Bài tập đại số tuyến tính

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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

doe:

Gicim

NGUYEN VAN THOA

Tong bien Op: Bien tap:



NGUYEN THIEF N GIAP

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 \

G

I

a2

G

n} len chinh no duet goi la

3

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. Ki hieu sgna =

1 neM s la phep the chan

-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 = 12 , 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 ••• , ik) phan tfch chive thanh tfch 0 1 ,

xfch

§ 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)a mo) 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

• phan tfch thanh tong, thi c) Ngu mot Bong (hay mot dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6: f

an

de

= det

al;

a 21

21

a2„

a,,, + ani

‘ a n„

a ll

an,

ail a21 +alci

...a1,„ + de t

all a21

—a 1111/

d) Cho A = (Ito)

E

...a2 n

" S ' Ill " S IM /

Mat(n, K), thi

= b)

a do

= aij &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 aii 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),

va Oh

ik),

1 s i, a b va a + (n-Gb O. Goi B = (bii ) la ma tran nghich (lac) cila A = (a„). 1 Ta Wet rang b i d = detA . A„ etia phan

a do A.; la phAn Phu dai so

trong ma tran A. a b

Vol mdi i thi

=

b

a b b

la nh thiic cap (n-1). a

Theo phan a) thi A i = (a-b)" -2 . (a + (n-2)b). 34

Vay b„ -

A a+(n-2)1( det A (a 1- (n -1)1).(a - b)

= (-1)H , d do kl„ la chub thue e6p n-1, Vol i # j co dupe ba' lig each x6a &Ong thu i va cat tha j eila ma tran A. Do = Gia sU rang i < j, khi do cot thfi i va A dovi ming nen 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=

Nha v(6y b

= (-1)'34 b.(a-b)" -2 .

-b dot A (a +(n -1)b)(a - b)

Do do a +(n -2)b

-b

-11

a +(n -2)b

1 (a +(n -1)b)(a -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

35

1 a)

b)

2 3 4 5 6 7 8

'

8 1 3 6 7 5 4 2 ) (1 2 3 4 5 6 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 nghich the. . Chiing minh cong thile truy 146i sau:

n ce dung k

(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 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, 36

k.

1.8. Xac dinh davu cua cac phop the' sau: 2

n

n +1

a) I 1 43 6 ... 3n b)

2

n + 2 ...

3n

1

3n-2

2n 2n-1,

1

2 1 ... 2n

2n +1

5

n+1 n+2

1 2

2n

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 di; = b

dO

0

vai i= j yen i + j= 2n +1 i=j vet i+j 42n+1 1j

- . Chung minh rang A2 =1,1 . kHrn-k)!

(9, la ma Iran chin vi). 91

1.26. Gth sa X = ())) e Mat(n, R),

a do x ii 4- (-1)")!

1-1 01- 4,

Chitng minh X' = I.

Chit Se voi aeR k e N, ki hiqu )a, 1.27.

Gia

a(a - 1)... (a - k +1) ())) ) 0 ; - )- 1 k! a 0 = yXx„ x.„„

x„) vdi (i = 1, 2, ..., n) la ham

cua cac bi8n doe lap x 1 , x2 , .. x„. Ma tran J = J(Y, X) = [ a Yi 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= (yi ) , 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 r u de 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. 0 1.31. Tim tat ca cac ma tran A e Mat(n, K), A= (d j), 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 = j, T (j) = Xet T e S„ (n 2 2) gie sii i < j va T 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 f

i =1,n la n!

a(i) # i vai moi

n

E (- 0 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 n +1 . NMI yky A, (i =1, 2,..., n-F1) a n , a n+1 ) ma ai = 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 n ghich 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 d k phan tich dude thanh dk -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

a) DS:

vi du 1.1

- 1-1(n+D (-1) 2 . r(1121)

h) DS: (-1)

2

1.10. Iasi trien cot dau, to ca. D ykk =(a Do

D2 =

1.11.

-13.2 nen

D 2n =

A = (ai d , detA =

2_ "

" --1

2n-2 •

(a 2

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) nen i + j 1e, thi cfic tich ha; le. VI Vey khi thay 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.y

{x = 20

= 200

{x =10

hoar

{y=10

{y=20 20 14 \

a) x = 20 va y = 10 o BA=

14 10 Ta co ABA = A

20 14'

5

11

14 10 ,

11

25 2

Tit do ta tinh dude A = va B = A -

5

5k

-14X

-11k

0

A.

,

AE

R

11

11 25, b) x = 10 va y = 20

10 14

BA=

:14 20, ABA = A

10 14 14 20

5 4

11

11 25,

X

X

1.15. a) D i, = 1.

b) A n = -A„. 1 + 4,1 = Tit do suy ra A 2p = 1, A 2p+, = 0 46

-3 5 l

A= P.

1.

55 32

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, D2 = (1 + x') 2 - x' = 1 + x 1 + x4 . Taco: Da -D„-1= x 2 (D„_i -Dn _2 )= x 1 (13 1-2 - D n _3 ) x 201-2) (D9

Tct

_ D1). x 21

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 o. n A(x) then (long tha nhat, to co: A(x) =

P(x) D(a 9 ,... a n ) x - a l -

+ ( 1)"7

P(x)

I3(x) x -a n

;

a 46 D(a 2 ,

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 x (a 1 -a.).

ISIS“

47



1.18. J In ma tran Vandormonde vat (Me s6 biet nen dot J * 0. Ta a+b+c a+ bj+ cj 2 a + bj 2 + ej A . J = a+b+c c+aj+bj 2 c+aj 2 +bj • a+b+c b+cj+aj 2 b+cj 2 +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 + Ta co: D, = a +

a0D«-s.

13, 1/2 = az + an + 13 2,

k

Gia sit D k = Ea i f3 k-i (k = 1, 2, =o n-1

Ta co

= (a +p)

a ir

-1-i

- an Ia'0 2-2-I i=o

i=o

=

n-1

a i+lpn-i-

1.0

=

n-1

or

k ira .2

i=0

r3 n i=1

=E

=o

Mtn tray D„ = Ear . i=o

48

n-1).

n-2

a al n n-i-1

i=0

n-1

n-1

]=1

ct

i v _i



b) Cheng minh qui nap thee n. 1

n = 2 o D2 =

I, = x2 - x, kheng phi thuec y,.

+ y i x, +y i

D„= (x, - x,)(x,, - xi)( 119 - x1)-

n = 3 Gia stir

1

Dk

=

11(X j -x i ) vdi moi k = 2,

n-1.

i Si q sao cho the ph&n tii taring ung cad AP va A P bang nhau thee modk. +(detS).0 , C e Mat(n, Z).

Do do AP = Tit do:

= +(detS).C.(Aql . that m = p - q. 1 A"'S=1„ +detS.S -1 .C.A -4 S;

Rhi do: I -

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, n=2, A=

a ll a l p a21

=a n a 22 -a 21 a t2 .

22

Neh cho track a 2 , . a 1 0 t 0, to chon a, 1 = a 22 = 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 d. n tit an . Theo gia thi6t quy nap, to da Hein dude a22, A, 1 0.

a„„ de?

55

DAt a ll = x, khai trien theo clang thu nhAt ta co: detA = x. A, 1 + Za li 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 (ali > 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, ding not p bieu thi tuy6n tinh qua cac vac to a t , ..., an„

Ta

MOt he vac to la,, ..•, tuyeIn tinh nen c6 cac s6

caa V dude goi la he phu thmic = 1, m) kh8ng deng thoi bang

kheng cna ]K sad cho

= 0 . Met Oat hkeu along &king:

1=1 ka he pha thuec tuy6n tinh nen c6 met vec Lo nac am) he (st„ 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 am} la he dOc lar he doe lap tuyen tinh. Nhu \ray, he {a„ tuyen tinh ned.1 moi t6 hop tuy6n tinh

k i a i =0 to suy ra

k, = = km = 0.

Cho he 14„ aid cac vec to doe lap tuygn tinh caa kh8ng gian vec to V ma m6i vec to caa no la t6 hop tuyeIn tinh caa ca( thi k < 1. vec td cim he ([3„ . 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 ak 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

, en} the vec td doe lap tuyen tinh am killing Met he {e„ 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 , a

,

, ej, yen vec tO bet kS7 a e V, ta co

, x„) dupe goi la cac toa doo cim vec td a del vOi

ed sa {e„ , e n}, ; la toa de thu i cim a doa vdi cd 56 do. Gia sit co ed sa khac c = {c,,

, E„} ciaa V, ma si = ]=1

n). Nen vec td a có toa di) (x i) trong cd s6 = 1, thi ta c6: de' (x1 1) trong ed sa = Ec oxl i , Neu ta ki hieu ma tran C = la cac ma tran cot, thi X = C X'.

va c6 toe

, n. va ma trail X = (x), X' = (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 hay Vect(X). Neu X = , bao tuygn tinh ciaa X thick ki hieu la . 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, Ta thay rang a e

,W„, ki hien Ia W, + + W„ hay

ZW; .

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

true tigp cua n khong gian vec hieu la W, e 60

W2

thi t8ng td

con

ED ... ®W„ hay S W,.

=1

W, ch.toe goi la tang I=t (i = 1, 2, ..., n) va ki

Gia sit V la klafing gian hitu han chikt,W, va khfing gian vec to con dm V, khi do:

W2

la hai

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 1119i k e IK, to ce:

e V.

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. a„ NMI le„ , e„) la co se, cOta khong gian vac to V va 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 f la don cdu khi va chi flomK(V, W) de f(e i ) = a, yea j = I, ... , a„} doe lap tuyetn tinh, f lA clang cdu khi va chi khi he EJ la cd = khi he M I , , a„) la co sa oda. W. Gia in

ciaa W, thi f(e,) = 62

E, =

vaaha tran A = (ad goi la ma

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 —> Km trong thc od sei chinh tdc. Khi do 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. tang

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 , t ron€

i=1

cd s6 6, f co ma tren B = (b ii) ughia

= Eb ii i=1 C = (co) chide goi la ma tren chuyen co sa. Ta ca:

Ma trar

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 dO aki , b k E

Ea ki xi = b k (k = 1: 2, ... , i=1 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



Ma trail Abs =

a ln a 2n

a 12

a ll

a 21

a 22

b

b2

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

la ma tran cot,

va B =

Neu ki hieu X =

bini

xni

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

va b(b„ clang

an}, ma a i = (aii ,a si ,...,a mj )

, b 0 ,) la vac to cat td do, the thi he viat dude dU6i =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 ado D la Binh thiic cim ma tran A. do, thi x = D I 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) i=1 ce nghiem khi va chi khi rangA = rangA hs.

(1)

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

ling during cl6i vOi he phuong trinh Cap Oen bi6n 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:

I

P

h ip

1

0

131

p+i m-p

0

p

phan t>i

b'm

n-13

a phan goch (*) co the khac 0.

Khi rid n6u 13,;+1 + + n6u

0

> 0 thi he ye nghiem,

= 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 f e End K(V), W la hieu la Hom K(V, V) hay End K(V). Gia 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 ling vat gia tri rieng X ngu f(a) = Ira, A

e

e

EndK(V)

K. Khi do khong gian

met chigu sinh bai vec to a Et met kh8ng gian vec con bgt bign cim f. Val A

E

1K, tap ker(f - AId) khi no khac {(5} la khong gian

con cim V, gam vec to khong va tat ca cac vec to rieng caa f ling vdi gia tri rieng X. Kitting gian nay goi la khong gian rieng cim f v6i gia tri rieng A, ki hieu Gia sit ta cl6ng eau f e End(V) trong met co sa Mao do cim V co ma tran A, thi det(A - XI n) IA met da fink bac n dovi v6i bign X, khong phu thuec vao vice, e chon co sa, va &lac goi la da thitc (lac trang caa ta (tang eau f (ta cling nOi do IA da that da, e trang caa ma tran A), ki hieu M.(x) = det I A - X I n I . Nha vay A la met gia tri rieng cim f khi va chi khi X11 nghiem &la da that dac trang eim f. GM sit

, Ak la cac gia tri rieng cigi met phan biet caa I

Px

Pik la cac khong gian rieng Wang ling via ale gia tri rieng do, thi t6ng Pr + Px 2 Pxk la tong true tigp. 2 -Ong gian rieng suy Ong GM sit V la khong gian vec to tren (Wong K, f e End K(V). Vdi mei A e K, xot tap {a e VI co s6 nguyen m > 0 de1 (f-XIcl,)m(a) = 6). DO la met khong gian con caa V, khi khac {0} no &arc goi IA khong gian Hong suy rang cua f ling vdi A va ki hieu IA Ta thgy rang: 68

+ Vdi moi khong gian rieng suy rang

3 k

7r

IA met gia tri

rieng cim f va Y, c R s . + 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 \ {0} va u, e

IA cac gia tri rieng phan biet tiing cap u k} doc = 1, 2, ..., k) thi he cac vec td {u„ ,

"f.k

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

(i = 1, , n-1) va Re p) = 0, thi hay linh bOg n va ed sa {ei, ..., en dude goi 11 cd so xyclic d6i v6i f. Trong cd SO xyclic ma trail cua f co clang: Re) =

}

0 0 1 0 0 1

0'

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

(f -

d) N'Eu 9 ) IA khong gian rung suy rang cua f dng vdi A, thi 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)sk

(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.? X2 e

Xk

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: - 2rang(f - kildv)s + rang(f -

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„, la m vec td thuee R". ]E la met khong gian vac td x„ x2 , ChUng minh rang tap IF the vec td ena K" dang con eila Zt i x i (t 1 , ,

e E Fa khong gian vec to con ena K', dimF =

dimE - dim(E phtlong trinh:

nN), trong de N la khbng gian cac nghiem tha

E ti ., =0

(a do t i ,

tn, la cac An).

i=1 Lai gidi

:

Xet anh xa tuyan tinh u c Hom(R m, 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(x1) = x1 , 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. sit menh de dung vdi moi he fx„ chimg minh menh d6 dung vdi k = n. Xet t6 hqp tuyen tinh Gia

xki; k < n - 1. Ta

ECiX i = 0 .

(1)

i=1

Khi do: ) = CIX ] i=1

Zcif(xi) = c,x, + Eci (x i +xi_1 ) = i=2

1=2

.

= ZeiXi i=1

Tit (1) va (2) to suy ra:

Ec ix,„ = i=2

=e2 =

n-1

= 0. i=1

Do gin thik quy nap hee, fx,, , C2

(2)

i=2

doe lap tuyen tinh nen

= cn = 0.

Tit do suy ra c,x, = 0 c i = = = c„ = 0 Nhu vSy he {x1 , , xj doe lap tuy6n tinh.

72



ak} cac vec to doe lap tuygn tinh ma m8i vec to dm no la t6 hop tuyen Hay ehung minh k < 1. tinh cua cac vec to cim he 113„ , Vi du 2.3. Trong khong gian vec to V cho he

la doe lap tuy6n tinh, neu Ta eó the' gia thigt ••• • 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 , ten tai cac vS hudng a.„ e K sao cho:

(1)

k)

; 0 =1

a ;

p, nen

ak phty thuec

Ta gia sit k > 1. Ta se cluing minh tuygn tinh. Xet t6 hop euyen tinh k

1

i=1

j=1

/Xiai

1=1 I

(k

=0

pi =o

j=1 .i=1

EauXi

=0

(2)

j=1 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/E2 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, + Xet

E2;

F F/E2 va k = 1-1/E, •

Do phAn 1) k la toan anh va kerk = E l C Kern = E l 11 Do do to co E l /E l fl 74

E2

ding can not E 1 2>E2/E2 .

E2.

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

q (pq — pq) (Imq) = 0

Er>p(q—).

.(=> (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 Vi du

as.

E

Kerp n Kerq.

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 tip i=1

ciaa khong gian F. Vbi mdi F, xet 2, = if Chung to rang Horn (E, F) = WL

E

Horn (E, F) I Imf c Fd.

2.

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

76

E -, K,

e

Horn (E, Fi) = 2 hi (x) = yi

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) = 1=1

h i (x) = 0, h,(x) e F, 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 k(x,) =

=

ingui=j

..— 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

EXi f,

= 0,

c Q.

1-1 Khi do vdi moi x E cam

Chon x= x„ thi

,

LW; (x) = 0; (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 D o 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)-112(xj) = 0 vdi moi j = 1 78

n.

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„ gn}, 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). -

h2 )t

hi

n+1. Bay gia

= g -

thi vdi j = 1,

n to co gi (xj )=

bin+, vdi MOt i = 1,2, ..., = 1, ..., n),

(xJ )= si; va gi (xn+1

)

=g. (x„. 1 ) - x; = 0 n6u 2, 1 = gi (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

sao cho g(Ci;( ) = f(

E

Ta phai chiing minh g c 93, nghia la c6 f g@t„i ) tren bb

).

d f

e

la co so cua R".

Gia sii node Lai, vdi m6i f

E

c6 khong qui (n - 1) sd the

phan biet xi cl6 g(d xi ) = * xi ).

Do hop &dm Mtge eac tap bop hfiu han phan to 1a mat to ma g(d x hap kh8ng qua deM dude, nen sod cle vee to d x e 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 = f(di xi ) vOi moi i= 1,n; nhu vay g=f

80

e

gi d

E t

der hied mot sa sri kien cilia khong

(rich 2. (Mtn}, cho doe

gnin metric). de" f(a) = g(a). nhu

der f e

Theo gia thiet vdi moi a e vay — g) rad= 0 => a e Ker (rig).

NInt vay U (Ker (f — g) f E gq} = R. Hia sir g

thi f — g x O. do vay

vSy vdi mai f E

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 e = le,,

trong cd ed tly &nen

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 81

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 Jrco ma Ran oat = (ad trong cd sa e = (e,,

cie'n

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

twang K, f

c

°W la hai khbng gian vac td treat

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

gthi: 1 2 3 4 -1 2 - 3 4 D=

= -4 0 1 -1 1 1

1

1

1

Day 1a he phudng Huh Cramer.

80 2 3 4

3 4 3 1 -1 1 10 1 1 I 1 2 30 4 - -1 2 10 4 Da = 0 1 3 1 1 1 10 1 1), =



10

2

-

1 30 3 4

D,=

-1 10 -3 4 0

3 -1 1

= -8

1 10 1 1 1 2 3 30

-12 D'=

-1 2 -3 10 = -16 0

1

1

1

1

3

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. va ta ce,

-

it +1 X +2

1 , x 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. x i x i

=1

2x, + x 3 g - 2 + 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:

)(5

xt + 3x, + x 3 + 2x 1

=2

3x 1 + 10x 9 + 5x 3 + 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 + 12x 5 = 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 X9 + 2X3

x4

+

X5 -

2

+ 2x5 =

0

2x, + 4x 3 + 4x 4 + 8x 5 = -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 dyeingx 1 + 3x 2 + x 3 + 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 + 6x 4

-3 (5)

85

Car phtfting trinh thit (3), (4). (5) trong he (III) la tudng during. Vi 04 he (III) tudng during vol. he x i

+ 3x 2 y 2

+ 2x 4 + 2x 3

+

2

x- I

+ 2x 5

=

x4

+ 2x 3

= -1

0

(IV)

Giai he (IV), to (little xn

= —1 — 2x 5

X2

=

1

— 2x.

x i

=

1

+ 5x i

a do x4. x

Inv 31

+ 3x 5

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) 86

dimlmg.

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: rangB < rang (A +

ningA -

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

a do tI/4xl = g(2), yin

XE

Kor(f g)/Kerg Kerf (1 Img 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 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(ei) = ° ; 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(ei) = [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 u3 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: X = Q .

'Ir 0 0 0

.P

a do

I, 0 00

la ma tran vuong cap n ,

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) uv 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. Min ntm, neAl X # 1 thi Id

E - u la

e

1K

a u2 =

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 n. Chimg minh rang t6n tai ma trail B, C e Mat moi i = 1, 2, (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 A 2 deing clang vet B2, 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 thCla man didu kien

e

End(V)

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„f n. 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 j Qej) = Sij =

1i=j

Chung minh rang (£ 1 , ...,1„1 la cd sa cua V* va do do.dim V* f„} ciudc goi la cd sa lien hop hay ed sa = dim V = n. Cd sa fb, d6i ngau N:i ed sa ••.,

fil, la nhfing dang tuy6n 2.23. Chang minh rang nen 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. 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* b)

trong cd sa (2) la ma trait chuye'n vi cua (ct ii)„ cua u trong Cd (1), nghia la Po = a3; vdi moi 1 5 i, j

sa

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 (1 1°)° # 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, ( fl y; IET

= I Vi° /

neat V Mitt han chi6u.

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 iE=.1

\ iel

b)

n

=

n6u V heti hen chiau.

ieL

c) NC, u can T,

N.2

va han, thi trong V* cia hai khong gian

(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).

103

a) Cluing minh rang co hai co sa {e„ e„) va {E D ..., en} 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; ail = 1 neu 1 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 +2x 4 + X 5 =0 5x 1 -8x. 2 + 5x 3 + 4x 4 +3x5 = 0 • 4x 1 -x 2 + X 3 + 2x 4 + 3X 5 = 0.

▪

-

2.35. Cho hee, phudng trinh thuan nhat: 5x, + x 3 + 2x4 = 0

2x 1

5x 1 - 9x 2 7X9

-3x1

2x3

7x4 = 0

X3 + 4x 4 = 0

6X2 + X3 - kX4 = 0

4x1

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)z2 = 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: -X1 1 +

xl +

X 2 4-

X3

x2

X3 + X

x 4

a

X9 + X3 + x4

X1 + X2 + X3 + x4

2.38. Giai va bien Man he phudng trinh tuyen tinh sau: 2x + y X + my

3x + y

z = a z mz = c

105

a

2.39. Tim digu kien din va du 4 dim A,(x l , VI), Ai(xi, 312), 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 i g e Mat (n. K) sao cho dot A = 0. Goi phan phu dal ce cUa phgn to a,,, gia thigt A 11 # 0.

la

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

+ 'Prong do ck PTO,) = (-2,)" + c,(-2,)"1 + + 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 e. m 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

0 -1 1 0 -1 1

1 2.47. Tim da the dac bung cua ma tran vuong cap n. 0

1 0

A=

1

0

1

a va a

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.

107

2.48. GM

V lk khong gian vec W n chieu troll trUang K,

End (V), X° la mOt gia tri rieng ena gyp,

cp E

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 ma tran A 108

e

e

Mat (n, K) co anh chgt: Kai moi

Mat (n, 1K), to Mon viet duye mot each duy nhfit

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, V„, = v„,+, va khi do V. = V,„„ vdi moi j x 1.

va t6n tai s6 m

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 va kha nghich tren W„,. Bien doi vdi f. Hon nua, f Iny linh tren 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: 'a 0 .

0

'

0 0 . 0 0

hoac 0

to p

0

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: 110

t„). (X . Xt) = (0, 0, .., 0). Neat Y = (t,, X.)4' .

t)

E

H Y. XX' = 0

= 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„ to cot la A l , A2, A„ thi

A2, ...,

A ,) (16 chi ma tran co cac vec 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

auo =

i

(j)

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 n a . Khi do AB = BA a uV = vu < uv(ej ) = vu(a) j = 1, 2, n

n

E bijecro) = Ebi,wei =

in

n Eb u(a)= v(e a0)), j = 1, n
dim P kn = n-r. Goi R 1 0la kitting git 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

= I A - (p+X, D)I I = Cdch 2. Dat B = A - kJ. Khi d6 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 + + C k (-11) "a C, a do Ck 11 tong cga cac Binh tilde con chinh cap k dia. B. Do rang Ck B = r nen cac dinh there con chinh cap k > r dgu triet lieu = 0 voi moi k> r. Vey I B - I = (-P)"+

+

C. (-1,)" 5, s < r.

Doclop=n-s?.n-r. 2 i cos 2.46. DS. Ak = 21

k n +1

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. G ' -A 1 1 -A Khai trim theo P(x) = det an-1

an-2

thing coot to cg: (

1

o'

0'

+(-1)"*2 a,,2 det

P(A)=(-1)""a2,2det :v 1 1, f

- k

1)

0

(-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 T k), 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„ x 0 1 -X 0 1 -X

-

Dk

=

0

0

= E lrk ()Lin k

0

a do Dk la da thile b•c m k dang: 0 0

1 0 =(— OnI k+I-P ( Ark -

1-1.

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) b, co 1 vec to rieng: t i = t2 = ... = t„ = 1. VS'.y c6 the than Cd sox, = (1, -1, 0 ... 0), ..., x2. 1 (1, 0, 0,..., -1) va xn (1, 1, ..., 1) gem cac vec to rieng cita 1

2.51. HD. Xet dang aide ma tran: -

(XIn - AB

A

0

AIn i

'In 0'

[ In B 0

B . In,

In

2

(

)Lin

A

O

XIn - BA,

a) Ta có det (XIn - AB) . An = 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. 0

(2`k 1 Xk Ak = 1 '

, cap Gila Ak

0

bang dim/Kt)

1 X i
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 de tren

a MS = 0. Trude het to sit dung be

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

2S.A 1 = S.A + A.S {

2S.A2 =SA -A.S

A l =j-k+S -1 .AS) A2 =

2

(A -51 AS)

Vi A1 . S = S.A1 nen to he thiic tren to co AS + S'A. 52 = SA + AS to do AS 2 = S2A. 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 +

va p(s) = sk aisk- , =

ak _,S + ak . In= O. Voi S # O. Ta nhgn

+ + ak 2 x + a k , k21,

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' + DM M =

S k22 + ± at.; - 1 0= 0 + ad1 1-1 +

, I„ # 0, to co SM = MS = 0.

2.54. a) DO (ang b) Nhan they V. n Wm = {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. q la se nguyen diking be nhal d5 f' = 0. 2.57. Cach 1. Gia Vi f9-1 x 0 neu co vac to x e V (I5 f^ - '(x) m 0, r(x) = 0. Theo hal 0-1 (x)} dOc lap tuyen tinh. TpY do q 5 n. tap 2.9, ta ce {x, f(x), Vi = 0 = 0. Cach 2: Dat W' = Ink(?). Ta co W' = W 2 = ... to do có m n vi ne'u kheng ta co n = dim V > dim W' > dim W' d5 W" = = dim W"+' < 0, ye 1.3"7. Mut Sy W" = >...> dim W6-1- Wm+i suy ra W" = 0 hay P"= 0 f" = 0. 2.58. HD: rang f = I

dim (Kerf) = n-1.

e,J la co sa cua Kerf xet-f(e l ) = ale, Chen e, e Kerf va + a 2e2 + ... + 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 = gigk gickf.f = k.gk:

ta suy ra

g k+1 gk.Eg = k.gk va

—

Tit do f.gk± 1 ,7, g'+' f + g (fgk - l132

e-,

.

f) g = 2k.gk .

Do gia thiCi guy nap: g k+1 , fkti (k-i) g k = 2k.gk

gk-'

—

= (k-1)

g k•2 nen: fel

(k+1).gk; nghia la he thtic (2) thing vdi hay 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 t ai cac ty dling lieu thaa man he thin: (1). Thai vz;)), , )(et P(x) = xP + a p.,xP-1 + + ao la da thdc tot lieu cua g; nghia la da thtic kluic khbng, cd bac nh6 nhet sao cho P to nhan dude: P(g) = 0. Khi viol he thug (2) vdi k = 0,1, 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 x. y)

0 (x, y) + 0 (x. z) 0 (x, y)

0 ( x, y) = 0 (x. y) 0 (x, y)

Neu vdi moi x, y

e

= 0 (y, x)

V ta co 0(x, y) = 0(y, x) thi 0 (bloc gyi la

dth Ngu vdi moi x xung.

134

e

V ta 66 0(x, x) = 0 till 0 dmic gni la phOn cl6i

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

.A.Y, hay yiet dual (bong ma tran 0(x, y) = neat cac tya dO cua x, y trong co sa (e i). ma trran Gia s& e = (E,), j = 1, 2,

a

do X, Y la

the

n la mOt cd sa khac ciaa V ma

CiiCi . Dang song tuygn tinh 0 trong cd so e = (au có ma ,=1 . A . C, trim A, va trong cd sa s = (8) có ma Ran A', to co A' = 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„ y Wong

xx ) va (y1 , yd IA toa dO cam x va

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 = (c i) 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 = trong co stl do, H co dang chinh tac. (0

a

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 0 vdi rani k = 1, 2, n. H le dang toan phuong xac dinh am khi va chi khi det vdi k than va det Ak < 0 vdi k Le.

Ak %

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 = 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 x, > 138

la chudn elm x e 1E, kr hiku ricfi ,

s6 thcic khong

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 = 0. Khi do to co: )1 2 = D1M 2 M2 •

3. Cd sa trip chua'n trong khong gian vec to dclit hitu han chi6u Gia sit E IA khong gian vec to delft n chigu. it vec td e l , e2, ...,e„ dupe goi la co so true chuitn cUa lE neu =

1i=j

0i#j

'Prong bluing gian vec td Gclit n chieu holt kY. loon ton tat mgt co sa true chuan. Dinh 19:

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:

co =

co

ME2M e 3 =

a do = et, - e l -

e9> e2

Mg3 M

139

n-1

= ,d a do a n =a n -I< a n ,e k >e k . en ka k=1 Thutit Loan chi ra a day &talc goi la qua trinh Hate chuA' hoa Gram - Schmidt co so {a„ a„}. 1-56 thky khong gian sin bai {e l , , ek} trimg vat khong gian sinh bai vat (a l , ak} vi moi k = 1, ..., n.

Nigu {et,. ai=1,2 m) la co so true chuL x c V thi x vdi = (x,

N6u co y =E yi e i thi = Ex i yi . 1=1

Gia s& F la khfing gian vac to con eim khong gian yea t Oclit E. We to a E E goi lk true giao vdi F neru = 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 = . 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> = - 2t 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 = 7r < x,y > = -t 1 I ya bat dang thif (1) co clang: t2 + 2t Vx, y>I

0

(3)

vdi moi t> 0. Kat hop (2) va (3) to t2 + 2t 11 + 0 vdi moi t e I 12
0 nen I = X hay X thee. i) Dinh 0: Cae vec td rieng Ung vat ode gia tri rieng phan biet caa met to deng au tai lien hdp la true giao vdi nhau. °ding minh:

Gia sit f la met tp deng au W lien hdp caa 'cluing gian Unita U; x, y la hai vec to rieng Ung voi hai gia tri rieng phan biet X 1, X2 . Ta co f(x) =

f(y) = X2y = = X i = = = 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 1 s(x, y) = - (y(x, 2 a(x, = 2

y(y, x))

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) = n2 . Vi S (ttiong ung A) dAng caIu vdi khong gian cox ma tram doi xiing (phan xiing) cap n. Do do dim S -

n(n +1)

2

. n(n -1) , dim A = 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, yo} la met cd set cua F va dim E > n. Khi do trong E et) he doe lap tuy6n anh gem n + 1 vac to {x„ x o , xo, xozz}. Dat 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„

E ma f(xo,

C„+1 ).

Khi do x„ = Ec i x i la vac to khac kheng cua

= 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 x 0 e E [le g(xo, x0) # 0. Ta hay °hung minh g doi x3ng. Vin m6i x

E

E. do g (xo, xo) # 0

nen co a e Yb de g (x, xo ) = a. g (xo, xo) . Khi do g (x - a x o, 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, x o ) # 0 thi ce aeRa g(x, y) = a g (x, xo) hay g (x, y - a xo) = 0 = g (y - a xo , x). g(y, x) = a g(x o , 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 , xo) = g(x, Y xo) Suy ra g (x - a xo, y + xo ) = 0 g(y + xo , x - a xo) = 0 149

g (y + x0, x) = a g (y + xo, x0) = a g(x„, x0) = g(x,y). NMI va. 37, trong moi &Jiang hop to clgu co g(x, y) = g(y, x). nghia la g dOl xQng. Vi du 3.4. Cho E la khong gian vac td thuc n chieu VA co Fa

Bang song toyan tinh dal xung xac dinh throng teen E. Gra sit x,, x2, xk la nhung vec td mia 1E. Dal a id = xi), 1 j s k. Ta goi dinh Ulric det (a 1 ) la dinh thuc Cram (Ma cac vac td x,, xk va ki hiOu la Gr xk). Chang minh rang Gr (x i, xk) 0 va Gr (x 1 , . xk) = 0 khi va chi khi xl , xk phu tha0c tuyeln tinh. Lidi brick

Ta chgng to rang ngu x i , Gr(x l , xk) = 0, con ngu Gr(xl , xk) .> 0.

xk phu thuoc tuyan tinh thi , xk clOc lap tuygn tinh thi

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(xr, = a, a lj + a2 aji + + nghia la thing thir r am ma trap (ad k tau thi tuythn tinh qua r-1 dOng dau. Ta do suy ra Gr (x l , xk) = det (adk = 0. Gia sr) x„ xk d'Oc lap tuyen anh. The thi fx,, la cd sa cua khong gian con F via E sinh bai he {x 1 , xk} va ma trail (adk la ma trail clic( (p i = (p1 F doi vdi cd sä do. Vi

150

xac dinh

during nen theo Binh lY Sylvester, to co det (a, i)k > 0 nghia la > 0. Gr{x l , Vi du 3.5: Cho A la ma tran yang del xiing dip n tren R. ( xi , trong do x i e It; u Xet khong gian R" cac vac to cot X = n

la phep bign del tuygn t;nh trong nhien

co ma tran trong co sa' to

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 n. Ta co AX = XX, AY = HY. Tii de rieng X, n; 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 = (µXt . 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 2X,

A"- ')C

LaPi

AX,

A"X phu thutic tuydn tinh, nen co mot vec to bik thi tuyin tinh qua cac the to dung bathe no. Gia sit k ad 152

h6 nhat de AkX bleu thi tuygn tinh qua X, AX, 0 tan tai the s6 thuc b k., ..., bo dg

Ak 'X. Khi

AkX + b k, Akd + + b iAX +130 X = O. Gia sit (x - p 1 ) xk +

(x - i.tk) la &tan tich cim da tilde

+...+bo thanh the nhan tit tuy5n tinh, vdi g 1 , gk E C.

Khi 0 = (Ak + bk., Ak - ' + + bo I) X = (A - hay (A -

... (A - ukI) X

= 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 g o , ..., g k 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 igng cua u. HO nhtt thg vdi s = 1 lam tan tai theo vi du 3.5.

td

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 Xs,1 cern thueic khong gian con sinh bei X, AX, A"Nhung cac tree to AIX true giao \el cac X, (i = 1, s) vi tray 3 clang true giao yea Nhtt tray trong co X„ X„ trip g gem toan vec to tieing ciaa u. Dat

1

-

X• thi he Y1 ,

Ya

.X,

la he cac vec to rie

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 V, = X, va vie j thi Y,' A Y, = X; Yit Y; = O.

. A. Y;

Do do PAP=A trong do A la ma tren chdo yea cac Olen eh& a I A - XIn I =

..., X. Vi P = -

Pu do suy ra cUa A, Ire ca bOi. ,

nen A thing dang vdi A, suy

I = (X, - X) •-•

-

3.„ chinh la tat ca Cite nghiem (lac tn./

Cugi clang, do p AP =

A

suy ra A the Binh decing (am)

va chi khi A xac dinh duong (am) ngl âa. 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 Ct. A. C=A va C'. B. C= In, 6 de A la ma trait suy Bien C 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 AX = XX, tit de (AX)* = X*.A* = 7r X*. Nhan vg vat eet X # 0 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 1 det (- 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 +

ar z _

Ta say dung cd se {v,} (i = 1, Intang cim E nhu sau: v, = e, + er,„ v, = e - e , v,= 156

ctsr_fri _

_ coo_s.

n) g6m nhung vec td clan 1 0 va g > 0. Ta dua g v2 clang chua'n tAc

=

2

, yi = Eq 33 xi (1 = 1, 2, ..., n), det (n o) # 0. KM do i=1

E y3 i =1

g

)=E (f, i=1

Nhung

\ 2 = Ecniq ik x 3x k , nen (f,y3 )= Eaik kq u x i lkk xk ),

do f = Ia ik xix k . Do f xac dinh dtidng nen (f, n 2)

0. Neu X

j,k=1

qi; # 0 n8n ci n xy x 0, va 0, thi co; a x; # 0, tea do do có i o f xac dinh dudng nen (f, 37,2 ) > 0. Do \Tay (f, g) > 0. (xi)

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 to don vi. Ta {tat x= n 1+1 "In, . Hay tim goe giva vec to x - 1 1

va x -

U2 .

Ldi

Gth six {e1 } (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=11112 E1=1 va 14.u 2 = Ta b sung tr, nhu sau: u3 =

+ ).2u2 + X3e3, u3 u 1 = 113 . 1.12 = 2—1 , u32 = I.

X? 1 Xi Tit do co —21 = X, += 2 ' 2 2 + A2

Va vi Al + x.22 + 4 + Xi X2 = 1

1 A1 X2 - —3 .

X 3 = + IF —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; = — vdt j = 1, 2 tim duac = 1 2 = = An _ = —1 va I n 160

—

.;. n-1 va u,12 = 1, to n +1N hit vay to da 2n



u„} cac vec to don vi trong kitting gian xdy dung dude hee {m, 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). Bat x -n

1

+1

+ u2 +...+ ti n ), goi 01a gee gala x-u, va x-u 2 .

Ta ce 1 = (u 9 - 1102 = [(x-u 1) - (x-u 2)] 2 = (x-u 1 )2 + (x-u2)2 2) (x-u 2); -2(xu 1 tux + va (x-1.1 1 )2 = (x-u2)• = 2M +1) M +1r - Tit do cost/ = 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 del tuy6n tinh A cim V sao cho B(x, y) = yin phop moi x, ye V. gidi:

ta xet bit

Truisc

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

V, Ta co f(x)=

x= EXic ek e n

< Ix k e k , X=1

Exk k k=1

k =1

= xk hk = k=1

n

Eh k e k > = < x,h > . 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) = = vOi moi x c V. Tit d. = 0 vdi moi x e V. Dac biet, lay x = h, - h 2, 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) = . Xet anh xa A: V —> V y —> A(y) = h.

Ta chitng 6> A la anh xa tuyen tinh. Ta co B (x, y, + y 2) = B (x, y1) + B (x, y 2 ) = + = 0 vdi moi x A(Y)+372) = A(37 1) + A(Y2)-

c

V. Do v'OY

Tudng Lu B(x, 1y) = = y) = 7 = = 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) = = vdi moi x, y e V. Suy ra = 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, Binh during cap n.

xiing xac

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 = D2, 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 PD11 . 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 + Do D- ' B trot), to of):

.

B D-1 ).

del sung the dinh dming nen set dung k6t qua ,

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

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 a) Gial sit X =

al as j as •4

do {XI ,

= a, X, + a2 X2 + a3 X3 + ai X4.

X4} 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 0

A=

0

1

2 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. 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 A=

2

4 3 2 3-i -2 0 1 -2 - i 3 1 2 i 0 4

165

Zd gidi:

Ta co A = A, -

2 3 4 1 12 3 -2 0 a do Al = 3 -2 0 1 4 0 1 2

'

Ma trail AI 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 phttong Q tren

co sa cim R3 . Cho clang Loan

Q (x) = x 12 + x22 + x32 - x1 x2 - x2x3 ; x = (x 1, x2,

x2 ) la toa dO cua x trong co sa {e 1 , e2 , 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 '3 -1 1

sa

02, e3}:

'

A= 0 2 0 1 -1 3 Hay kie7 m 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, e3} co clang 166



Taco H, = 1, H 1 det H = — > 0. Vi H la ma trail xac dinh during nen Q 2 dang town phudng xac dinh during. Mut vay Q xac dinh mot la 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,, 1 2 12, x3) thi Q(x)=(x, -- x2 ) + (1 x3 - X 3 ) 2 + 2 1

x l =

' 1- 2

2

2

Hat

1 x3

x2 = 1

x3 = --)r2

+

X =

KM do



2

x3

x2

+ —x 3 2 Tit do, xet cd sa {e' 1 ,

0' 2 ,

e'a} ma

167

e



el

ez =



e3

e 3

—e + .ne 2 + 2 '

thi trong eel sd fe'„ e' 2, Bang Q

'

thdc cua clang town phdong Q eó

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 co ma tran H.

e2, e2),

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 hayrA t .H.Y=Xt.H.A.Y 1

(3 0 That Vey At H =

-1 2 -1 1

=

3

-2

1 :`

-2

3

-2

1

-2

3

1'

0

3 i

2

0

2 2 1 0 -- 1 2

= H . A.

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: 2 \ H i (X)= Xj. +54 +144 -F2XI 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

dang cue eda dang toan phuong H 1 . Vi H, a) Gia sii )(lc dinh dudng, nen W 1 siuh ra mOt tich vo huang trail V: y). Ta bigt rang, trong khong gian vectd dclit V co co so = clg clang toan phudng H2 co dang chinh trip chan Trong có so true chua'n do, H 1 (x) =O 1(x, x) =

the H2(X)=-

n

=

0

Exr 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 vdi cat toa clO (x, y) dang chinh the. Chang han xOt trong trong co so chinh

tae

, e2 }. 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. = e ll e l +C 21 e 2

GM ad:

62 rc

).

Khi do

Goi (x', y') lk toa dO trong co sä

+ C22 62

IX --c'Cii11+ Cl2

(y= C21x'+ C22

r'

Ta nhan thdy H, c6 dang chinh the trong ca sa {6,, E 2} thi C„ . C 1 2 = 0; H2 c6 clang chinh tac trong co so fe„ E2 ) thi CI, C22 + C.2 1 C 1 2 = 0.

Tit do suy ra ma tran C =

Cu Ti12

suy hign. C 21

C 22

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 .

{YI = X I + x 2 + x 3

xl = Y1

Pat bign mai y 2 = 2x2 + 2x3 hay x2 3x 3

Y3 =

=

X3 =

1

1

2 Y2 1 Y3

1

3 y3

thi voi die toa dO (y„ y 2, ..., y3 ), H, co dang chinh tac: 2

2

H1 = + Y2 + y3

(2)

Trong ho toa do, do, I-I 2 co bia thfic: H2 = 17 22 + 170

213373

(3)

'o 0 1' Ma tran maa

H2 CO

(4)

dang A = 0 1 0 ,1 0 0

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& 1 . A. T = 0 0 1 116 T= (5) 1 1 0 1. 0 yl

Dung phep

toa (.10

=-

1 1 r zi + z2 V2 a/2

Y2 =

(6)

3

1

+

Y3="'

1

22

Trong do ca hai dang town Oaring se c6 clang chinh tAc. H, (z , z2 z3) = + H2 (Z , z2, 23) — -

x Tii (1) va (6) ta co

x a — HS {el, e2, C31 7a

9

+

+ z3

(7)

2

2 2 +22 +22

(8)

1 1/2 '

= --,z,

x2

9

9

1 3.h z1

+

1 .5 z2

1

2 1 1 z2 1 + 3-,,& 3,5

1 —

2 z3

+ lz 2z 3

co sa trong K3, trong do H, va

H2

(9)

ce dang

chinh ta.'c (7) va (8). Khi do: 171

1 el

1 $

=

1

3

1 " 2 -

+ 1 e2

ea

1 ,a

1 i) 3-,N -2 + 3I_ -3

1 +,7, 02

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

c2 , ...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 1 1a bao nhieu? quantihcdgpynfhac6treF

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 + F1 ) 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 , ..., ep) tren d6 f xac Binh ducing. Ta co dim F2 (1 E, z p - 1, nhung dim (F 2 n El) o p - 1 5 p', nhu vay = p-1 hoac p'=q.

Tit de

Tuong to q' = q-1 hoac q' = q. Ta nhan thgy ea ban trudng hop P‘= P

q -1

IP P =

1

k r = '11

I P r = P -1 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 ,

ep, e p+ 2,

to co

le1 = 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, + en+2, ... en} thi p' = p-1, q' = q-1.

e2 ,

e,,

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,x 9 +4x1 x3 +2x 2 x3 b) H (x 1 , x2, x3) = x,x2 + x,x3 + x2x3. 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 +2Xxi x, +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

a) Q(x l ,

Q(x 1 ,

9

x„)= Dx i t + Exi x j , 1 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 (E2) trong co sa trite chuan fe l , e2} co 1 2 1

ma tran A = 2

Hay tim co sa true chuan trong do ,

ma tran cha f co clang cheo. 3.23. Hay dua ma trail dot xiing sau vg clang cheo nha ma tran trhe giao '00 B=

010 10 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. 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. hitting 3.25. Gra. sit' E la khong gian vet to aclit vdi tich < , >. Tv ddng eau f E End (E) duoc goi la phan doei 'ding nefu vdi moi e, y E E, to co = - . a) Chung to rang f phan asi 'ding khi va chi khi = 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. f la tti ddng c'au phan ddi xang. ChUng to rang Imf c) Gia 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

1-

1 x 3 ) -(a33 3.1. a) H = (3, + x2 + 2302 - [ 3-x2 + 73

y i = +x 2 +2x3

fat

2 =-

Y3

2

+

I

3

3x3

179

32

Taco H= b) H

-37F2

-3 1

_2 2 32 Y3

c) H = 3.2.

a) Khong co 1. nao th6a man b) Kh6ng

nao th6a man

c) A> 13. 3.3. a)

Q-602 -i612)2 + 16 673)2 +---+ n+1 2n 67 :32

1 That \ray, dat y, = x, + —(x2 + 2

x„), thi

2 -3-x,x ) ;2 Dat y2 = x2 + 1 (x3 + + xn), tu do f

=

+71 1374) + - Ex; +—xi x ; 3 3 4 .i0 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 i x + ltyx 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: jai , ..., 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 jo t '

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 u2 (a) = u(4a) = 4 u(a) = 141 2 a=X. -

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 12 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 u12 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 Pill ', lap lai qua trinh tren to có (t - )02 la ink Gila Pu l a s6 muQuatrinhy vd 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; t2 (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. Pu2(t) =(%14)a' • 0.2 - 0'2 muyen during, ..., Irk e R; Gin)

- oak x

dO sn ,sk

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 >=== 187

Nhu vay thole hay thitc \TM moi x e U. Node lai n6u moi x e U, to có 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 = + i . Nhung ft, 1, la nhang to thing chin to lien hop nen