Bài tập về học phần Đại số tuyến tính

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Tài liệu tham khảo dành cho giáo viên và học sinh cao đẳng đại học - BÀI GIẢNG ĐẠI SỐ TUYẾN TÍNH ĐẠI HỌC THĂNG LONG - Học kỳ I.Tài liệu tha khảo đại số tuyến tính về Ma trận - Định Thức. Đại số tuyến tính là một ngành toán học nghiên cứu về không gian vectơ, hệ phương trình tuyến tính và các phép biến đổi tuyến...

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Nội dung Text: Bài tập về học phần Đại số tuyến tính

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: NGUYEN THIEF N GIAP




Bien tap: HUY CHU
DOAN 'MAN
NGOC QUYEN


Trinh bay Ilia: NGOC ANH




BAI TAP DAI sq TUYEN TINH
Ma s6: 01.249.0K.2002
In I .501) cudn, tai Xtiiing in NXI3 Giao thong van tai
S6 xuat ban: 49/ 171/CXS. S6 Inch ngang 39 KH/XB
In xong va Opt [Yu chi& CM/ I narn 2002.
Lai NOI DAU



Mon Dai s$ tuygn tinh dude dua vao giang day a hau hat
cac trUnng dai hoc va cao dang nhtt 1a mot mon hoc cd se; can
thigt d@ tigp thu nhUng mon hoc khan. Nham cung cap them
mot tai lieu tham khao phut vu cho sinh vien nganh Toan vi
cac nganh Ki thuat, chting Col Bien soan cugn "BM tap Dai so
tuygn tinh". Cugn each dude chia lam ba chudng bao g6m
nhUng van d6 Cd ban cna Dal so tuygn tinh: Dinh thfic va ma
trail - Khong gian tuygn tinh, anh xa tuygn tinh, he phticing
trinh tuygn tinh - Dang than phttdng.
Trong mOi chudng chung toi trinh bay phan torn tat lY
thuyat, cac vi du, cac hal tap W giai va cugi mOi chudng c6 phan
hudng dan (HD) hoac dap s6 (DS). Cac vi du va bai tap &roc
chon be a mac an to trung binh den kh6, c6 nhUng bai tap
mang tinh 1± thuygt va nhUng bai tap ran luyen ki nang nham
gain sinh vien higu sau them mon lice.
Chung toi xin cam on Ban bien tap nha xugt ban Dai hoc
Qugc gia Ha Nei da Lao digt, kien de cugn sach som dude ra mat
ban doe.
Mac du chting tea da sa dung 'Lai lieu nay nhigu narn cho
sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang
khi bier, soon, nhUng chat than con có khigm khuygt. Cluing
toi rat mong nhan dude nhUng y kin clang gap cna dee gia.
Ha N0i, thcing 3 !Lam 2001

NhOni bien soan
3
rvikic LUC

7
Chubhg .1: DINH THOC - MA TRA:N


7
A - Tom tat ly thuyeet
7
§1. Phep th6
§ 2. Dinh thitc
10
§ 3. Ma tram
12
B - Vi dn
35
C - Bei tap
43
D HtiOng dein hoac clap so


Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH
57
PHUGNG TRINH TUYEN TINH


57
A - TOrn tat ly thuyeet
57
§1. Kh8ng gian vec to
61
§2. Anh xa tuyeen tinh
64
§ 3. He phydng trinh tuy6n tinh
67
§4. Can true caa tai ding cku
71
B Vi dtt
96
C - Biti tap
96
§1. 'thong gian vec to va anh xa tuyeen tinh
104
§2. He pinking trinh tuy6n tinh
106
§3. Cau tit cna melt tu thing calu
110
D. Illidng sign ho(tc clap s6
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

n} len chinh no duet goi la
Met song anh o tit tap 11, 2,
met phep the bac n, ki hieu la
'1 2 3
a2 G
G 3
I
\

15 del a, = a(1), 0 2 = a( 2),..., a„ = a(n).
Tap cac phep the bac n yeti phep nhan anh xa lap thanh
met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen
t3 cua nhom S„ bang n! = 1, 2... n.

Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich
the cem a n6u s6 - j) (a, a) am. Phep the a &foe goi la than
-




ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6
nghich the ciaa a le.
1 neM s la phep the chan
Ki hieu sgna =
-1 net} a la phep th6 le

va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the
= sgn(a) . sgn( ).
cling bac, thi sgn(a
Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„
coo = 1 2 , coo = i3, a(ic) = i1
• - • , i k doi mot khac nhau dr

7
va a(i) = i vdi moi i x i„ i k . Vong )(felt do dttoc ki hieu IA
i k ). M9i phep th6 dau &tan tfch the thanh tfch nhung
yang xfch doe lap.

Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong
xfch ••• , i k ) phan tfch chive thanh tfch 0 1 ,


§ 2. DINH THUG

I. Gia sit K IA met trueng (trong cuan sich nay to din yau
xet K la &Ong s6thvc K hoac truang s6 phitc C). Ma tran kidu
(m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim
m hang, n cet cac phan tit K, i = 1,m, j = 1,n. Tap cac ma
tran kidu (m, n) chive kf hieu M(m, n, R). Ma trail vuong cap n
IA ma tran co n dong, n cot. Tap cac ma trail vu8ng cap n vdi cac
phan tit thuoc truong K ki hiOu IA Mat(n, K).
2. Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, ..., n.
Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K
dude xac dinh nhu sau:
detA = zsgn(a)a mo)
Sn
E


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:
an,
an ail
f

a21 +a lci a2„
de
a,,, + an i
‘a n„

a ll
a ll a l; ...a 1 ,„
a21 ...a2 n
a 21 21 + de t
= det

—a 1111/ " S ' Ill " S IM /


a do
d) Cho A = (It o ) Mat(n, K), thi = b) = a ij &toe
E

goi la ma tran chuy6n vi cim A.

Ta co detA = detA t .

4. Cdch tinh dinh that

a) Cho ma tran A Mat(n, K). Kf hi'911 M i; la dinh that cua
E

ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot
thu j cut ma tram A vb. Aij = (-1)H M u clucic g9i la pha'n phu dai
s6cUa phgn to a ii cna ma trait A. Ta có CAC tong thtic:
O ngu i k
det A ngu i = k
ngu i x k
O
det A ngu i = k

Nhu fly detA = EamAki (k = 1, 2, ... n)
1=1
heat detA = Z a ik A ik
/=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

1 s i,
nghich the'. Vi vay, do so' nghich th6 caa a la k < ,
a co
nen ten tai i o dg oc ii) < a i0+1 ;

0„) trong do 111 i = a ngu i # i„, i„ + 1,
Xet hoan vi p = ,




con p io = a ;0+1 , p i+ , =a,„ thi HI rang 13 co nhigu hon a met
nghich the". Nghia la s6 nghich th6 ciga p la k + 1.
17



Nhan cot thu nhal ciia ma tran A vdi -k rdi cOng vac) cot
this k, ta dude:
1 -1 -2 ... -(n-1)
- (n - 2)
1 0 -1
det A = 1 0 0 .. - (n - 3)

1 0 0 0
Khai trio'n the() dung Ulu n, ta ea:
-1 -2 ... -(n -1)
-1
= (-1)" +1. (-1)"-'=1



Cdch 2.

Ta tha'y A= B. ca do
1 t1
1 0
vi C=
B=
1
11

ma detB =1, detC = 1 nen detA = detB. detC = 1.

Vi du 1.9. Hay tinh

cosa 1 0 0 0
1 2cosa 1
= 0 1 2cosa 0 0
2cosa 1
1 2cosa
0 0 0


21
Lai gidi:

Khai trim dinh thac Lheo cot cub" to co

-
D„ = 2cosa . 17,1 _2

De thay D, = cosa.

1
cosa
= 2coi2 a - 1 = cos2a.
2cosa,

Gia sa D 1 = cosia \TM moi = 1, . k.

Ta có

2cosa .
Dk,I = Dk - Dk_ i

= 2.cosa . coska - cos(k -Da.

= (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a.

Nhn vay D„ = cosna

Vi do 1.10

Hay Dull
1 0
+ a -9
0
1 e" +e 1
A„ = N x0
1
1
1 eP + e -(1)
0

a do the phan to tren &tang choo chinh bang nhau va band
e q) +e -9 ; the phan tit tren hai &tong xien Win nhat \TM (Mang
the() chinh bang 1, con the phAn ta khac bang 0.

22
Khai trin theo cot tht nhEt, to c6:

A n = (e P +e - P)A n _ i:

e 21' - e -2(P
Nlinn xet rang 4 1 = 6 9 +C c =
-

36 - -39
6
e
e ro
((i
A2 =
e (P -


e (1+1)6 - e -(lrv1),p
, n - 1.
k - 1, 2,
Girt sit ,
AR -
e (-0 -e (P



Ta c6 An = (e c e - P)A n _ i -A n _,

(:+1)o - e -(n+1)4'
nip - e npe( n-1) n4) e
- e
=(eP +e w)e

ew e q) -

-(n+1),p
e (n+1)p -e
Nhu v 6}.:
. An =
e 1 -e

Vi du 1.11

Tinh:

an
a,
ll a1
an
a l: +h i a,
1
an
D = dot 1 a, +b.,
a1


a n +b n
a1 a.9 ...



23
Lai gidi:

LAY ciOng dAu nhan vOi -1 r6i Ong vao cac thing con
1ai to có ngay D = b, 1) 2 ...b„.
,




Vi du 1.12

Cho da thric P(x)=x(x+1)...(x+n)

Hay tinh Binh thdc:
P(x) P(x +1) P(x + n)
P(x) P(x +1) ... P(x +n)
d=
P (n-1) (x) P th-1) (x+1) P th-1) (x + n)
P (n) (x +n)
P thl (x) P thl (x +1)

gidi:

Ta b6 sung de' dude ma Han dip (n+2):
P(x) P(x +1) P(x +n) 0
P(x) P4x+1) P4x+n) 0
D=
P ( n ) (x) Pthd(x +1) P 0P(x + n) 0
pg+0( x +n ) 1
P („+l) (x) 1301+1) (x +1)

RO rang det D = d
(x +n
Nhan dOng 1111 k cua ma trail D vdi dc-ix( 1) k-1 r6i
(k-1)!
Ong vao clang Hirt nhgt vgi tat ca k=2, .. n+2).
Khi do, phAn tii dung dau có clang:
poc + 0 + P (k) (x +0.(x +11.) k = n).
ok
k!
k=1



24
n; con phan tit a cot cuoi bang
(-1) 1T1 (x +n) n+i
nghia 13 clang thg nhaa c6 dang:
(n+1)!

(0, 0, , (+1)"+1(x+n)ni ).
(n+1)!

Do do
P(x+n)
Pkx+1.)
(x+n)°+i
d = del D -
(n+1)! P T Ux +1) . P (n) (x+n)
P T+I kx+n)
P T+I kx+1)

Ta ki hieu dinh thge a v6 phai bai C va ma tr5n Wong ring

hai (6-1 Vi da thge P(x) = n(x + i) Ken P'" 0 (x) (n+1) !, vi vay
i=0

the s6 hang a dOng cu6i &au bang (n+1) !. dgn gian ki higu
va each viek ta dirt x k = x+ k, k = 0, 1, n. d dOng thg hai tit
&leg len ciaa ( ta co:

P T) (x.,)) ((n+1) hco + a l , , (n+1) tx„+
(Pw(x 0 ), P ( "ax,),

al
a do a l la hang s6 &do do. Khi nhan (long cue"' ding voi
(n+1)!
r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang

P (x i) )
P(x 0 ) P(x ] )

(*)
P (11-1) (x 0 ) P th-e (x l ) P T-1) (x n )
(n+1)!x n
(n+1)!x 0 (n+1)!x 1
(n+1)!
(n+1)!
(n+1)!

25
Deng this ba tit dUdi len caa ma Dan (*) co dang
(n+1)! 2
(n +1)! 2 x n +a i x n +a.,
x o +a i x o +a,,..„ 2
2
Cong vac) (long nay hai clang °Ma sau khi nhan voi cac s
a, ta nhan dude clang
a1
va

(n+1)!
(n +1)!
(n +1) 6 9
(n+1)!
(n+1)! 2
2
xo ,
2
2
2
Bang each bidn ct6i nhu vay vdi cac dong con lai, ta clan m
Dan ( '61) ve clang sau ma khang thay d6i clinh thfic caa no.




c = det = det




x o Xn
n

x n1
n-1
X0
(n+1)!
= (n+vn
k=1 IC !
X0




n(n+1)
.((n +1)!6' 1
2
(-1)
.D n .
J k!
k=1



26
6 do D„ la dinh thiic Vandermonde cua clic so" x„, x„
n-1
D6' thay D„= n(x k - x i )= n (c_i) = 11(n-o!= 1.
k>1 k=1
k>i>0 1=0


Vay

d = detD = (-1) 2 [(n +1)! i n .(x +n) n+1 .

Vi du 1.13

a do a li = 0 vdi moil= 1, 2...,
Mat(n, K), A= (a u )
Gia sU A e

con aij bang 1 hoac 2001 \Ted i t j. Chung to rang ne"u n chan,
thi det A # 0.

Lbi gidi:

Nhan xet rang neh to them vim mit phan to a jj nao do cha
ma tran vuemg A mot s6 than, thi dinh thfic cha ma tran nhan
dticic se sai khac vat dinh thiic cha ma tran A mot s6 than. Vi
the ne'u to hat di 2000 don vi a nhUng phan t,i bang 2001 cha A,
thi tinh than le cha dinh thfic cila A khong thay d6i, nghia la:
detA = detB (mod 2) a do B = (14

h i; = 0 n6u i= j va = 1 ngu it j.

Ta có:

01
10
det B =

0
1 1






nhan Bong ddu veli -1 fee cling vao cac dOng con lei ta dttuc:




detB =


-1
1 0 0

Deng vao cot thu nhat ca car cot con lai ta cO:
. (n-1).
detB =

Vey khi n than thi detB la sidle, do vey detA # 0.

Vi du 1.14

Tinh Binh theft:

1 1 1
1
'




0 C3 CI Ci
n+1
2
D = det q C.
, C 2412
Cn+2
...
c n-1
c n-1 C.!; 1-1 1- . ) C2-11 >
2n-1
n n+1

LtlL gill

Vi CI, +Clr = C74, nen hieu cUa moi phan tit vol phfin I
dUng b8n trai no thi bang phen td thing ngay tren n6. De tir
D, ta ldy cot thit n tr3 di del n-1, r6i ldy cot n-1 tie( di cot
2,..., ldy cot the] 2 trU di cot thU nhat, ta co:




28
0 0
0
1 0
1 1
1
1
C1
2
c?c,
Cy C1 1
D = det „
C1

"•

cn-2
C 112
011 C 1,1 12
, 2n-2


lai lam nhtt tren, to co

0
0
1 0
0
0
1
C 2-
l
1 1
C3 C13
D = det

n-3
c^ -3 2n-4


sau n -1 budc nhit v'ay, to

0'
0
1 1 0
0 0
1
= 1.
0
3 1
CI
D = det

Lin-2 cn-3
cn-1 CI 1

VEty D = 1.

1.71 du 1.15
cos9 - sin9
fray tinh A n .
Cho A=
sin9 cow ,

Lai gidi:
cos2p -sin29
cos9 -sine
cosy -saw
Ta co A 2
sin2c cos29
sing cosy,
sine cost°

29
= cosky -sinkp vOi k = 1,
Gia Ak n-1.
sink cosk(p
Ta ttnh
cos(n - lap - sin(n cosy - situp
A" = A n-1 .A =
sin(n -1)9 cos(n -1)9 2 \ sing cosy ,

cos (n -1)9 cosy - sin(n -1)9 since -cos(n - lap since- sin(n -1 9 cosy
sin(n - lap cosy + cos(n - lap situp cos(n - lap cosy - sin(n -1)9 sirup

cosmp - sinmp
sinmp cosny

= cosny - sinny
Vay vdi 11191. n E N.
sinny cosny ,

Vi (11..t 1.16
0 1
Cho A = hay tinh A 200 "
0

Lbi gidi:
(1 0
0 -1
-1 0
Ta c6 A 2 =
0 -1 1 0 0 1
ma 2000 chia het cho 4

(I, la ma tran thin vi cep hai).
\ray A 200° =(A) 500 =I,

Vi du 1.17
Ma Iran vuong A e Mat(n, K), A = (ad throe goi la Ina tran
netu a ii + = 0 vdi moi i, j = 1, n.
phan del xang

30





Hay chang minh: 'rich caa hai ma tran phan doi xang A va
la mat ma tran phan doi 'ding khi va chi khi AB = -BA.

Lai gidi:
Gia sid A = (ad, B = (b i d d do ait + a i , = 0, b y =0
di mm L j = 1, ..., n.

=
Dat C = A. B = (c,);
=1

= Ib ij a jk
D = B. A = (c1,0
id]
Ta ca.: c, k = Za ji b ik - Eb ki a ji - d ki


V i k.
Nha vay AB phan del xang c, = ,




vet mot i, k c> AB = -BA.
tt=> =
Vi chi 1.18
minh
Cho A, B la hai ma tran vuong c9p n. Hay
let(A.B)= detA. detB.
gidi:
G1a. sit A = (a,,) , B = (It o ) yea i, j = 1.
Ta lap ma tram
00
a n a1.,a l n
0 0
a 2n
a91 a22



a n11 0
ant
C=
bth
1 0 ..
-




b2n
0 b21



0
0


31
Khai tri6n theo n clang dal) (thee dinh 12) Laplace), ta co
detC = detA. detB. (1)
Mat khac, bi6n &it ma Iran C bai phep bi6n ct6i sd ca) sau:
Nhan cot thu nhal vat b 1 , cot thil hat Null cot thin n vet btl;
r6i Ong vac) Ot thu n + j (j = 1, 2, n), ta dude ma tran D
Bang sau ma dinh thiic cua D va cua C bang nhau:
a l a a d
d ta
a 21


a m a m ... a nn d m d , ... d nn
,


D=
-1 0 0
0 -1 0


0


a do = ia ik b ki , nghia la (dij) = A. B.
r
k=1


Khai tri6n theo n cot cuoi (theo dinh 157 Laplace) ta có

detD = det(dij) = det(A. B) (2)

Tit (1) va (2) va do detC = detD nen ta co:
det(A0 B) = detA . detB.

Vi rtu 1.19

Cho X la ma tran vueng cap n. Chung minh rang X giao
hean vOi moi ma tran vueng Ong ca) = X co clang XI,,, a do I„
la ma trail dun vi cap n.


32
Lari gidi:
1 thi re rang X giao ho an vdi moi ma Han
Ni111u X =
\along cling cap.
Ngnoc lai, gia sit X = (X„) giao haul vdi moi ma tran
yang cdp n.
VOi i„ # j 0 , to chiing minh x inio = 0. Mudn stay chon A = (ad
trong do a joio =1 con one phan tit khac ddu bAng Thong. Phan to
a thing i o
ding i o cot j„ cim ma tran XA Wing x. •' can phan tit
cotj,imAXla0.Trd6uken=XAsyraO.Nhu
y X co dang:
kr 0
X= „



0 k„
Cho ma tran A = (a) a do a ] , = 1 vii moi i, j. Khi do phan tit
o dung i cat j cim ma Han XA la X , con phan to 0 ding i cot j
dia ma tran AX la 2. nen k = ,




Vi vay:
X= 7. I,,.
Vi du 1.20
Cho ma Iran cap n:
ab
ba
A=

b

a) Chung minh detA = (a-b)° (a + (n-1)b).

b) Trong trudng hOp detA s 0. Hay Huh ma trdn nghich dao
A+ oda ma tran A.
33
Lai giai:
a) Gang the clang vao dOng this nhat ro'i nit
a + (n-1)b a clang dau, ta (Woe
1 1 ... 1
b a ... b
detA = (a + (n-1)b) x

b b ... a
Lay dOng chin aria (Anil th.fie tit nhan voi -b r6i Ong vao
the dong sau, ta eó:
1 0 ... 0
0 a - b ... 0
detA = (a + (n-1)b) x

0 0 ... a - b
detA = (a + (n-1)b) . (a-b)" -i .

h) A kha nghieh detA 0 .(=> a b va a + (n-Gb O.
Goi B = (b ii ) la ma tran nghich (lac) cila A = (a„).

1
a do A. ; la phAn Phu dai so
Ta Wet rang b i d = detA . A„

etia phan trong ma tran A.

a b b
a
Vol mdi i thi = la nh thiic cap (n-1).

b b a

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

34
A a+(n-2)1(
Vay b„ -
det A (a 1- (n -1)1).(a - b)

= (-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 M o gain town nhang phan t& b. N6u d6i eh 6 .
Bong j - 1 len tren Ming dau (Mu nguyen cac dung khac), rei lai
MS( nit i len 6;4 thu nhai (va van gib nguyen the cat khac), thi
to (Moe:



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




-b
Nha v(6y b
dot A (a +(n -1)b)(a - b)
Do do
a +(n -2)b -b
1
(a +(n -1)b)(a -b)
-11 a +(n -2)b


BAI TAP
C

(n 2 2) lh
1.1. Chung to rang mai phep chuyan tri dm
mot phep the le.

1.2. Hay phan Lich eau pile') the! sau thanh tieh ene phep
chuy6n tri
35
2 3 4 5 6 7 8
1 '
a)
8 1 3 6 7 5 4 2 )
(1 2 3 4 5 6
b)
651243
1.3 Tim s6 tat ca the phep the a e S„ sao oho a(i) x i vol
mm i = 1, 2, ...n. Cluing t8 rang khi n chan, so" one phep th6
clang tren la m54 3616.
1.4. Ki hi8u (n, k) la secac hoan vj rim 1, 2 n ce dung k
nghich the. . Chiing minh cong thile truy 146i sau:
(n+1, k) = (n, k) + (n. k-1) + + (n, k-n).
9
vdi guy vac (n, j) = 0 nevu j < 0 hoac j >

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, k.


36
1.8. Xac dinh davu cua cac phop the' sau:
3n
2n +1
2n
n + 2 ...
n n +1
2
a) I 1
3n-2
1
2
43 6 ... 3n 5
2n
n+1 n+2
12
b)
2n-1,
1
2 1 ... 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 vai i= j
yen i + j= 2n +1
d i; = b
dO
i=j vet i+j 42n+1
0
1141

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.

Fr (a 1 -a.).
Tit do: A(x) = (-1) 11 ' x
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 + a0D«-s.

13, 1/2 = az + an + 13 2 ,
Ta co: D, = a +

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

n-1
-1-i
Ta co = (a +p) a ir - an Ia'0 2-2-I
i=o i=o

n-1 n-1 n-2
= a i+lpn-i- k ira .2 a al n n-i-1
or

1.0 i=0 i=0


n-1 n-1
i v _i
= r3 n ct
-
i=1 ]=1




=E
=o


Mtn tray D„ = Ear .
i=o


48



b) Cheng minh qui nap thee n.

1 1 I,
n = 2 o D2 = = x 2 - x, kheng phi thuec y,.
+ y i x, +y i

D„= (x, - x,)(x,, - xi)( 11 9 - x1)-
n = 3

= 11(X j -x i ) vdi moi k = 2,
Gia stir n-1.
Dk
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 =

= +(detS).C.(A ql . that m = p - q.
Tit do:

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,

a ll a l p
=a n a 22 -a 21 a t2 .
n=2, A=
a21 22

Neh cho track a 2 , . a 1 0 t 0, to chon a, 1 = a 22 = 0.

Con nei n a 21 . 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.

Xet A„ la phan phu dai so' caa
Gia silt A = (ap) e Mat(n
a„„ de?
ph dn tit a n . Theo gia thi6t quy nap, to da Hei n dude a 22 ,
.
A, 1 0.

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 u A 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 -1 A = 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 i 2 cua A -1 tY le vdi nhau. Diu do clan deIn mau
thuan.

NMI vAy m61 cot cim ma tram A co dung met s6 clueing (con
lai dgu bang 0). Dicing te, m6i deng cila ma tran A clang có
dung met s6 throng. Giao hoan cac demg (hoac cac get) ta duoc
ma tran chg. °.

Nguoc lai, tat ca the ma tran 'Then dude tt ma tran cheo
(a li > 0) bang each chuygn clang host cot deu kha nghich. Ma
trAn A - I nhAn dude to ma tran A bang each Iay nghich dao cac
phan to khac kitting rdi chuyAn vi.

1.32. DS det(A) = ± 1.

56
Chuang 2

KHONG GIAN VECTO - ANH XA TUYE -N TINH
- HE PHUONG TRINH TUYEN TINH

A - TOM TAT Lt THUlt

§1 KHONG WAN VECTCS

1. Dinh nghin: Gin sit K la mot traong. Mt tap hop V
kirk rung cimg vdi hai phep town "+" : V x V —> V
(a , p) 1—* +
va phep than " •" : K x V —> V
(k, a) 1--> k. a
dime goi la mot kheng gian vec to tren twang K n6u no thOa
c K va moi a, p, y, c V,
man cac tinh chgt sau. mot k, /,
to CO:
a) a +p=p+ a
p)+ y= a+(p+ y
b) (a+ )




c) CO phlin tit 0 e V sao cho:
a + 0 = 0 + a = a vin moi a c V.
ton tai (-a)
d) Vdi a e V, V sao cho:
e
a + (-a) = 0
e) k(a + is) = ka +103.
g) (k + /)a = ka + la.
h) (k. / )a = It (/ a)
ta don vi cita truang K.
k) 1. a = a, 1 la phan


57
tren throng 1K con &foe goi K -
Mot kh8ng gian vec CO

kKong gian \ecto.
to &loc goi la kh8ng gian vec to
Khi 1K = R, kh8ng gian vac

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

to 13= k,a, + + k„,a„„ (a, c V, k e K)
Vec

Ta
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 , ..., a n „
caa V dude goi la he phu thmic
MOt he vac to la,, ..•,
m) kh8ng deng thoi bang
tuyeIn tinh nen c6 cac s6 = 1,

= 0 . Met Oat hkeu along &king:
kheng cna ]K sad cho
1=1
a m ) ka he pha thuec tuy6n tinh nen c6 met vec Lo nac
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
a m } la he dOc lar
he doe lap tuyen tinh. Nhu \ray, he {a„

k i a i =0 to suy ra
tuyen tinh ned.1 moi t6 hop tuy6n tinh

k, = = k m = 0.

a id cac vec to doe lap tuygn tinh caa kh8ng
Cho he 14„
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 (a 3) jedcI goi la he con de(

58
lap tuyen tinh t6i dai cim he da cho neu n6 la he dec lap tuyen
tinh, va nen them bet kST vec td a k nao (k E I \J) thi ta dude met
he phu thuec tuyen tinh.

Cho he hau han vec td {a, , , a m } trong khong gian vec td
V, thi s6 phain tit cim moi he con dee lap tuyen tinh t6i dal ciia
he tren deal bang nhau, so / do duoc goi la hang cim he vec td
{ao ...,
va so clueu cua khong gian vec td
3- Cd sd

, e n } 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.
, ej, yen vec tO bet kS7 a e V, ta co
Cho co sa e = le i ,
, x„) dupe goi la cac toa doo cim vec td a del vOi
,
a

ed sa {e„ , e n}, ; la toa de thu i cim a doa vdi cd 56 do.

, E„} ciaa V, ma s i =
Gia sit co ed sa khac c = {c,,
]=1
va c6 toe
n). Nen vec td a có toa di) (x i ) trong cd s6
= 1,
thi ta c6:
de' (x 1 1 ) trong ed sa
= Ec o xl i , , n.

va ma trail X = (x), X' = (x')
Neu ta ki hieu ma tran C =
la cac ma tran cot, thi X = C X'.

59
4 - Klaang gian vec tei con va klaing gian vec td thacing

Tap con khong r6ng W ciaa khong gian vec to V duoc goi IA
khong gian vec to con ciaa V nen W la khOng gian vec to, voi cac
phep toan ciaa V han dig tren W.
Tap con khong rang W dia. V la khong gian vec to con ciaa V
khi va chi khi W on climb dna vdi hai phep toan dm V, nghia la
vdi a, 13 E W va k E K, thi +p E W va k a E W.
Cho W,.... W EI la cac khong gian to con cria khong gian
vec
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

ZW ; .
khong gian con W, ,W„, ki hien Ia W, + + W„ hay


EW; khi va chi khi a = Za i , a, e
Ta thay rang a e
i=1

/113/4 , a vigt chicle mat each duy nhat a
Neu moi e
i=1
dang a = a, + + a„ , a, thi t8ng W, ch.toe goi la tang
e
I=t
true tigp cua n khong gian vec con (i = 1, 2, ..., n) va ki
td

hieu la W, e ED ... ®W„ hay S W,.
W2
=1

60
Gia sit V la klafing gian hitu han chikt,W, va la hai
W2
khfing gian vec to con dm V, khi do:
dim(W,+ W 2 ) = dimW, + dimW 2 - 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 V va k e K, lam thanh mat klihng gian vac
E td,

durie got la kitting gian vec to thtfong (ciia V chia cho W):

dimV = n, dimW = in (0 m n), thi dim V/W = n - m.
Anh xa it : V -> V/W ma a(a) = [al (Inge goi la phdp chi6u
tdc.

§2. ANH XA TUYEN TINH

1. Dinh nghia. Cho V va W lh cac khong gian vec to tram
twang K; Anh xa f: V -> W duo° goi la anh xa tuye"n tinh (hay
itIng ca'u tuy6n tinh, hay toan tit tuyan tinh) n6u no bao
phop toan cua khong gian vec to, cu the' la: veil moi a. p e V.
1119i k e IK, to ce:

P) = f(a) f(P)
f(k a) = k . f(a).
Anh xa tuy6n tinh dude goi la don din nett ne la don anh,
:oan can n6u n6 la than anh, va dang 6.'11 n6u n6 la song anh.

61
Hai khong gian vac td V va W chive goi la clang cdu vol nhau
nAu ce met ding cau f tla V len W.
2 - Cac phep than tren cac anh xa tuy6n tinh

Ta ki hieu tap cac anh xa tuy6n tinh tit khong gian vac to V
dAn khong gian vac to W la Hom(V, W) hay Hom K (V,W) de chi
rd K la tniang co sir.

Hom K (V,W) la met khong gian vec to tit truong K vbi hai
phep town nhtt sau:

Hom K (V,W); anh xa f + g e Hom K (V,W) the dinh
Vai f, g e

ben (f + g) (a) = f(a) + g(a) vdi moi a e V.
Vol k e K, f e Hom K (V,W), thi kf: V —> W xac dinh boi
(kf)(a) = kf(a) vdi moi a e V.
Anh xa f + g dupe goi la tong Kin hai anh xa f va g.
Anh xa k. f ddoc goi la tich eda anh xa f vdi vo hdong k.
3 - Di6u ki6n xac dinh anh xa tuy6n tinh
Anh xa tuydn tinh f: V —> W hohn toan chidc :Mc dinh khi
Mei anh cOm met co so.
NMI le„ , e„) la co se, cOta khong gian vac to V va a,, a„
la n vec to cent MI:Ong gian vec to W (V, W la cac khong gian vec
to tren clang rant trnong K), thi Kin tai duy nhdt met anh xa f e
f la don cdu khi va chi
flom K (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
vaaha tran A = (ad goi la ma
E, =
ciaa W, thi f(e,) =

62
tan dm anh xa tuy6n tinh f dOi vOi co sa {e,) va {EJ. Nhg vay
16u cho cd sa E cua W va co sa e cua V, thi dinh 19 tren chring to
:Ang co mOt song anh girla tap Hom K (V, W) va tap 1),E#t(m, n, K).

FlOp thanh cua hai anh xa tuy6n tinh la mot anh xa tuy6n
rhh, nghia la nOu f: V —> W va g: W —> Z la the anh 'ea tuygn
thi g f: V --> Z cling la mot anh xa tuygn tinh.

4 - Anh ca hat nhan cua anh xa tuygn tinh
Cho f: V —> W la thing thu tuy6n tinh gifla cac khong gian
7ec to, neh X la khong gian vec to con cua V, thi f(X) = { f(a) I a E X)
a khong gian con cua W, va n6u Y c W, Y la khong gian eon
W thi f-'(Y) = e VI f(a) e Y} la mot kWh:1g gian con cua V.
Ta goi Kerf = f-1 101 la hat nhan cda anh xa f va Imf = f(V) la
inh cua anh xa f. S6 dim Imf doge goi la hang cua anh xa f, kf
lieu rang E

Gia sit f e Hom K (V, W),

la don eau khi va chi khi Kerf = {0}. Nth dimV Fa hfiu han thi
IimV = dim Kerf + dim Imf.
Cho ma Wan A e Mat(m. n, 1K), xem A nInt ma tran cua
inh xa tuy6n Unh f: K n —> K m trong thc od sei chinh tdc. Khi do
cua ma trail A (da dude dinh nghia trong chudng I) hang
tang
rang cua f va chinh la hang cua he vec td cot cua ma tran A.

5 - Ma tran aim t‘i thing cgu trong the cd ad khac nhau

Cho f E Hom K (V, W), trong co se: e = (e„ , e n ) f cO ma Wan

= (a d ), 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 Ma trar
i=1
C = (c o ) chide goi la ma tren chuyen co sa. Ta ca:
B=C' AC

Hai ma tran A ya B dude goi la deng deng ne iu 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

Ea ki x i = b k (k = 1: 2, ... ,
He
i=1
dO a ki , b k E K cho trUoc, k = 1, , m; it= 1, , n.
x i 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 = (a k; ) goi la ma tram cac he sg.
64





a ln
a 12 b
a ll
a 2n b2
a 21 a 22 goi la ma tran
Ma trail Abs =


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

a n}, ma a i = (a ii ,a si ,...,a mj )
Cdch 2: Xet he vec td cat
, b 0 ,) la vac to cat td do, the thi he viat dude dU6i
va b(b„
=b. N6u ta goi D i la dinh thiic cda ma trail nhain
clang

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

(Gauss hay Kronecker - Cape 11i)
a) Dinh b./

He phudng trinh Za ii x j = h i (i= 1,..., m) (1)
i=1
ce nghiem khi va chi khi rangA = rangA h s.
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 131

P
h ip
1
0
p+i
m-p
0 0 b' m

n-13
p

a phan goch (*) co the khac 0.
phan t>i

> 0 thi he ye nghiem,
Khi rid n6u 13,; +1 + +
= b'„,= 0 thi he có nghiem phn thuOc n-p tham s6.
n6u

§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 cal u 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 End K (V)
e

ling vat gia tri rieng X ngu f(a) = Ira, A K. Khi do khong gian
e

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

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

con cim V, gam vec to khong va tat ca cac vec to rieng caa f ling
vdi gia tri rieng X. Kitting gian nay goi la khong gian rieng cim f
v6i gia tri rieng A, ki hieu

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

GM sit , A k la cac gia tri rieng cigi met phan biet caa I
Px Pik la cac khong gian rieng Wang ling via ale gia tri
rieng do, thi t6ng Pr + Px 2 Pxk la tong true tigp.

2 -Ong gian rieng suy Ong

GM sit V la khong gian vec to tren (Wong K, f e End K (V).
Vdi mei A e K, xot tap {a e VI co s6 nguyen m > 0 de1
(f-XIcl,)m(a) = 6). DO la met khong gian con caa V, khi khac {0}
no &arc goi IA khong gian Hong suy rang cua f ling vdi A va ki
hieu IA Ta thgy rang:
68
IA met gia tri
+ Vdi moi khong gian rieng suy rang 7r
3k


rieng cim f va Y, c R s .

+ V6i X IA gia tri rieng cua f, dim2h x bAng bei cern nghiem 7.
da thitc dac trung /. f(X).
cua

+ Mot g?",, la met khring gian con cila V bait bi6n qua anh xa f.
IA cac gia tri rieng phan biet tiing cap
+ Gia sf1 , " .k
f

u k} doc
= 1, 2, ..., k) thi he cac vec td {u„
va u, e \ {0}

lap tuy6n tfnh.

3 - Tit citing clu luy linh

a) Ta not ring f e End K (V) IA tu ding caiu luy linh nau tan
# 0 va
tai s6nguy'en k > 0 de'f = f o .. of= 0, hdn nua, nau
k hin

0' = 0 thi k goi la bac luy linh cua E
, e n } sao cho
Tu dang udu f e End K (V) ma co cd sa te„
(i = 1, , n-1) va Re p ) = 0, thi hay linh bOg n va ed sa
Re) =
{e i , ..., e n dude goi 11 cd so xyclic d6i v6i f. Trong cd SO xyclic ma
}




trail cua f co clang:

00 0'
10
0
01
1
00
10
..




69
b) f e End K (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 End K (V), dimV = n, thi V phAn tich dude thanh
e
tang trip tiEp cua cac khong gian vec to con xyclic doi vdi f. Vdi
mdi so nguyen s 2 1, sE cac khong gian vac to con s chiEu xyclic
doi vdi f trong moi each phan tich dEu bdng nhau va bAng:
ran g(f 8 ') - 2rang(f s) rang(f

d) N'Eu 9 ) IA khong gian rung suy rang cua f dng vdi A, thi
A.Ick) la td d6ng eau Iuy linh cern V.
(f -


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 End x (V) ma da that ddc trung 94(X) cd dang:

i (X) = (>1 (X -X)` 2 ... -X) s k
,




(cac &Ai mat phAn biet, i = 1, 2, ..., k), khi do V la King true tiEp
tha cac khong gian riAng suy rang V = x G3.? X 2 e 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„ x 2 ,
ChUng minh rang tap IF the vec td ena K" dang
con eila
e E Fa khong gian vec to con ena K', dimF =
Zt i x i (t 1 , ,
nN), trong de N la khbng gian cac nghiem tha
dimE - dim(E
phtlong trinh:
E t i ., =0 t n, la cac An).
(a do t i ,
i=1

Lai gidi :




a do u(t,, t„,) =
Xet anh xa tuyan tinh u c Hom(R m , R")

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.
n N).
Nhtt vat dimF = dimE - dim(E



71




Vi du 2.2. Cho x i , , x i , la nhUng vec to khac khong trong
khong gian tuy6n tinh V. Gia sit c6 Oleg bi6n d6i f e End V sao
cho f(x 1 ) = x 1 , f(x k ) = x k + 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„
Gia x k i; k < n - 1. Ta
chimg minh menh d6 dung vdi k = n.
Xet t6 hqp tuyen tinh
ECiX i = 0 . (1)
i=1
Khi do:

Zcif(xi) = c,x, + Ec i (x i +x i _ 1 ) =
) = CIX ]
i=1 i=2 1=2



.
= ZeiXi (2)
i=1 i=2

n-1
Ec i x,„ =
Tit (1) va (2) to suy ra: = 0.
i=2 i=1

Do gin thik quy nap hee, fx,, , doe lap tuyen tinh nen
= c n = 0.
=e2 =
C2

Tit do suy ra c,x, = 0 c i = = = c„ = 0 Nhu vSy he
{x 1 , , xj doe lap tuy6n tinh.


72




a k } cac
Vi du 2.3. Trong khong gian vec to V cho he
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„ ,


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,, , p i }
g6m m vec to va se elnYng minh k s m < 1.
p, nen
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 ;


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

/Xiai =0
j=1
i=1
1=1

I (k
p i =o
j=1 .i=1


(2)
=0
EauXi
j=1
p i leriic lap tuyen tinh.
1 do he
vdi j =

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 nE 2 .

L of Rich:

1) a) X& E —> E/E 2 la anh xa chidu chinh cac, h =
h toan anh a ME,/ =11(E).

E, + Kern = E + Ker n = E

E, + E (vi kerb = E2)
E2 =

n
b) Ta en Kerh = E, r1 Kern = E, E2

Nhtt vey h don anh a Kerh = 0 a E, E, = {O}

2) GiA. se F = E, + E2;

Xet F F/E 2 va k = 1-1/E, •

Do phAn 1) k la toan anh va kerk = E l C Kern = E l 11 E2.

Do do to co E l /E l fl ding can not E 1 2>E 2 /E 2 .
E2


74
Vi du 2.5. GM. sit V la khong gian vec td tren truing K. Ty
Bong can p: V -+ V ducic goi la met phep chien ngu p 2 = p.
1) Chung to rang ngu p, q la hai phep chigu, thi p + q la
phep chigu khi va chi khi pq + qp = 0.
2) Chiang t6 rang p.q IA phep chi gu khi va chi khi [p,qI = qp
- pq la anh xa tuyern tinh chuye'n Imq van Kerp.
3) Vol p,q Fa hai phep chigu sao cho p+q la phep chigu, hay
cluing to Im (p+q) = Imp + Imq va Ker(p+q) = Kerp n Kerq.

Lei gied:

1) p+q la phep chigu ra (p+q) 2 = p+q
= p+q p.q + q.p = 0
p 2 + p.q + q.p +

2) p.q la phep chigu (p.q) 2 = p.q.p.q = p.q = p 2 q 2

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 Kerp n Kerq.
E


as. Gis sit E va Fla hai khong gian vec td tren
Vi du
trulang K; Horn (E, F) IA tap cac anh xa tuyeal tinh tit F Mn ]F;
F t la khong gian vec to con cna F.

a) Chiang to rang £ = Horn (E, F) / Imf c F 1 ) la khong
e
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 Horn (E, F) I Imf c Fd.
E



2.
Chung to rang Horn (E, F) =

gia'i:
WL



a) Do g ding kin vai phep town tong anh xa va nhan anh
xA vdi mat vet hriong thuac K, nen hiel n nhien la khdng gian
vec to con crIm Horn (E, F).
b) Vdi h Horn (E, F), vbi m6i x e E, to phan tich h(x) = y
E


E yi , yi e F1
theo eec thinh ph'An y = h(x) =
id

Xet E -, K, Horn (E, F i ) = 2 h i (x) = yi
e




76
Eh ; (x)
Va vdi moi x e E, thi h(x) =
i=1

Do 05 h= ,e


Gia six Eh ; = 0 nghia la vial moi x e E,


( ) (x) = h i (x) = 0, h,(x) e F,
1=1 1=1

to do suy ra 11,(x) = 0 voi moi i vi F = e K.
I=1

Nhu \ray Horn (E, F) = e g.
=1

a 7. GO Q la Huang s6 h> u ti va WI la mot tkp hap
Vi du

khong rkng. Ki hiku E la tap cac anh xa tif c32 vao Q. E la
Q - khong gian vac to viii hai phep than: f, g e E, a e cc9, k E Q the

(f + g) (a) = f(a) + g (a)

K. f) (a) = k. f (a).

Gia sif V la khong gian vac to con ciaa E.

1) Chang to rang: nku f e V va co n dim x„ x„E Ca
sao cho
ingui=j ..—
k(x,) = = ej=1,n
Ongui#j
thi he 10, •••, f,,} dOe lkp tnyen tinh.

77
Goi W la khong gian con cea V sinh bed f„), khi do
m81 g E V 11611 viet dude met each duy nhat dudi clang: g = h, +11 2 ,
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:


EX i f,
Xet to hop tuyen tfnh = 0, c Q.
1)
1-1

LW; (x) = 0;
Khi do vdi moi x E cam ,




Chon x= x„ thi (x i ) = 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 g frA .
lei 1=1

Dat h a =g - h, eV
thi h 2 (xj ) = g(xj )-11 2 (x j ) = 0 vdi moi j = 1 n.

78
2) Theo phan 1) neu b) clime th6a man, thi he (g„ g„) doe lap
tuyen tinh, do \ay dim V 2 n; nglila la menh de b) a) thing.

Ta chung minh a) suy ra b) bang phydng phap quy nap
theo n.
Vdi n = 1; n6u dim V 2 1, tan tai f i x 0 va vi vAy có x, e c14

de f i (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 g i (x,) = 8,, (i, j = 1,2, ..., n).
Goi W la khong gian con sinh bid {g„ g n }, to cd. dim
W< dim V.
Theo cau 1), vdi f e V \ W, to cd f = h, + h 2 , vi h, e W nen
h 2 0; vi th6 c6 x,,, d'e' h 1 0; (x,„., x x, vdi i = 1, 2, ..., n).

h2
- hi bi n+ , vdi MOt i = 1,2, ...,
)t
= 1, ..., n),
n+1. Bay gia = g -

n to co g i (x j )= (x J )= s i; va g i (x n+1
thi vdi j = 1, )




=g. (x„. 1 ) - x ; = 0 n6u 2, 1 = g i (x„,)_

Nhu co n4-1 ham {g- i } th6a man digu kien bai town.
Nty


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) :0304 vdi
E

moi a e K. Chung minh rang g c PC.

79
Lo gidi:

Ccieh 1. \TM m6i x e R, xet vec to d x (1, x, x 2 ,

H9 eac vee to d x c6 tinh chat:

doi mat phan biet, thi ta xi dPe ld
a) vdi x„
tuy6n tinh, vi Binh



# 0 (Dinh thite Vandermonde)




sao cho g(Ci; ( ) = f( ).
b) V6i m6i d x e R', tan tai f E


df
Ta phai chiing minh g c 93, nghia la c6 f e

la co so cua R".
g@t„ i ) tren bb

c6 khong qui (n - 1) sd the
Gia sii node Lai, vdi m6i f E

phan biet x i 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

gi d
Nhu vay phai co co sa Vt xi ,...,a x ) oda an vti 06 f Et


= f(di xi ) vOi moi i= 1,n; nhu vay g=f e




80
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.


thi f — g x O. do vay
vSy vdi mai f E
Hia sir g
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). r g." = ra n 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).



trong cd ed tly &nen
Xet f e End (IR") co ma trhn

e = le,, e„). i(e,)= apie

Theo gia Hired to cd f2 = f.

khi de Y c Kerf, nen Yea a e
Bat Y = - f(a) I 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„) Kerf, thi trace f= trace al= r.
e


Cho Nhaat lai rang n6u f la anh toa tuyeat tinh cie'n

Jrco ma Ran oat = (ad trong cd sa e = (e,, nao do ciaa Tthi
so' a„ +,...,+ a„„khong phu thuoc vita vice chop cd so oda °L va
clack goi la vat ciaa anh xa tuyan tinh f va chiac kf hieu la trace f.

Vi du 2d0. Gia sa °W la hai khbng gian vac td treat
twang K, f Hom K CP; °Tf).
c


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:

1234
-1 2 - 3 4
D= = -4
0 1 -1 1
1 1 1 1

Day 1a he phudng Huh Cramer.

80 2 3 4 1 30 3 4
3 4
10 -1 10 -3 4
2

D,=
-


1), = = -8
3 1 -1 1 0 3 -1 1

10 1 1 I 1 10 1 1

1 2 30 4 1 2 3 30
- -1 2 10 4 -1 2 -3 10
-12 D'=

Da = = -16

1 1
0131 0 3
-




1 1 10 1 1 1
1 10

NMI vey , = 1, x 2 = 2, x 3 = 3, x 4 = 4.

83



Vi dii 2.12. Gihi va bran luhn theo thaw s6

x, + x• 1
A.X1

Xx, + x. 2,
+ 4x 3



I) =




a) Veil X s 1 vic a x -2. Day la he Cramer.

it +1 1
,x-
va ta ce, -
+2
X +2 2, + 2

b) vdi A = 1, to c6 he Wring during vdi:

x i + x 2 + x,, =1 hay x 3 = I - x, -x.,

do x i , x, lay trtyl,

c) Vol X = -2. He co dang:

-2x 1 + x., + x. =1
2x, + x 3 g - 2
x i
x i + x, - 2x 3 1


GUng vg still ve curt ba planing trinh Ln c6 0 = 3, nhn vhy he
voi nghiam.



84





Vi du 2.13. Dung phitong phap khn. hay giai he phuang trinh:
)( 5 =2
x t + 3x, + x 3 + 2x 1
3x 1 + 10x 9 + 5x 3 + 7)( 4 + 5)( 5 = 6
2x 1 + 8x, + 6x 3 + 8x 1 + 10x 5 = 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):
+ 2
x + :3x 2 + x,, + 2x X5 -

0
X9 + 2X3 x4 + 2x5 =

2x, + 4x 3 + 4x 4 + 8x 5 = -2 (II)
3x, + 6x 3 + 4)( 1 + 8x 5 = -1
2x, + 4x 3 + 5x 1 + 10x 5 = -3

Nhan pIntong trinh thit hai cna he (II) vdi gag s6 thich hop
roi tong viva du; phuong trinh kink cita he, ta &toe he Wong
dyeing -

x 1 + 3x 2 + x 3 + 2x 1 + x 5 = 2 (I)
x, + 2x 1 + x, + 2x 5 = 0 (2)
2x 4 + 4x 5 =-2 (3) (Iii)
(
x 1 + 2x 5 = -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

+ 3x 2
x i + 2x 4 2
y 2 + 2x 3 + + 2x 5 = 0 (IV)
xI
-
x4 + 2x 3 = -1

Giai he (IV), to (little

= —1 — 2x 5
xn
a do x 4 . x
= — 2x. Inv 31
1
X2
x i = 1 + 5x i + 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) dimlmg.

86
Tit de suy ra:
rang(A + B) < rang A + rangB.
Mat khdc:
rangA = rang(A+B-B) < rang(A+B) + rang(-B)
suy ra: rangA < rang(A + B) + rangB
Tit do:
rangB < rang (A +
ningA -

ta: rangB - rangA < rang(A + B). VI yay
rangA - rangB 15 rang (A + B)

K" la hai anh xa tuy6n
2) Ta co f: K" , g:
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).

Kor(f g)/Kerg Kerf (1 Img
Mat khde 26t anh xa

a do tI/4xl = g(2), yin Ker(f 0 g). De they (To la (Tang au tuyeal
XE

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 E xAc dinh bai p(x) = y.
E la phep
Ching minh rang cl8ng cad tuyan tinh p: E
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).
End(V).
2.31. Cho V Ea khong gian vec td tren tnidng K va f E


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 ..., e n } cna
V via soi nguyen s (0 s n) sao cho u(e,) = 6, vdi i i< s va u(e) = 0
vii s < i
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 to p 0
'




00. 0 0 0
hoac

0 0 \ 0 0 ... 0
0

Neu f co ma Den a dang thu hai, thi f lay linh va c2 = O.

2.59. Chung to rang neu hai tp deing cau L g gem kh6)
gian V Dacia man he thew f.g—g f = Id (0 thi vai moi k e
ta — &c.f. = k.g" (2).

Tit d6 suy ra rang khong the t6n tai the tp dgng cit'u tiv
man he theic (1).

2.60. Chung minh rang, d6i yeti ho bat kY CAC torn to tuy
tinh giao hoan vdi nhau Ding doi mgt trong khong gian vac
hem han chigu- tren trang so" Ode, tan tai yen to rieng chui
cho tat cA the Loan to tuyen tinh cria ho do.

D. HU6NG DAN HOAC DAP S6

§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINI
2.1. GM sU V, W Ian hurt 1a cac kheng gian sinh bai A„
ka B 1 , ..., D r Veli mei = 1, 2, n thi B, = a,,A, + a 2 A 2 +
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,, t) H Y. XX' = 0
E

X.)4' . = 0o Y . X. X' . = (Y.X) . (Y.X)' = 0

Y.X = 0. Nhtt vay Y e M la khOng gian nghiem cim
phuong trinh t„) . X = 0 to do dim W=n—cln— dim M
= rang X = dim V.
Do do W = V.
la the vec to cot. (-Ala ma tran X hang r.
2.2. Gia su A„ ...,
C6 th6 gia thi6t A„ (lac lap tuy6n tinh. Khi do cac A i ; r
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 , ..., e n } cern C", to ca u(e) = e,„„ Vay A = (ad la ma trait cua u
trong co so {e„ thi

111
{1 i (j)
au =
o
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 e n : v(e i )= Eb n a . Khi do AB = BA a uV = vu
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

k o ), ma Pxo c R ko 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
B = r nen cac dinh there con chinh cap k > r dgu triet lieu Ck
= 0 voi moi k> r.

C. (-1,)" 5 , s < r.
Vey I B - I = (-P)"+ +
Doclop=n-s?.n-r.

k
2.46. DS. Ak = 21i cos
2
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.
1 G'
-A
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, 1)
- k 0
f



(-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 {e 1 , ..., e n } la cd s6; tn nhien cua Ca, tt
u(e) =e a . 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
(m k 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

a do Dk la da thile b•c m k dang:
I A - kin I= D I ... D„

x0 0 1
-



1 -X 0 0
0 1 -X
=
Dk =(— OnI k+I-P ( Ark
-




0 0 0 1-1.

= E lrk ()Lin k Nhu vay

A-xi n 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
+ + t o = O.


128



VOi it = a + (n-1) b, co 1 vec to rieng: t i = t 2 = ... = t„ = 1.
VS'.y c6 the than Cd sox, = (1, -1, 0 ... 0), ..., x 2 . 1 (1, 0, 0,..., -1)
va x n (1, 1, ..., 1) gem cac vec to rieng cita 1
2.51. HD. Xet dang aide ma tran: -




(XIn - AB A 'In 0' [ In 0
B A
)Lin
(




0 AIn i B . In, In XIn - BA,
O
2


a) Ta có det (XIn - AB) . A n = A" det (Mn - BA).

+) N a3i X # 0, Ta có det (AB - Mn) = det (BA - Mn) nen AB
va. BA c6 cling gin tri rieng khac khong.
+) Vol X = 0, do dot AB = det BA, nen no cling cO nghiem
chung X = 0.
b) Do det (AB - Mn) = det (BA - Mn) vdi moi A, nen hai da
thnc dac trung eim AB va cna BA trung nhau.
2.52. Xet cci so (6,, ..., e n } ciia V, trong do ma tran f duac tao
heti m khung Giooc clang Ung vdi m gia tri rieng phan hi@ A l , ...At m .
0
(2`k
1 Xk
Ak = 1 ' bang dim/Kt)
, cap Gila Ak

0 1 X i
1, thi
\ay AI co phAn tit cheo Wang 0 vdi mm N, digu de trod gia
thigt. Vi \ray n k = 1 moi k. Nghia la ma Pearl cim f trong cd sd
(or e„.} co dang then, hay {e l , ..., e n } 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
a MS = 0. Trude het to sit dung be
tr6n M e Mat (n, K), M 0
de chung minh bai than.
de tren
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, SA 2 = - 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 2 S , SA k -SA 2


A l =j-k+S -1 .AS)
2S.A 1 = S.A + A.S
2S.A 2 =SA -A.S (A -5 1 AS)
A2 =
{


2

Vi A 1 . S = S.A 1 nen to he thiic tren to co
AS + S'A. 5 2 = SA + AS to do AS 2 = S 2 A.

130
mm A e Mat (n, 111)
VOi 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 = x k + + + a k 2 x + a k , k21,
va p(s) = sk a i sk- , = a k _,S + a k . In= O. Voi S # O. Ta nhgn
they k> 1, vi neu k = 1 thi P(S) = S + a, . I„ = 0

S = a, . I„; do S suy bien nen S = 0.

Ta lai thely a k = 0, vi neal a k # 0 thi S kha nghich, trai gia
thiea. Nhu vay:

S (S 1 ' + S k22 + ± at.; - 1 0= 0

+ ad1 1-1 +
DM M = , I„ # 0, to co SM = MS = 0.
2.54. a) DO (ang

b) Nhan they V. n W m = {0}. That vay, gia se x e V„, n vv„,
[moo = 0 va co y E E de fth(y) = x, tit do el(y) = 0 yeV 2m = 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.

dim (Kerf) = n-1.
2.58. HD: rang f = I

e,J la co sa cua Kerf xet - f(e l ) = ale,
Chen e, e Kerf va
+ a 2 e 2 + ... + a ue„, phan biet hai truong hop a, = 0 va a, x 0.

2.59. He tinic Ig k — 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.
g k+1 gk.Eg = k.gk va
ta suy ra
Gia sit co f.gk — gk.f =
gigk gickf.f = k.gk:





e- ,
Tit do f.gk± 1 ,7 g'+' f + g (fgk - l- f) g = 2k.g k .
, .




132
g k•2 nen: fel
= (k-1)
Do gia thiCi guy nap: gk-' —




(k-i) g k = 2k.gk
g k+1 , fkti

(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 + + a o la da thdc tot lieu
cua g; nghia la da thtic kluic khbng, cd bac nh6 nhet sao cho
P(g) = 0. Khi viol he thug (2) vdi k = 0,1, P to nhan dude:

f 0 P(g) — P(g) f P' (g). d do 13 ' la dao ham dm. P. VI P(g)= 0
nen P'(g) = 0, trai vdi gia thiet P la da thne t6i tint' clan g.

Chi) y: CO the nhan xet rang khong the ton tai cac tq d8ng
ceu f, g thea man fog-g0f= id vi trace (f ug-go I) = 0, trong
khi trace(Id) = n = dimV # 0.




133
Ch uang III
DANG TOAN PHUONG - KHONG GIAN VEC TO
OCLIT VA KHONG GIAN VEC TO UNITA

A. TOM TAT lit THUYET

DANG SONG TUYEN TINH DOI XUNG
VA DANG TOAN PHUONG
1. Dinh nghia

Gia sit V la khong gian vele to trial truang so th0c Wit
dang song tuyeal tinh xac dinh tren V la mat tinh xa

0:VxV—>R
(x, y) H) 0 (x, y)
sao cho vdi 662 kjc x, y, z e V va 7. e 7f& ta ca:

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

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

Neu vdi moi x, y V ta co 0(x, y) = 0(y, x) thi 0 (bloc gyi la
e
dth

Ngu vdi moi x V ta 66 0(x, x) = 0 till 0 dmic gni la phOn cl6i
e
xung.

134
2. Bieu thik toa do cita clang song tuyen tinh

Gia sn dim V = n, e = (e) = 1, 2, n la cd sa cila V. Anh xa
song tuy6n tinh B hohn toan (Liao the dinh bai ma trail A = (a 0 ),
a do a i , = 0 (0,, e i ).

Khi do voi x= , y= ta co 0 (x, Y)= Lao
y e„, xi Y1
,



i= 1
a do X, Y la
.A.Y,
hay yiet dual (bong ma tran 0(x, y) = the
ma tran cac tya dO cua x, y trong co sa (e i ).
r neat
n la mOt cd sa khac ciaa V ma
Gia s& e = (E,), j = 1, 2,

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 y i , thi


H(4= Za ii x i yi a do (x„ x x ) va (y 1 , yd IA toa dO cam x va

y Wong N6u trong 100 cd sä nao do a = (e), dang than

phudng H co dang Ii(4= D i x?, LW cd sa a = (th dime goi la cd

sd chinh tac d6i vdi H, va to not trong cd so e = (th, H co dang
chinh the. Ta cling not cd sa e = (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
a
sa a = trong co stl do, H co dang chinh tac.
(0




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 V 2 thi
a n6i V la t6ng trttc ti6p trtIc giao cern V, va V2 (d6i vei 0) va ki

riOu la V =V, ® V 2 . 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 V o 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 V o , V,, Vs
V = V. CS V. ff) V o 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 (A k 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 Ak %
vdi k than va det Ak < 0 vdi k Le.
Chu 9: Khi H có ma tran dti). 'clang A, n6u H )(lc dinh duct
to cling noi ma trail A 'the dinh ducing.

§ 2 KH6NG WAN VEC TO OCLIT

1. Dinh nghia

Cho E la khong gian vec to tren truiing s6 tlnic R. Ta
mat tich ve hudng a tren E la mkt anh xa song tuy6n tinh,
xilng va xac dinh dining lien E, ki hiou ban hay (. , .), ngh
la to co: < >; E x E R la anh xa song tuy6n tinh
thOa man (x, x) > 0 vdi moi x 1E va = 0 suy ra x = 0.
E


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

la chudn elm x e 1E, kr hiku ricfi ,
a) Ta goi s6 thcic khong
x, >

138
Ta có halt clang theic sau, goi la bat clang these Cos Bunhiacopski:

2 < N 2 MYM 2 •

b) Hai vec td x, y &tee goi IA true giao vol nhau n6u = 0.
Khi do to co:

)1 2 = D1 M 2 M 2 •

3. Cd sa trip chua'n trong khong gian vec to dclit
hitu han chi6u

Gia sit E IA khong gian vec to delft n chigu. it vec td e l , e 2 , ...,e„
1i=j
dupe goi la co so true chuitn cUa lE neu =
0i#j

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

Chung minh: gia sti la„ a , i l3 mot co sa nao do dm
khong gian vec td (kilt E. Khi do co the ray dung mot en sa true
chuan { e,, e 2 , ...e„} nlut sau:



co
a do = et, - e l -
e 3 = vei e 2 = e9> e2
Mg3 M



139
n-1
= , a do a n =a n -I< a n ,e k >e k .
d
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 , , e k } trimg vat khong gian sinh bai vat (a l , a k } vi
moi k = 1, ..., n.

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

vdi = (x, N6u co y =E y i e i thi = Ex i y i .
1=1
Gia s& F la khfing gian vac to con eim khong gian yea t
Oclit E. We to a E E goi lk true giao vdi F neru = 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 F i = 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 F 1 .

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 W i 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
t 2 + 2t 11 + 0 vdi moi t e Tit do suy ra
I 12 < . Ta co dieu phai chting minh.
.

c) Voi moi x, y thuec khong gian tree to Unita U, to co:

il x+ Yll = 11x11+11.
3. Toan ter tuy6n firth tit lien hdp
to (tit &Mg eau) lien hop
a) Khdi niem Loin

Cho f la to deng cau cem kheng gian vac td Unita U. Tt
dOng eau g cem U dtioc goi la lien hop vdi 1, n6u yea moi x, y e U,
to có = .
b) Dinh 5i: MOi mot tv deng cau cUa khong gian vec tc
Unita co duy nhlt met tit deng cau lien hop.
c) Tinh chat cua tit deng cau lien hop. KY hieu la to tieing
call lien hop dm 1. Ta co
1°) Id* = Id.
2°) (f + f* + g*
3) (k g )* n. g*

4) (f*)* = f
5) (g•f)* = r . g*
144
d) Ti doting can f dna khong gian vec to Unita U die goi la
td lien hop, nett f = f*.

f la t-L7 d6ng cdu cua khong gian vec to
e) Dinh lj: Gin sii
Unita U. Khi do to có f = f, + i fa , ado f, va fa la cac tg d6ng cdu
WI lien hop, &too goi tticing nag la (than thnc va ph'Un a() cda td
d6ng cdu I

Chiing minh:

Goi PIA td d6ng cdu lien hop end f.

ff
Wit f
2 =-i(f-f*)
Khi do = , = fa va f = f, + if,

g) Dinh Gia sd A la ma tran ciaa tti d6ng cdu f e End(V)
trong mot cd sa true chud'n cda V, a do V la khong gian vec to
Unita. Khi do f la tti (long cdu tti lien hop khi va chi khi A t =:(c

Chti 9: Ma tran phirc A co tinh chdt A t =A throe goi la ma
trdn Hecmit hay ma trail tn lion hop.

Cdc gia riling cita tst d6ng cdu W lien hop
h) Dinh lj:
dell tilde.

GM sd U la khong gian vec to Unita, f e End(U)
Chung minh:
la tq doing cdu td lien hop cua U, x e U la mat vec to riling cda f
nng vdi gia tri rintralt
Ta c6 f(x) = Xx,
= = X =
= = = 1r

145
Tit do: - X) = 0. Do > 0 nen I = X hay X thee.

i) Dinh 0: Cae vec td rieng Ung vat ode gia tri rieng phan
biet caa met to deng au tai lien hdp la true giao vdi nhau.
°ding minh:
Gia sit f la met tp deng au W lien hdp caa 'cluing gian
Unita U; x, y la hai vec to rieng Ung voi hai gia tri rieng phan
biet X 1 , X 2 .
f(y) = X 2 y
Ta co f(x) =
= = X i =
= = F
E x
la anh xa song tuygn tinh phan del xfing.

(E, It) la khong gian cac dang song tuyin tinh
2) Ki hieu
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, y(y, x))
2

a(x, = 2 y) - 9(Y, 9)

kieI m 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) = - 4 1 (3', = - W(x,
suy ra w(x, y)=0= = 0 to do s = s' va a = a'.

2) Ta bigt rang .2'2 (E, R) clang cku vdi khong gian cac ma
trAn vuong cap n tren K, (n = dim E). Do do dim 9z (E, K) = n 2 .
Vi S (ttiong ung A) dAng caIu vdi khong gian cox ma tram doi
xiing (phan xiing) cap n. Do do
n(n +1) . n(n -1)
dim S - , dim A =
2 2
Vi du 3.2: Gia sit E va F la hai khong gian vec to tren
trueing se" that K. Dang song tuy6n tinh f tren E x IF (Woe g9i la
suy bi6n trai (phai) netu có x e E, x # 0 (Wong ung y e ]F, y x 0)
sao cho f(x, y) = 0 van moi y e ]F (Wong ung f(x, y) = 0 vdi moi
147
x e E). f &roc g9i la kheng suy bign ngu n6 khong suy bin trai
va khong suy bign phai.
Oiling minh rang:

1) Ngu f kh6ng suy bign trai va F hi u han chigu thi E cling
Mtn han chigu va dim ]E < dim F.

2) Neu f khong suy bign phi va E huu han chieu thi F
cling huu han chigu va dimF < dimE.
3) Ngu f kheng suy bin va met trong hai khong gian ]E,
có s6 chigu Min han, thi khong gian kia cling co s6 chi6u hilu
han ve. dim E = dim F.

Lo gidi:
1) Ta chang minh Wang phan chetng. Gia sit f kheng suy bign
trai, (y o y z , y o } la met cd set cua F va dim E > n. Khi do trong
E et) he doe lap tuy6n anh gem n + 1 vac to {x„ x o , x o , x ozz }.
Dat a o = yeti 1 < i < n+1, 1 < j < n.
He thuan nhal:
n+1
=0


do s6 8n nhigu hon s6 plutdng trinh nen co nghiem khong tam
n+1
Khi do x„ = Ec i x i la vac to khac kheng cua
11111611g C„ +1 ).
(C„


E ma f(x o , = 0 vol nail j = 1, ..., n. Vi 1.Y1, --= y i j 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.
E. do g (x o , x o ) # 0
Ta hay °hung minh g doi x3ng. Vin m6i x E


nen co a e Yb de g (x, xo ) = a. g (xo , x o) . Khi do g (x - a x o , xo) = 0.
TIT gia thi6t suy ra g (x o , x - ax o ) = 0, do do g(x o , x) = g(x o , ax o ) =
a g(x o , x o ) = 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, x o )
hay g (x, y - a x o ) = 0 = g (y - a x o , x).
g(y, x) = a g(x o , x) = a g(x, x o ) = g(x, y).

Tug:Mg -St neu g(x o , y) z 0 thi to cang c6 g(x, y) = g(y, x).
Cual cung, gia sit rang g(x, x o ) = g(x o , y) = 0, khi
g(x, y) = g(x, y + x o ) va g(y, x) = g(y + x o , x).

Ta c6 g(x o , y + x 0 ) = g(x o , x o ) # 0 nen g(x, y) = a g (x o , y + x o )
= a g(x o , x o ) = g(x, Y xo)

Suy ra g (x - a x o , y + x o ) = 0
g(y + x o , x - a x o ) = 0

149
g (y + x0, x) = a g (y + x o , x 0 )
= a g(x„, x 0 ) = 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,, x 2 , x k la nhung vec td mia 1E. Dal a id = xi ), 1 j s k.
Ta goi dinh Ulric det (a 1 ) la dinh thuc Cram (Ma cac vac td x,,
x k va ki hiOu la Gr x k ).

Chang minh rang Gr (x i , x k ) 0 va Gr (x 1 , . x k ) = 0 khi
va chi khi x l , x k phu tha0c tuyeln tinh.
Lidi brick

Ta chgng to rang ngu x i , x k phu thuoc tuyan tinh thi
xk) = 0, con ngu
Gr(x l , , x k clOc lap tuygn tinh thi
Gr(x l , x k ) .> 0.
Gia su x,, x k phu thul)c tuy6n tinh, th6 thi co vac. td
(2 < r 5 k) bigu thi tuygn tinh qua x,, x j+ : x = a„ x i + +
.



cc„,

Khi do a ji = u(x r , = a, a lj + a 2 a ji + + 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 , x k ) = det (ad k = 0.
Gia sr) x„ x k d'Oc lap tuyen anh. The thi fx,, la cd
sa cua khong gian con F via E sinh bai he {x 1 , x k } va ma trail
(ad k la ma trail clic( (p i = (p1 F doi vdi cd sä do. Vi xac dinh


150
during nen theo Binh lY Sylvester, to co det (a, i ) k > 0 nghia la
> 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
co ma tran trong co sa' to
la phep bign del tuygn t;nh trong
la A. Chung minh rang:
nhien
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 = (µX t . Y)` =N. Y` . X.
to do (X - . YtX = 0 YL . X = 0

2) Gia sit X + in la nghtem dac trung cim A. The thi ten tai
vee td cot (phitc) X + iY trong do X. Y la nhUng vac to cot that
khong dong that bang khong

151
A. (X + Y) = (7. + ip) (X + iY).
6 day i la don vi ao.

So sanh cac phan Uwe va pfin no a ca h ai v6, ta ducre
AX = XX -
(1)
AY = XY + p X. (2)
Tit (1) va (2) ta co:

VA X = Y`X - p Y'.Y
X`AY=i.- p
Nhung Yt. A . X= (XL. A Y)L nen ta có:
AY`.X-pYt .Y=XYLX+gr.X

p (XL . X + . Y) = 0. VI X, Y khong dang that bang khong, nen
r. X + Y.Y>Oviy*yn=0,docloA,+ip=X
e R.
3) Gia sit A la nghiem (lac trung cith A, theo tren X
e
Khi do co vec td cot X = 0 de' AX = XX, suy ra Xt.A.X = Vi
X = 0 nen Xt.X > 0. TU do, ndu A the dinh (lacing thi X`AX > 0
nen X > 0. Ndu A the dinh am thi XtAX < 0 nen X < 0.
Vi du 3.6. Cho A la ma tran thong tithe ddi ximg cap
u e End (R") có ma trgn A trong co sa tti ten. Chung minh
rang ndu X lit vec td khac khong tha R", thi ton tai vac to thong
Y cua u thuOc khong gian con sinh bai X, AX, A 2 X, A" - ')C
LaPi

AX, A"X phu thutic tuydn tinh, nen co mot vec to
bik thi tuyin tinh qua cac the to dung bathe no. Gia sit k ad
152
h6 nhat de AkX bleu thi tuygn tinh qua X, AX, Ak 'X. Khi
0 tan tai the s6 thuc b k ., ..., b o dg

AkX + b k , Akd + + b i AX +13 0 X = O.

Gia sit (x - p 1 ) (x - i.t k ) la &tan tich cim da tilde

xk + +...+b o thanh the nhan tit tuy5n tinh, vdi g 1 , g k E C.

Khi
0 = (A k + b k ., Ak - ' + + b o I) X = (A - ... (A - u k I) X
hay (A - = 0 vol Y = (A -1.1 2 I)... (A - ti k I)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 td
igng cua u. HO nhtt thg vdi s = 1 lam tan tai theo vi du 3.5.


153
Ta cheing to rang có vec td rieng X„, trip giao vet cac
i = 1, 2, s. That tray, gia sat X x 0 la mOt vec td true giao
s). Vei mai i= 1,
Xi (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 X i . Theo kat qua trong vi du 3.6, co vec
rieng X s , 1 cern thueic khong gian con sinh bei X, AX, A" -
Nhung cac tree to AIX true giao \el cac X, (i = 1, s) vi tray 3
clang true giao yea Nhtt tray trong co X„ X„ trip g
gem toan vec to tieing ciaa u.

1
X• thi he Y 1 ,
Dat - la he cac vec to rie
Ya
.X,
sao choY,` . Y, = 8,, ki hiOu Kronecker). Vi tray ngu dal
IA ma tran ma cam cot la Y„ thi P IA ma trait trvc giao.
Mat khac, AX ; = 3,X ; nen AY, = X ; lc ti/ do . A. Y ;
V, = X, va vie j thi Y,' A Y, = X ; Y i t Y ; = O.

Do do PAP=A trong do A la ma tren chdo yea cac Olen
..., X. Vi P =
eh& a nen A thing dang vdi A, suy
-
I A - XIn I = I = (X, - X) •-• -

Pu do suy ra 3.„ chinh la tat ca Cite nghiem (lac tn./
cUa A, Ire ca bOi. ,

Cugi clang, do p AP = suy ra A the Binh decing (am)
A

va chi khi A xac dinh duong (am) ngl â a. IA khi va chi khi 3 1 ,
dtiOng (am).

154
Vi du 3.8: Cho A, B la hai ma tran that d6i xiing cap n, hen
nem B the dinh during. Chung mink rang ten 41 ma teen khong
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'e tu 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
X* X. Vi X x 0,
v6 hai clang thtic tren, to co X*A* A X =
nen X* X x 0, vi A* A = In nen:
h.. X*X (i. . - 1) X*X = 0
X*X =
.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 1
det (- 1 - A*) = 0 det (- I - A*) = 0

1
det (A - — 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(e i ) = -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 _ cts r _fri _ _ co o_ s.
Ta say dung cd se {v,} (i = 1, n) g6m nhung vec td clan
Intang cim E nhu sau:
v, = e, + e r ,„ 1 0 va g > 0. Ta dua g v2 clang chua'n tAc
= 2
, y i = Eq 33 x i (1 = 1, 2, ..., n), det (n o ) # 0. KM do
E y3
i=1
i =1


)=E (f,
g
i=1

\
2
= Ecn i q ik x 3 x k , nen (f,y 3 )= Ea ik kq u x i lk k x k ),
Nhung


do f = Ia ik x i x 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
(xi)
f xac dinh dudng nen (f, 37, 2 ) > 0. Do \Tay (f, g) > 0.



159
Vi du 3.13: Chung to rang trong kh6ng gian vec to Odclit
chieu, c6 the tim dudc ho n vec to don vi u l , u 2 , sao cho di
vai tat ca cap s6nguyen phan biet i, j, yen to - 11 j cling la of
to don vi. Ta {tat x= n 1+1 " . Hay tim goe giva vec to x - 1
In,
1
va x - U2 .


Ldi

Gth six {e 1 } (i = I, n) la mot co sa bye chuSn °Cm khan
gian vec to dclit E. Ta xay dung hen vec to {u„ u„) nhu sat
he (ti t , u 2 ) thuOc khong gian sinh bai {e„ e a} va {0, u,, u 2 } lar
thanh tam gine deu, nghia la ria l 11=1111 2 E1=1 va 14.u 2 = Ta b
sung tr, nhu sau:
1
u 3 = + ). 2 u 2 + X 3 e 3 , u 3 u 1 = 11 3 . 1.1 2 = 2 , u 3 2 = I.


Tit do co — = X, += 1 Xi
1 X? 1
2
A1 X2 - — .
3
2 ' 2 2 + A2

X 3 = + IF
Va vi Al + x.22 + 4 + Xi X2 = 1 —
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, n-1 va u, 1 2 = 1, to
2
1
tim duac = 1 2 = = A n _ = — va I n n +1N hit vay to da

2n
160



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).
1
+ u 2 +...+ ti n ), goi 01a gee gala x-u, va x-u 2 .
Bat x -n
+1
Ta ce 1 = (u 9 - 110 2 = [(x-u 1 ) - (x-u 2 )] 2 = (x-u 1 ) 2 + (x-u 2 ) 2
2 ) (x-u 2 ); -2(xu
1
tu x + -
va (x-1.1 1 ) 2 = (x-u 2 )• =
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 sau:
Truisc
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>.
de: Net cd sa true chua'n te,, e 2 , ..., e a } trong
Ghiing minh
V. ret phdn tit h e V, h = Eh k e k , a do h k = fk). Gia sii
k=1

Ex k k = xk hk =
V, Ta co f(x)=
x= EXic ek e
k=1
k =1 k=1
n n
< Ix k e k , Eh k e k > = < x,h > .
k=1
X=1
161
Ta chfing minh phan tit h la duy nhat. Gia sii co 11 1 , h 2 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 h 2 = 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, y 1 ) + B (x, y 2 ) = +
= 0 vdi moi x V. Do v'OY
c
A(Y)+372) = A(37 1) + A(Y2)-
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, xiing xac
Binh during cap n.

Chung minh rang det (A + B)?, det A + det B.
Lai gicii:
Try& Mt ta giai bai toan trong twang hop rieng, A = In la
ma tran ddn vi. Vi B la ma tran doi 'ding xac climb during nen
162
ton tai ma trail truc giao C sao cho C 4 B C c6 clang cheo ma cac
phan tii tren duang cheo chinh la cac gia tri cna B.
Vi vay det (I + B) = det C - ' (I + B). C= det (I + C -1 B C)
= (1 + ... (1 + a 1 + X, ... X„ = det I + det B.
Ret truong hop t6ng gnat, vi ma tran A xac:dinh duong,
nen c6 ma tran D sao cho A = D 2 , a do D the dinhducmg. That
vay, có ma trail D, de2 D,' . A D, = A co clang cheo, D, la ma tit)
true giao (vi du 3.7) A la ma tran cheo ma cac pith) tit tit)
cluong cheo chfnh la cac gia tri rieng cim A: p„ n„ > 0. Dat P
la ma tran cheo mA cac phan to trot) during cheo chfnh la
P 2 = A va A = D I A D l d = D,P 1 D1 = D I PD1 1 .
D,PD 1 -1 = D 2 , vei D = D, . P . D, - ' = D,PD,t, D la ma tran d61
xang.
de A + B = D 2 + B D (I + D - ' B D - ') D
Td .




B D -1 ).
Vi vay det (A + B) = (det D) 2 . dot (I +
Do D - ' B del sung the dinh dming nen set dung k6t qua
,



trot), to of):
Det (A + B) Y. (det D) 2 (1 + det D - ' B . D - ') =
= det D 1 + 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

al as j
a) Gial sit X = = a, X, + a2 X2 + a 3 X3 + ai X4.
as •4

do {X I , X 4 } la cd so to nhien dm M.

f(X) = a,a, - a 2 a s . Day la mot dang than phudng tren M.
Trong cd so tg nhien cem M, f có ma tran
1
0 0 0
2
0 0 0
2
A=
0-1 0 0
2
1
0 00
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 . X d ) = 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 2 4
3
2 3-i -2 0
A= 1
-2 - i
3
12-i
0
4



165
Zd gidi:

23 4 '



1 12 3 -2 0
Ta co A = A, - a do A l =
3 -2 0 1
40 1 2

Ma trail A I la ma trAn thvc dai xting cap 4, nen tat ca 4 gin
tri rieng dm A, deo thvc, (xem vi du 3.5), nghia IA da thfic POO
= det (A, - ce, the nghiOm deu thuc, ter do P (i)* 0, nghia lA
det Ax 0.
Vi du 3.18

a) cis. sit {e h e 2 , e 3 } IA mot co sa cim R 3 . Cho clang Loan
phttong Q tren Q (x) = x 1 2 + x 2 2 + x 3 2 - x 1 x 2 - x 2 x 3 ; x = (x 1 , x 2 ,
x 2 ) la toa dO cua x trong co sa {e 1 , e 2 , e 3 }.

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 (R 2 ), co ma trap A trong co sa 0 2 , e 3 }:
'3 -1 1 '




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 , e 2 , e 3 } 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 R a (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, -- x 2 ) + ( 1 x3 - X 3 ) 2 +
2
1
x l = 2
' 1- 2
2


1
Hat x3
x2 =
1
x3 = --)r2


+ x3
X=
2
KM do x2

+ —x 3
2

e' a } ma
Tit do, xet cd sa {e' 1 , 0' 2 ,




167

el
e
e3
ez =
e 3 —e + .ne 2 +
2' '
thi trong eel sd fe'„ e' 2 , thdc cua clang town phdong Q eó
Bang Q x,, x 3 )- x 12 + x'; +

Nhtt vay {e' 1 , 8'3} la co so true ehuan.
b) Goi n la dang eve cua Q, khi do trong co sa e2, e2),
co ma tran H.

ife" chdng minh f la tv d6ng ciiu del xdng, ta phai chtIng
minh n(f(x), y) = n(x, f(y)) vdi moi x, y E R. Goi X, Y la ma tran
cot vac toa de) cua x, y, ta phai el-Ong minh:
(AX)'1-1.Y=Xt.H.A.Y
hayrA t .H.Y=Xt.H.A.Y
1
0
(3 0 1' 2
That Vey At H = -1 2 -1
2 2
1 1
0 3i
0 -- 1
2
3 -2 1 :`
= -2 3 -2 = H . A.
1 -2 3

Vi du 3.19. Cho V 11 khang gian vec td tren tnlang s6 thvc
R. Gin' su H i 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.
tat
c) Cho hai dang than phudng tren R2 3 trong cd so chinh
có clang:
2
\
H i (X)= Xj. +54 +144 -F2X I X9+2XIX3+10X2X3

H2(X)= 4x 2 +104 + 6x,x 3 +14x 2 x 3
2

sa trong do ea H, va H 2 dgu co dang chinh
Hay tim mot co
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
Ex r Nhti \Tay, trong cd so {v 1 , Vi co ca hai
=
i=1
clang toan phudng da cho dgu co clang chinh tilt.

b) Kid ca hai dang dgu khong xac dinh &king, thi co th6
khong ton tai mot cd so de? ca hai dang toan phudng dgu co
vdi cat toa clO (x, y)
dang chinh the. Chang han xOt trong
, e 2 }. Hai dang toan phudng H 1 (x, y) = x 2 ,
tae
trong co so chinh
169





H 2 (x, y) = xy. GM a co mot co so te l. , dg' trong cd sã do H, va
H 2 dgu có ding chinh tac.

= e ll e l +C 21 e 2
GM ad: Goi (x', y') lk toa dO trong co sä
62 rc + C22 62
IX --c'Cii11+ Cl2
Khi do
).
r'
(y= C21x'+ C22

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„ E 2 ) thi CI, C22 +
Cu Ti12
C.2 1 C 1 2 = 0. Tit do suy ra ma tran C = 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 + x 2 + x % ) 2 + 4 (x 2 + x 2 ) 2 + 9x 3 2 .

1
xl = Y1 2 Y2
{YI = X I + x 2 + x 3
1 1
Pat bign mai y 2 = 2x 2 + 2x 3 hay x2 = Y3

3x 3
Y3 = 1
X3 =
3 y3

thi voi die toa dO (y„ y 2 , ..., y 3 ), H, co dang chinh tac:

2 2
H1 = + Y2 + y3 (2)
Trong ho toa do, do, I-I 2 co bia thfic:

H2 = 17 2 2 + (3)
2 13373


170
'o 0 1'
(4)
dang A = 0 1 0
Ma tran maa H2 CO

,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 1.
0 0


1 1
=- r zi + z2
yl
V2 a/2
(6)
toa (.10
Dung phep Y2 = 3
1
1
+ 22
Y3="'


Trong do ca hai dang town Oaring se c6 clang chinh tAc.
9
9
9
H, (z , z 2 z 3 ) = + + + z3 (7)
2
2 2
(8)
H2 (Z , z2, 23) — - +22 +22

1 1
1
= --,z, +
x
2 z3
.5 z2 —
1/2 '
1 1
Tii (1) va (6) ta co + lz 3
x2 (9)
2z
2
3.h z1
1 1
1+
x a — z2
3-,,&
3 ,5
ce dang
co sa trong K 3 , trong do H, va H2
HS {el, e2, C31 7a

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

171
1 + 1
1
el 3 ea
e2
"-
2




1 $ 1 ,a 1 i)

3-,N -2 + 3I_ -3
1 1
= +,7, 0 2

Chu if: Da dila dang thai hai clang town phydng H„ H2 trong
d6 H, xac dinh diking v6 dang chinh ta c, ta lam cac btafic eau:
l

Bieck 1: Dila dang toan phtidng H, v6 dang cheo, nghla IA
tim mat co sa c 2 , ...c„} true chuan dai vdi dang cue 0, cUa H„

aide 2: Vik biau iliac maaja 2 trong cd sa fe l , c 2 ,

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„ c rib e„+„fri, •.., e„I ma

f(e,) = 1, laiap

172
f(e,) = -1, p+1 < j 5 p+q
1> p+q.
f(e / ) = 0
Bat F, la khong gian con sinh bai cle vac to {e„,, e pop .
•.., e n ), thi f/ F, o 0. Nhung dim F n F, = dim F + dim Ft -
dim (F + F 1 ) nen dim Fri F l >p+ (n-p) - n = 0, matt thuan vdi
gia thiat f/F > 0, f/F, o 0.
Gia se dim E l = n - 1 va p', q' la cac chi s6 dicing quart tinh
va chi s6 am quail tinh maa f han the troll E l . Ki hieu F2 la
khOng gian sinh bai le l , ..., e p ) tren d6 f xac Binh ducing.
Ta co dim F2 (1 E, z p - 1, nhung dim (F 2 n El) o Tit de
p - 1 5 p', nhu vay = p-1 hoac p'=q.
Tuong to q' = q-1 hoac q' = q.

Ta nhan thgy ea ban trudng hop


I P r = P -1
P‘= P IP P = 1
q -1 kir= q -1
k r = '11
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„ 4 1, thi p' = p, q = q.
b) neu q > 1, thi vdi E l sinh bai he fe l , e p , e p+ 2, to co
le 1 = P
q = q-1
c) neat p > 1, chon E, sinh bai e„) to &lac p' = p-1,
q'=q .



173
d) n6u co p z 1, q 1, chnn E, sinh bai le, + e 2 , e,,
e n+2 , ... e n } thi p' = p-1, q' = q-1.

C - BAI TAP

3.1. Gia sii H la clang than phuong tren khong gian vec tc
R 3 . 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„ x 2 , x 3 ) = 4 -24 -Fx .g +2x,x 9 +4x 1 x 3 +2x 2 x 3
b) H (x 1 , x 2 , x 3 ) = x,x 2 + x,x 3 + x 2 x 3 .
c) H (x„ x 2 , x 3 ) = 44+,,..3+4-4x 1 x 2 +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.,x 3

b) 24 +24 +4 +2Xx i x, +6x 2 x 3 +2x,x 3

c) 4 +24 +21/4,4 -2x,x 2 +6x 1 x 3 -2x 2 x 3 .
2

3.3. Dua dang than phtiong sau v6 dang chinh tAc:
11 9

a) Q(x l , x„)= Dx i t + Ex i x j , 1 3, va t&t ca can
3.21. Gia sit A = (ay) e Mat (n, R),
ph&n tit a n # 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 E 2 la khong gian vec to dent hai chigu. Tu
citing din dot xitng f e End (E 2 ) trong co sa trite chuan fe l , e 2} co
1
2
ma tran A = Hay tim co sa true chuan trong do
1
2 ,




ma tran cha f co clang cheo.

3.23. Hay dua ma trail dot xiing sau vg clang cheo nha ma
tran trhe giao

'00
B= 010
10 0

3.24. Trong khong gian vec to R 3 , cho hai clang town
phttong:

= 2x 12 - 24 - 3x 3 -10x 2 x 3 +2x 1 x 3
2

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.
3.25. Gra. sit' E la khong gian vet to aclit vdi tich hitting
< , >. Tv ddng eau f E End (E) duoc goi la phan doei 'ding nefu
vdi moi e, y E E, to co = - .
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.

SO
D. NCTONO DAN HOC DAP
32
9

1-
3.1. a) H = (3, + x 2 + 230 2 - [ 3-x 2 + 73x 3 ) -(a3 3
1


y i = +x 2 +2x 3

I
+
fat 3
2
2 =-



3x3
Y3




179
2
Taco H= 3

-3 1
b) H -37F
2
_2 2
c) H = 32 Y3

3.2.
a) Khong co 1. nao th6a man
b) Kh6ng nao th6a man
c) A> 13.
3.3.

Q-602 -i61 2)2 + 1 67 3)2 +---+ n+1 67 :32
a)
6 2n
1
That \ray, dat y, = x, + —(x 2 + x„), thi
2

2
--x,x ) ;2
3


Dat y 2 = x 2 + 1 (x 3 + + x n ), tu do

f
+7 137 4) + - Ex; +—x i 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) = a 2 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 + l t yx 0. Theo gia thiet f(a, a) = 0 k, 2 f(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 (n, c Mat (n, C). Vi A phan di51. ximg .
M:11.

len ma Iran iA Hemnit - Min vt ao). Ta co det (A - kin) = 0
let (iA - ixIn) = 0. Nhung mot nghiem da' c trung caa ma tran
iecmit den thuc. Tif do suy ra mkti agh*n dac trung cila ma trail A
a thuAn ao hac bang khong. Gia this cat nghiem khac khong
as da thud dee trung PA la: ja i , ..., ta t , -ia k , (cti E K, ai x 0).

Chi do PA N=


+ a ?)
PA W )
{0 vain >2k
Do do det A = P A (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 u 2 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 u 2
vdi u 2 (a) = Aa. Khi do u(a) va a la dec lap tuygn tinh trong V
That vay n'elk u(a) = -4 a, 4 e C thi u 2 (a) = u(4a) = 4 u(a) = 141 2
a=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:V c —>

[x] —> u,[x] = [u(x)]

Ta c6 u, IA anh xa nUa tuygn tinh, tit do u 1 2 la anh xa tuygr
tinh va u1. 2 [x] = [u 2 (x)]. VI vay, ki higu PIO va Pu, 2 la cac dz
thtc dac trung cliatt 2 c End (V) va u1 2 c End (V 1 ) ttiOng ling thi
Pu 2 (t) = - A.) 2 Pu 1 2 (t). Niu y lai IA nghigm cf.a da thug 4(
thing Pil l ', lap lai qua trinh tren to có (t - )0 2 la ink Gila Pu l a
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° —> C 11 , u (x) = Al; u la Anh xa n&a tuyen tinh;
Ta co Pu 2 (t) =
t 2 (x) = u(u00) = u (Al) = K Ax voi moi x e

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.
dO sn ,sk
- oak x
Pu2 (t) =(%14) a ' • 0.2 - 0'2
Gin)
muyen during, ..., Irk e R;

Q e R Q khong co nghiem thne
Vi Q khong có nghiem thile nen deg Q = n -(s, + s 2 + ...+ s k)
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 (X i + nsi z 0. N6u 7u z 0, thi
,




re rang (X,+1) 31 >0.Nhv vay Pu 2 (-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
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