e
' + .ne 2 +
'
thi trong eel sd fe'„ e' 2 , thdc cua clang town phdong Q eó Bang Q x,, x 3 )- x 12 + x'; +
Nhtt vay {e' 1 , 8'3} la co so true ehuan.
b) Goi n la dang eve cua Q, khi do trong co sa e2, e2), co ma tran H.
ife" chdng minh f la tv d6ng ciiu del xdng, ta phai chtIng minh n(f(x), y) = n(x, f(y)) vdi moi x, y E R. Goi X, Y la ma tran
cot vac toa de) cua x, y, ta phai el-Ong minh: (AX)'1-1.Y=Xt.H.A.Y
hayrAt .H.Y=Xt.H.A.Y
That Vey At H =
( 3
-1
1
0
2
0
1 '
-1
3 i
1
2
0
2
--1 2
0
2
1
3 -2 1 :`
= -2 3 -2 = H . A.
1 -2 3
Vi du 3.19. Cho V 11 khang gian vec td tren tnlang s6 thvc R. Gin' su Hi va. H2 la hai dang than phuang tren kheng gian V.
168
a) Chung to rang ngu H , la xac dinh (Wong, thi ten tai met
cd so de trong co so do ca H, ve. H 2 dgu co clang chinh tilt.
b) Neu vi du chUng to rang kheng phai bao gio cling clang
thai dua &toe hai dang toan phudng da cho vg clang chinh tat.
c) Cho hai dang than phudng tren R2 3 trong cd so chinh tat
có clang:
\ H i (X)= Xj.2 +54 +144 -F2X I X9+2XIX3+10X2X3
H2(X)= 4x 22 +104 + 6x,x 3 +14x2 x3
Hay tim mot co sa trong do ea H, va H 2 dgu co dang chinh
tac.
L.& gidi
a) Gia sii dang cue eda dang toan phuong H 1 . Vi H,
)(lc dinh dudng, nen W 1 siuh ra mOt tich vo huang trail V: <x, y>
= y). Ta bigt rang, trong khong gian vectd dclit V co co so
trip chan clg clang toan phudng H2 co dang chinh
the H2(X)=- Trong có so true chua'n do, H 1 (x) =O 1 (x, x) =
n 0 <x, x> = Ex r Nhti \Tay, trong cd so {v 1 , Vi co ca hai
i=1 clang toan phudng da cho dgu co clang chinh tilt.
b) Kid ca hai dang dgu khong xac dinh &king, thi co th6
khong ton tai mot cd so de? ca hai dang toan phudng dgu co
dang chinh the. Chang han xOt trong vdi cat toa clO (x, y)
trong co so chinh tae , e 2 }. Hai dang toan phudng H 1 (x, y) = x2 ,
169
H2 (x, y) = xy. GM a co mot co so te l. , dg' trong cd sã do H, va H2 dgu có ding chinh tac.
GM ad: = ell e l +C21 e 2
62 rc + C22 62 Goi (x', y') lk toa dO trong co sä
IX --c'Cii11+ Cl2 ). Khi do
(y= C21x'+ C22 r'
Ta nhan thdy H, c6 dang chinh the trong ca sa {6,, E 2} thi C„ . C 12 = 0; H2 c6 clang chinh tac trong co so fe„ E2) thi CI, C22 +
Cu Ti12 C.2 1 C 1 2 = 0. Tit do suy ra ma tran C = suy hign.
C 21 C22
Didu nay mall than vi C la ma trail chuygn cd so.
c) Ta nhan thdy H 1 la clang tan phudng xac Binh throng.
H, (x) = (x 1 + x2 + x %) 2 + 4 (x2 + x2) 2 + 9x32 .
1
Y3 = 3x3 1
{YI = X I + x2 + x3
Pat bign mai y2 = 2x2 + 2x3 hay
Trong ho toa do, do, I-I2 co bia thfic:
H2 = 172 2 + 213373
xl = Y1 2 Y2
1 1
x2 = Y3
(2)
(3)
X3 =3 y3
thi voi die toa dO (y„ y 2, ..., y3), H, co dang chinh tac:
2 2 H1 = + Y2 + y3
170
'o 0 1'
Ma tran maa H2 CO dang A = 0 1 0
,1 0 0
(4)
Bay gia ta (Ilia ma tran (A) vg dang the() bai ma tran tryc
;iao. B&ng phnang pho.p quen thu0c, ta tim &No
1 1 0 -1 0' n&
T= 0 0 1 116 . A. T = 1 (5) 1 1
0 0 1 .
1 1 z2 yl =- + r z i
V2 a/2
Dung phep toa (.10 Y2 = 3 (6)
1 1 22 Y3="' +
Trong do ca hai dang town Oaring se c6 clang chinh tAc.
H, (z
H2 (Z
Tii (1) va (6) ta co
, z2 z 3)
, z2, 23)
x =
x2
xa —
9 = + +
— - 2 +22
1 --,z, 1/2 '
1 3.h z1
1
9 +
2 +22
+
+
9 z 3
2
1 1 — 2 z3
lz + 2z3
(7)
(8)
(9)
.5 z2 1
2 1
1 3,5
z 2 3-,,&
HS {el, e2, C31 7a co sa trong K3, trong do H, va H2 ce dang
chinh ta.'c (7) va (8). Khi do:
171
1 1 1 el 3"2- e2
+ ea
1 $ 1 ,a 1 i)
3-,N -2 + 3I_ -3
1 1 = +,7, 0 2
Chu if: Da dila dang thai hai clang town phydng H„ H2 trong
d6 H, xac dinh diking v6 dang chinh tal c, ta lam cac btafic eau:
Bieck 1: Dila dang toan phtidng H, v6 dang cheo, nghla IA
tim mat co sa c 2 , ...c„} true chuan dai vdi dang cue 0, cUa H„
aide 2: Vik biau iliac maaja 2 trong cd sa fe l , c2 ,
Bade 3: Dua clang Loan plincing H2 vi dang cheo bang ma
tran true giao, nghla la tim cd sa {a„ a„} true chuan doh vdi
de' H2 = co dang cheo. Khi do trong co sq {a„ um} ea hai dang H i , H2 den có dang cheo.
Vi du 3.20: Cho dang town phtiong f tren khOng gian thuc n
chit E, có cac chi s6quan tinh throng va am Ian Itto la p va q.
Gia sa E l la khong gian con cern E ma dim E, = n - 1 thi chi see
quan tinh cua dang than phydng f han ch6tren F1 1a bao nhieu?
Lo gidi:
Trude 116t ta nhan thay, neu F c E lh khong gian con cila ]E
ma f I F xac dinh diving (tticing tang xac dinh am) thi dim F < p,
(tticing tang dim F < q). That vay, gia s 1 f/F xac dinh &icing va dim F > p. Xet co sa (e r , e„ crib e„+„fri, •.., e„I ma
f(e,) = 1, laiap
172
f(e,) = -1, p+1 < j 5 p+q
f(e/) = 0 1> p+q.
Bat F, la khong gian con sinh bai cle vac to {e„,, epop . •.., e n), thi f/ F, o 0. Nhung dim F n F, = dim F + dim Ft -
dim (F + F 1 ) nen dim Fri F l >p+ (n-p) - n = 0, matt thuan vdi gia thiat f/F > 0, f/F, o 0.
Gia se dim E l = n - 1 va p', q' la cac chi s6 dicing quart tinh va chi s6 am quail tinh maa f han the troll E l . Ki hieu F2 la khOng gian sinh bai le l , ..., e p) tren d6 f xac Binh ducing.
Ta co dim F2 (1 E, z p - 1, nhung dim (F 2 n El) o Tit de p - 1 5 p', nhu vay = p-1 hoac p'=q.
Tuong to q' = q-1 hoac q' = q.
Ta nhan thgy ea ban trudng hop
P‘= P IP P = 1 I P r = P -1
q -1 k r = '11 kir= q -1
den co the xay ra. Ta xet cac vi du sau:
a) n'au p + q < n va E, la kh8ng gian vac to sinh bai le„ e„41, thi p' = p, q = q.
b) neu q > 1, thi vdi E l sinh bai he fe l , e p , e p+ 2, to co
le 1 = P
q = q-1
c) neat p > 1, chon E, sinh bai e„) to &lac p' = p-1, q'=q.
173
d) n6u co p z 1, q 1, chnn E, sinh bai le, + e 2 , e,, en+2 , ... e n } thi p' = p-1, q' = q-1.
C - BAI TAP
3.1. Gia sii H la clang than phuong tren khong gian vec tc R3 . Dang chuan titc cua dang toan phtiong H la dang chinh rac
trong do cac hG s6 deu bang ±1 hoac bang 0. Hay tim dang chuan tac cua H trong cac trtiang hop sau:
a) H (x„ x2, x3) = 4 -24 -Fx .g +2x,x9 +4x1 x3 +2x 2 x3
b) H (x 1 , x 2 , x3) = x,x2 + x,x3 + x2 x3 .
c) H (x„ x2 , x3 ) = 44+,,..3+4-4x1 x2 +4„„(3 -3x 2. 3 .
3.2. Xac dinh X de dang toan phtiong sau xec Binh cling.
a) 4 + 44 +4 + 2Xx, x 2 +10x,,x 3 + 6x.,x3
b) 24 +24 +4 +2Xxix, +6x 2 x3 +2x,x 3
c) 4 +24 +21/4,42 -2x,x 2 +6x 1 x 3 -2x2x 3 .
3.3. Dua dang than phtiong sau v6 dang chinh tAc:
11 9
a) Q(x l , x„)= Dx i t + Exi x j , 1<i<j<n i=1 i<j
Q(x 1 , x„)= Exi x, <n
3.4. Cho V la khong gian cua vac to tren trueng s6 thuc
va f la dang than phdOng tren V, p. q la the chi so throng quan tinh va am (pan tinh cua f. K la non clang Inking cna f (xem vi
174
du 3.10 va 3.11). Chung minh rang khong gian con t6i did &la E nam trong non clang hueing K có chigu la n - max (p,q). Ngu f khOng suy bign thi s6 ehigu do la min (p, q).
3.5. Cho E la khong gian vac to (Debt, fo„ e„} la co so trUc chuan cua e. Vol {a,, a„} la he n vac to tny y ena E, goi A la dinh fink dm he tan ..., an} dgi yea co sa trite chuan fen •••,
nghia la A = det (N) a do ai = Za ij e i (j = 1, n). K1 hieu
Grier, an} la dinh thile Gram cna {a„ aj. Hay chUng minh Gr {ct„ ct„} = A2 .
3.6. Gia A 1 , ..., An la the doan thong trgn during thong thvc h (khOng gian vee to (kilt mc5t. ehigu R). Ki hieu a y la do
dal dont]. A, 11 A; 1 j s n. Pat A = (a u) e Mat (n, R). Hay
chUng minh det A a. 0.
3.7. Gia sit K,, K„ la nhring hinh troll trong mat phang. KI hieu b„ la then Lich cue giao 1{, fl 1 j Got ma trim B = e Mat (n, R.). Hay chiing minh det 0.
3.8. Chiang minh rang moi ma tran Unita dgu eheo hoa duoc, va moi gia tri rieng ena no co modun bang 1.
3.9. Cho ma trait Heemit A, nghia la A t = A, Chung minh rAng co ma trail Unita C d@ C' .A .0 la ma tran cheo.
3.10. Chung minh rang ngu A = (a„) la ma tren deri xthig, xae dinh (Wong cap n thi (tr A) (tr A -') n2 .
3.11. ChUng minh rAng cac nghiem Jac trong mia ma trait del 'tang thvc A nhm trail doan [a, b] khi va chi khi clang Wan
175
phudng e6 ma trAn A - Xth the dinh &tong vdi moi Xo < a va xac dinh am vdi moi Xo > b.
3.12. Gia sit A e Mat(n, R) la ma Win thud del xfing va u,
la vec to rieng (cot) eim A vdi gia tri rieng X,. DM B 1 = A - a I UI .Ut
a) Chung 6) rang c6 the chon dttOc U, de B,. U, = 0.
b) Xet met cd sa true chua'n Up U9, Un giSm cac vec tc
rieng dm A, ting veil the gia tri rieng 4
ChUng td rang
A= .
3.13. Cho E la kh6ng gian vec td tren truang s6 thtic R, f lII
Bang song tuy6n tinh del )(ling tren E. Gia sit ton tai vec tc
khac khong a e E sao cho f (a, = 0 va to clang that f(x, = ( suy ra x ding phudng vdi a hoac f(x, x) < 0.
Chung minh rang n6u x * 0 va n6u co y de? f(x,y) = 0 Nq' f(y, y) > 0 thi f(x, x) < 0.
3.14. Cho ma tran A E Mat (n, R), A phan dol xOng. Chiim
minh rAng del A a 0.
3.15. Cho ma trAn A e Mat (n, C ), A, la ma trAn lien hqi
cim A. Chung minh rAng det (AA + I) a 0.
3.16. Cho U la khong gian vec td Unita, f e End (U). K lieu Pr la to &Ong cgu lien hop vdi f. Hay chiing mink
a) Id* = Id.
b) (f+g)* = + g*
176
c) (Xt)*= ti f* yak mai A e C.
d) (CI)* = f
e) (or = f*og*.
3.17. Cho U la khang gian vec td Unita va f e End (U). Chung minh rang f la to citing ca.0 tg lien hop khi va ehi khi <1(x), x> thgc \Tdi moi x c U.
3.18. Chang to rang gin tri rieng haft kg X aim tp (long tau to lien hop f caa khang gian vec td Unita U bang <f(x), x>, a do
x la vec to nal] do oda U thOa man Pc11 = 1.
3.19. Cho U la khang gian vac to Unita, f la to citing ca:u ty lien hop tren U. {X„ la sac gift tri rieng cim f, {e„ la cc; sa trge chua'n go'm nhung vec td rieng cim f sao cho f(e 1 ) = ki e i ; i = 1, n. Xet Pk; U U, Pk(x) = <X, ek> ek.
Hay chatng minh:
a) Pk =1)k
b) Pi oP k = 0 nen k # j
c)f - XkPk k=1
d) Gia sii Q(x) = Eck x, la da thac tag § vdi he 86 philc. k=0
Pat Q(f) = riCi r e End (U); f' = Id ]=4
Chiang minh rang Q(f) = Q (4)Pk k=1
177
3.20. Hay cluing minh dinh lY Hamilton - Kelli cho &Jiang hop ma Whit A la Hecmit: Cho A la ma tran Hecmit va P(k) = det (A - XI) = (-X)" + C 1 (4.)"-. + + Ck(-X)" + + C. la da thuc dac thing cim ma tran A. Hay chitng minh P(A) = (-A)"
C 1 (-A)°4 + + C„ . I n = O.
3.21. Gia sit A = (ay) e Mat (n, R), a do n > 3, va t&t ca can
ph&n tit an # 0. Chung minh Ang oh ma tran Be Mat (n, > 0
Nth mci i , j ma C = (a 1 . bd la ma tran suy bign.
3.22. Gia sit E2 la khong gian vec to dent hai chigu. Tu citing din dot xitng f e End (E 2) trong co sa trite chuan fe l , e2} co
1 2
1 2
,
ma tran cha f co clang cheo.
3.23. Hay dua ma trail dot xiing sau vg clang cheo nha ma tran trhe giao
' 0 0
B= 0 1 0
1 0 0
3.24. Trong khong gian vec to R3 , cho hai clang town phttong:
= 2x 12 - 24 - 3x32 -10x 2 x 3 +2x 1 x3
9 = 2x12 +34 +2x32+2xiX3.
ma tran A = Hay tim co sa true chuan trong do
178
a) Chung to o la clang tnan phuong xac Binh during
b) Bang Olen biOn deit toa do thfch hip, hay dua ca hai
dang toan phudng tren vg dang chinh tat.
3.25. Gra. sit' E la khong gian vet to aclit vdi tich hitting
< , >. Tv ddng eau f E End (E) duoc goi la phan doei 'ding nefu
vdi moi e, y E E, to co <x, y (y)> = - <1(x), y>.
a) Chung to rang f phan asi 'ding khi va chi khi <f(x), x> = 0
vdi moi x e E.
b) Trong Wit co so true chua'n bat Icy, ma trait cait to ddng eau phan d61 ming la ma trail phan 'ding. Tit do suy ra moi to ddng ca."u phan d8I xiing caa khang gian delft s6 chigu to de).1
khang kha nghich.
c) Gia f la tti ddng c'au phan ddi xang. ChUng to rang Imf
va Kerf la hai khOng gian con ba trip giao vdi nhau. Chung to
rang hang caa f la mot s6 than.
D. NCTONO DAN HOC DAP SO 9 32
3.1. a) H = (3, + x2 + 2302 - [1-3-x2 + 731 x 3 ) -(a33
y i = +x2 +2x3
fat 2 =- + I 2 3
Y 3 3x3
179
Taco H= 23
b) H -37F2 - 3 1
c) H = _2 2 32 Y3
3.2.
a) Khong co 1. nao th6a man
b) Kh6ng nao th6a man
c) A> 13.
3.3.
a) Q-602 -i612)2 + 1 67 3)2 +---+ n+1 67 :32 6 2n
1 That \ray, dat y, = x, + —(x2 + 2
x„), thi
2 -3-x,x) ;2
Dat y2 = x2 + 1 (x 3 + + xn), tu do
f = +71 1374) + -
6 Ex; +—xi x ; 3 <1
;>3 4 .i<j
Gia si c1641 blidc thtt p:
E xVE 2 Exi x j j , i=p+1 P ± 2 i<j
Q = 671) +
adO(p.4i<j)
p+1(_ , )2 + p+2 2p VP! 2p + 2
180
Da, t = x,,, 1 , + (x l,„2 + + x„) ta dude tong thiic tren, p + 2
dng voi p+ I_ Do do sau (hsj) buoc ta co ket qua. Ta ce the' viet gon quy tacd6i bie'n nhu sau:
1 --
1 1 1
2 2 2
0 1 1 1
3 3
(Yu, 0 0 0
b) Dua ve a) nhu sau
Q= x i x 2 +x l Ex; +x2 Ex i +Exi xi ; 3<i< j i=3 i=3 i<j
= (x, + x s + + xs) (xs + x, + + x s) -
1 t
Q =-1
+ x s + 2Z x,1
4 4 >3
Dang (n-2) bin x3, ..., x i, E .1 3.5+=j
dttqc xet trong phein a). Nhu vay ta co
Q=(Y1)2 -6,7)2 L 2k
—2) -1 (Y1J2
1(23 (k
1 s V = —kx, + x 2 ),- Lx,
2 =3
YI
y2
/
x1
9
X9
181
1 Y2=2 1 + x9)
1 Y3 = x3 + --kx 1 +
' 2
1 Y4 = X4 + —2
/ + .
xi
x n
yfi =
3.5 Gr (a l , an) = det (<a1, ak>); 1 < j, k 5 n
a.j =Ea-e• ak = Ea ik ei . Do ha {ea trite chuan nen ,=1 1=1
< a ja k >=Ea,a ; ,, ; Dat A= (ao).
Ta có ma train (<ci, ak>) = At A.
Tit do Gr (a„ an) = (det A)2 =
3.6. Gia s8 doan [a, hi chita tat ca cac = 1, n). Xet
e [a, hi la khong gian cac ham lien tuc (co the trii mat s6 frau
han diem) tren loan [a, N. WI* [a, 131 la kheing gian vec to vo
han chiau Den truang sathac K.
Goi bla,b1= C a,b de "—" la quart he Wong throng f g
caf= g tren [a, hi tril mat s6 hvu han x E [a, b].
182
NMI T,i e C [(Lb], (tat = cf(t)g(t)dt la tich vo httdng
tren C la,b1 (khong phu thuec vao dai dien f, g x;
{.1n6uteA; la ham dac trdng caa tile la X i(t) =
Oneitto6„
Thl > va det A = Gr (R I %^) >_ o.
3.7. Thong to bat 3.6.
3.10. Vi ma tran A d6i xfing, xac dinh &mug nen ton tai ma
trail true giao C C -1 . A . C co clang cheo:
'a do x,, ... la citc gia X2
0
tri rieng caa ma tran A. 1
xt 1
12
0
k„
0
= A-1 C. Ta có: 131 =
1 NMI vay A-' có cac gia tr rieng —,..., —
1
ki
n
Va trace A = Ek ; trace A-' E - i=1
n 1 2 Ta có (tr A) (tr 21/2 - ') = E-
LI Xi)
183
3.11. I-ID. Ta thky rang: rigu I la nghiem dac trong cda ma tran A thi - X(,) la nghiem dac trong ciia A - X0I. (xem giai bei 2.45, each 2). Nhu vay A Ia nghiem dac tning cim A, X e [a, b], thi k - ?to IA nghiem dac cna A - - X0 e [a - k, b - X]. Neu moi nghiem dac trong cda A deli thuec [a, b] va I, < a thi do a - 10 > 0 nen moi nghiem dac trong caa A - X0I, deo duong, vay A - Xid xic dinh duong vdi X < a (xem vi du 3.7). N6u b <10
thi b - < 0, nhu vay moi nghiem (lac thing cna A - 7.01 dgu am. Do vay A - XeI xac dinh am. Phan ngvdc Iai chting minh Wong tv.
3.12. a) Vol B, = A -1 11dUI, La c6 B I U, = AU, -1,15 1 .U1 .U1 = X 1U, - X,U 1 • IlUd1 2 = X(1 -111-11 111 2)Up Do do, n6u 7L1 = 0 thi
= 0 vdi mm vec to rieng U 1 ; ngu X1 # 0 thi B I U, = 0 khi va chi khi U, la vec to rieng co chua'n dclit bang 1.
b) Gi sii X la vec to true giao vdi U, trong (vdi tich ye
Inking chinh tic). Khi do .X = 0 va B,X = AX - X1U 1 .1.1 X = AX. Nhu vay, n6u U2 la tree to Hong don vi Gila A trip giao vdi U 1 , thi U2 cling la vac to rieng cim B 1 . Xet clang eau 11 2 = A - a,U 1 . Ui -
12U2 .M2 , ta co KerB2 chtla khong gian vec to sinh bai U2}; va thu hep cita B2 tren khong gian con true giao vdi Vect(IJ I , U2)
trong vdi thu hep cila A tren do (coi ma trait A la ma trail cua
to clang eau trong khong gian vec to dclit R5. Sau n bade ta nhan
dude B,1=A- EX .111 , c6 tinh chat Ker(Ba) Vect(IJ I , i=1
Nhu vay B„ = 0 hay A = E)<lU i .U1 . i=1
U0)= an.
184
3.13. Gia sit co xx 0, yx 0 ma f(x, y) = 0 va f(y, y) >0 nhung (x, x) 0. Ta thky x, y dec lap tuy6n tinh va vat z thuec khong
;tan vec to sinh bat x, y thi fez, z) = f(ax I by, ax + by) = a2 f(x, x) + f(y, y) + 2ab f(x,y) ?0, (").
Theo gia thiet n6u f(x, a) =0 thi x cling phitong n6u f(y,a) = 0 hi y cane phtiong voi a. Nhung x, y khong tang phuong, nen (x, a) va f(y, khong deng that bang khong (**). Do do co hai
6 thvc k, I cK sao cho k' +1' > 0 va k f(x, + I f(y,a) = 0.
TU do f(kx + ly, = 0. Theo gia thiet kx + ly ding phucing 'di a, trudng hop f(kx + ly, kx + ly) < 0 Coat do nhan 'cot (*). Do to a= 'K ix + l tyx 0. Theo gia thiet f(a, a) = 0 k, 2f(x, x)-11, 1 f(y, y) = O. )o f(y, y) > 0, f(x, x) '2. 0 nen to co 1, = 0 Ira f(x, x) = 0. Nhtt t = k, x va ter do [(a, y) = 0, f(a, x) = O. Mau thuan 'got (*").
3.14. Xet A e M:11. (n, c Mat (n, C). Vi A phan di51. ximg .
len ma Iran iA Hemnit - Min vt ao). Ta co det (A - kin) = 0 <=> let (iA - ixIn) = 0. Nhung mot nghiem da' c trung caa ma tran iecmit den thuc. Tif do suy ra mkti agh*n dac trung cila ma trail A a thuAn ao hac bang khong. Gia this cat nghiem khac khong
as da thud dee trung PA la: ja i , ..., tat , -iak , (cti E K, ai x 0).
Chi do PA N=
PA W ) + a?)
{0 vain >2k
Do do det A = PA(0) = k 2 II ot i nefun=2k
t jo '
185
Do do det A a 0. TU day, de thAy detA = 0 ngu n le. Di& nay cling có the suy ra ngay bang cluing mirth trkic tigp.
3.15. Bo dg: Cho V la khong gian vec to tren truang C, U li
anh xa nem tuygn tinh: V —> V, nghia la u (ax + py) = u (x)+ 5 u (y) vol mgi x, y e V, a, p E C. Khi do u2 IA huh 3u.
tuygn tinh. Gia sit X la mgt gia tri rieng thkic, am cua u 2 , khi do
X la nghiem bOi than cim da thfic dac trung Put .
Chung mink be, dg: De thy u 2 e End (V). Gin). six 11 mg gia tri rieng time < 0 cim u 2 va a x 0 la vec td rieng cua u2 vdi u2 (a) = Aa. Khi do u(a) va a la dec lap tuygn tinh trong V
That vay n'elk u(a) = -4 a, 4 e C thi u 2(a) = u(4a) = 4 u(a) = 141 2
a=Xa.-
Do do X = 141 2 a. 0, trai vdi X< O.
G9i W la khong gian vec td con hai chigu cna V, sinh bai a u (a). De thAy moi vec td cim W den la vec to rieng vol gia tri rieng X va u (W) c W. Dal V 1 = V/W, xet anh xa can sinh.
u:Vc —>
[x] —> u,[x] = [u(x)]
Ta c6 u, IA anh xa nUa tuygn tinh, tit do u 1 2 la anh xa tuygr tinh va u1. 2[x] = [u 2(x)]. VI vay, ki higu PIO va Pu, 2 la cac dz thtc dac trung cliatt 2 c End (V) va u1 2 c End (V1) ttiOng ling thi Pu2(t) = - A.) 2 Pu 1 2(t). Niu y lai IA nghigm cf.a da thug 4( thing Pil l ', lap lai qua trinh tren to có (t - )0 2 la ink Gila Pu l a
Qua trinh nay huu han vi dim V = n huu hart. Nhti vay s6 mu
186
na. (t - A) trong phan b.& Put la s6 than. B6 de' dude cluing
ainh.
Chung mink bai Man:
Xet u: C° —> C11 , u (x) = Al; u la Anh xa n&a tuyen tinh;
t 2(x) = u(u00) = u (Al) = K Ax voi moi x e Ta co Pu2 (t) =
let (A. A — tIn). Viii moi t e R, ta en det (A.A -t In) =
let (A A - t In) = det k -t In) (xem bat 2.51). Nhu \Tay Pu 2 (t)
a da dine vol he so' thve.
Gin) Pu2(t) =(%14) a ' • 0.2 - 0'2 - oak x dO sn ,sk
muyen during, ..., Irk e R;
Q e R Q khong co nghiem thne
Vi Q khong có nghiem thile nen deg Q = n -(s, + s 2 + ...+ sk)
Alan. He se cao nhat cua Q(t) la (-1)"" -- sk) , nghia la bang 1.
Do do Q(t) > 0 vdi moi t e R, to do Q(-1) > 0.
Bay gid ta xet the nghiem (i = 1, 2, ..., k). N6u CO < 0,
thl boi s, ena nghiem ?9 chart, khi do (Xi + nsi z 0. N6u 7u, z 0, thi
re rang (X,+1) 31 >0.Nhv vay Pu2(-1) a 0, nghia la det (A A+ In) a a
Chu $: Deu bang co th6 x637 ra, chamg han xet A =
3.17. Vol f e End (U). Neu f la t0 Tang caM tor lien h0p thi
vdi moi x e U; ta co:
< f(x), x >=<x,f(x)>=<x,f*(x)>=<x,f(x)>
187
Nhu vay <x, f(x)> thole hay <f(x), x> thitc \TM moi x e U.
Node lai n6u moi x e U, to có <f(x), x> thoc, to cheini minh f to lien hOp. Than Lich f thanh tting can hai to deing cM to lien hop:
f = ft + i . ft,, khi do
<f(x), x> = <f,(x), x> + i <ft,(x), x>.
Nhung ft , 1, la nhang to thing chin to lien hop nen <Ii(x), xi va <f,(x), x> thoc. Vi vay <fa (x), x> = Im <f(x), x>. N'eal <1(x), xi thitc thi <fa(x), x> = 0 yen moi x, ta do If„ = 0 va f, = 0. Do vay
= f1 nghia la f la to lien hqp.
3.18. Gia sa y la gia tri rieng c>ia f e End (U), ado f la tit long din to lien hop caa khong gian vec in Unita U. Khi do co e U, z # 0 d' f(z) = Az.
D4t x z
khi do f(x) = ax, x = 1 va <f(x), x> = <Xx,
=GI4=X.
3.19. a) va b) hir3n nhi'en
c) 1(x) = ek > e k = E< x, f(e k )>e k k=1 k=1
n
= Ex], < ek > ek = Xk Pk (X) k=1 k=1
\ray f = Ek kPk . k=1
188
d) Theo con trot) to co f = Xi, Pk, do vgy f 2 = 14 pk va k=1 t=1
Is = En Ask Pk k=1
Nhu vgy Q(f)= iOi k )Pk k=1
3.20. Gia sit f e End (C") la titan tit tit lien hdp trong kh8ng
;Mit vec to Unita C" vdi tfch vo hthing Hecmit tti nhien, f c6 ma
rdn A trong c6 so chinh tag cria C. Gia sit X„ la cac gid
ri Hong (thtic) cua f, tfic 2., la nghiem mad da thitc dgc trung det A - = P(X). Goi (e t , co so trite chugn gam nhiing vec 0 Hong ring voi cac gid tri rieng f(g) = ).1 e 1 (i = 1, n).
Hat Pk: C" C", Pk(x) = <x, ek> e k . Theo bat 3.19. Ta co
P(f ) = I P (21 k)• Pk • Nhung P(1.k) =0 171c nen P(f)= 0, to do P(A) = 0. k=1
3.22. Platong trinh flag trung ciia ma trdn c1 c6 dang:
1 x
2 2 .F3 1
2 2
Cac toa do cud vec to ring tim dupe tit he:
= 0 c5i — 1 = 0 <=> = -±1
189
o -1
(2 ' - X)X 2x2 =0
1 2 1 2
1761 k = 1 to co x1 = f x2 , churl vec to rieng
2
1
.2
e E2 =
\TM X = -1 to co x 2 =-Z xl , chon vec to rieng
(
X2= 2 e 2 1/3
2
N6u lKy cd sd true chan
,,75 ix, = 2 2 2 2
—e 1 + -1 e ; x 2 = le l -‘1e 2 thi trong cd so do, r
tran cim f co dqng eh& JJJJ
3.23. HD. Nghi'em dac trong cim ma tran B la: X = -1, =
(INN 2)
1
V2 V61 X = -1, có vac td rieng X 1 = 0
1
190
Vdi 7r = 1, cac toa do &la \roc td rieng thaa man: x, = x3 . Ta
6 hai vec td rieng Dap
1
giao
V2
IA: X2 =
1 .`
va X3 =
'0 '
1
o 0
42 Pat C = 0 0 1 thi C la ma tram tnic giao vi
1 1 0 .%&
-1 0`
B C = 1
0 1
3.25. c) HD. Xet F = Imf c E, F1 ® F = E khi do Kerf =
Ta c6 f/Imf: F F lA clang au, c6 hang mtc dai, f - phan del
rang nen theo b) dim Imf la s6 chan.
Chu 31: Lai giai cac bai tap 1.25; 1.26; 1.27; 1.28 clatong 1,
?,.53 chttong II, 3.15 clumng III do Le Thai Hang, sinh vien :K49, lop chgt ltiong cao, khoa Toln DEISP HA NOi de nghi.
191
TAI Lieu THAM KHAO
1. Nge TInie Lanh. Dai s6 tuy'en tinh. NXB Dai hoc va Trur hoc chuyen nghiep. Ha net 1970.
2. Doan Quynh Bien). Giao trinh Dal s6 tuyeln tinh hinh hoc giai Bch. NXB Dai hoc Qu6c gia Ha NOi, 1998.
3. Khu Que.c Anh, Nguyen Anh Kiet, Ta Man, Nguyen Doa Turin. BM tap Dai s6 tuy6n tinh va hinh hoc giii tic' NXB DHQG Ha mai, 1999.
4. R. Millman. Introduction to matrix analysis. MCGRAl book company, inc. New-youk - Toronto - London, 1960
5. J. Rivaud. Exercices d'algebre lineaire. Librairie Vuibei 1978.
6. Proxcuriacop I. V. Tuyk tap the bli tap Dai s6 tuye tinh. NXB "Khoo hoe", 1978 (Tielng Nga).
7. Cadovnhitri V.A. Cac bai town thi Olympic sinh vien. NX Dai hoc tong hOp Mockba 1987 (Tieng Nga).
192
