Bai tap dai_so_tuyen_tinh

PHAN HUY PHIJ • NGUYEN DOAN TUAN
BAI TAP
DAI SO TUYEN TINH
NHA XUAT BAN HAI HOC QUOC GIA HA NOI

Chin trach nhiem xual bcin
Gicim doe: NGUYEN VAN THOA
Tong bien Op: NGUYEN THIEF N GIAP
Bien tap:

HUY CHU
DOAN 'MAN
NGOC QUYEN
Trinh bay Ilia:

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

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

rvikic LUC
Chubhg .1: DINH THOC - MA TRA:N 7
A - Tom tat ly thuyeet 7
§1. Phep th6 7
§ 2. Dinh thitc
§ 3. Ma tram 10
B - Vi dn 12
C - Bei tap 35
D HtiOng dein hoac clap so 43
Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH
• PHUGNG TRINH TUYEN TINH 57
A - TOrn tat ly thuyeet 57
§1. Kh8ng gian vec to 57
§2. Anh xa tuyeen tinh 61
§ 3. He phydng trinh tuy6n tinh 64
§4. Can true caa tai ding cku 67
B Vi dtt 71
C - Biti tap 96
§1. 'thong gian vec to va anh xa tuyeen tinh 96
§2. He pinking trinh tuy6n tinh 104
§3. Cau tit cna melt tu thing calu 106
D. Illidng sign ho(tc clap s6 110
5

§1. Khong gian vec td va anh xn tuyin tinh 11(
§ 2. He phudng trinh tuyeit tinh 12';
§3. Cau trite dm mot tg ang cau 12Z
Chtedng DANG TOAN PHUONG - KHONG GIAN VEC TO
OCLIT VA KHONG GIAN VEC TO UNITA 134
A. Tom Vitt 1t thuyeet 134
§1. Dang song tuy6n tinh aol xUng va dang town phuong 139
§ 2. Killing gian vec to gent 135
§3. Khong gian vec to Unita 142
B. Vi du 14E
C - Bai DM 174
D. Hitting dan hotic ditp so 179
Tai lieu them khan 192
6

Chuang 1
DINH THUG - MA TRAN
A - TOM TAT Lt THUYET
§1. PHEP THE
Met song anh o tit tap 11, 2,
met phep the bac n, ki hieu la
'1 2 3
n} len chinh no duet goi la

G
I
a 2 G
3
15 del a, = a(1), 02 = a(2),..., a„ = a(n).
Tap cac phep the bac n yeti phep nhan anh xa lap thanh
met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen
t3 cua nhom S„ bang n! = 1, 2... n.
Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich
the cem a n6u s6 - j) (a, - a) am. Phep the a &foe goi la than
ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6
nghich the ciaa a le.
1 neM s la phep the chan
Ki hieu sgna =
-1 net} a la phep th6 le
va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the
cling bac, thi sgn(a = sgn(a) . sgn( ).
Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„
•-• , ik doi mot khac nhau dr coo = 12, coo = i3, a(ic) = i1
7

va a(i) = i vdi moi i x i„ ik . Vong )(felt do dttoc ki hieu IA
ik). M9i phep th6 dau &tan tfch the thanh tfch nhung
yang xfch doe lap.
Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong
xfch ••• , ik) phan tfch chive thanh tfch 0 1 ,
§ 2. DINH THUG
I. Gia sit K IA met trueng (trong cuan sich nay to din yau
xet K la &Ong s6thvc K hoac truang s6 phitc C). Ma tran kidu
(m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim
m hang, n cet cac phan tit K, i = 1,m, j = 1,n. Tap cac ma
tran kidu (m, n) chive kf hieu M(m, n, R). Ma trail vuong cap n
IA ma tran co n dong, n cot. Tap cac ma trail vu8ng cap n vdi cac
phan tit thuoc truong K ki hiOu IA Mat(n, K).
2. Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, ..., n.
Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K
dude xac dinh nhu sau:
detA = zsgn(a)amo)
E Sn
3. Tinh eh& ceta Binh that
a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram
A, thi dinh auk cim no ddi da:u.
b) N6u them veo met dong (hoac met cot) cim ma tran A
met to hdp tuygn tinh cim nhUng thing (hoac nhung khac,
thi dinh auk khong thay ddi.
8

f an ail
de a21 +alci
‘a n„ a,,, + ani
all al; ...a1,„ all
= det a 21 21 +det
a21
—a1111/
an,
a2„
...a2 n
" S ' Ill " S IM /
c) Ngu mot Bong (hay mot phan tfch thanh tong, thi
dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6:
d) Cho A = (Ito) E Mat(n, K), thi = b) a do = aij &toe
goi la ma tran chuy6n vi cim A.
Ta co detA = detAt.
4. Cdch tinh dinh that
a) Cho ma tran A E Mat(n, K). Kf hi'911 Mi; la dinh that cua
ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot
thu j cut ma tram A vb. Aij = (-1)H M u clucic g9i la pha'n phu dai
s6cUa phgn to aii cna ma trait A. Ta có CAC tong thtic:
O ngu i k
det A ngu i = k
O ngu i x k
det A ngu i = k
Nhu fly detA = EamAki (k = 1, 2, ... n)
1=1
heat detA = Z
•
aikAik
/=1
9

CUT thac tit throe goi la cang thdc khai trim dinh tilde
theo (long hay theo cot.
b) Dinh 1ST Laplace
Cho ma Iran A = (a,J) c Mat(n, K). Vo; rn6i bQ
;2.••, ix), 1 s i, <12 < . <
ik
va Oh ik), 1 =11 < j2 < ••• <jk n, 1 s k< n, dat A. la11-11,
dinh thac caa ma tran vuong cap k nam d cae &nig i„ ik va cac
cot j,... jk cim ma tran A; M. la dinh thdc cita ma tran vuongh
cap (n - k) c6 dude bang each gach the clang thu ik va cac cot
thu j 1 , , jk caa ma trail A. Ta c6 kgt qua:
detA = ZED' Si k+31+

A.
ik
do j1.-- jk la k cot cgdinh. Tgng &toe lgy theo tat ca cac bQ ik)
sao cho < i2 < < ik Cong thdc troll dude goi la c8ng
thdc khai trim dinh thae theo k cot ji, 'along tV, to có gong
thdc khai trim theo k clang. Khi k = 1, to &roc gong tilde da not
trong muc a.
§ 3. MA TRANI
1. Ma trgn kigu (m, n) vgi cac phan tU tr8n trang K da dude
gidi thigu trong §2. Tap the ma tran kigu (m, n) vdi cac phan ti
tren tragng K dude ki hiQu la Mat(m, n, K) A E Mat(m, n K)
(Woe vigt A = (aii) i = 1, m ; j = 1, 2... n hay ro rang hon:
10

811 a19 aln

A= a,1 a99 aon

amt amt arnn,
2. Cac phep todn tren Mat(m, n,
Cho A = (a y ), B= (b,j ) thuOc Mat(m. n, K)
Ta có:
a) Ma tran C = (cg) a do cy = aii +
&toe goi la tong cua hai ma tran A va B va ki hien la A +B.
Ma tran D= (d,,) a do di; = aij -
dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B.
b) Vdi k E At, ma trail kA c8 cac phAn to la (kaii ) duoc goi
la tick cua ma tran A vdi ph&n td k cua trudng K.
c)Neu A = (aii ) c Mat(m, n, K) va
B = (bp() e Mat(n, p, K) thi
ma tran A . B E Mat(m, p, K) ma cac phAn tit &tele xac dinh INN
AB = (c,k), a do
e ik = Zaijbjk
5=1
&toe goi litich caa hai ma tram B ye. A.
Vol A, B e Mat(n, K), to có det(AB) = detA. detB.
11

d) Tap Mat(n, K) con ma tran yang cap n vdi phep toan
cOng lap thanh mot nhom giao hoan, con vdi phep Wan rang ma
tran va phep nhan ma trail lap thanh mat vanh khong giao
hodn, co don vi.
3. Hang ctia ma tran; Ma trim nghich ddo
Gig A E Mat(m, n, K), ta dinh nghia hang ciat ma tran A
la cap cao nhgt cua dinh thric con khgc khong rut ra W ma tran
A. KM A E Mat(n, K) va hang A = n (ta cling dung ki hi3u hang
A la rang A) thi ma tran A goi la khong suy bign, khi do
detA * 0 va ton tai duy nhgt ma tran B thuOc M(n, K)
A.B = B.A = I„; d do I. lit ma trail don vi. Ma tran B &roc goi 11
ma tran nghich dgo cna ma tran A va ki hi3u la A'.
Gig su A= (Au )la ma trail plw hpp cim ma tran A = (ad,
Ab la Olga Ow dal see mitt phgn ht aii; At la ma tran chuya'n
vi cua A . Khi do:
At .
detA
B- VI DTI
Vi cla 1.1. Xac dinh clgu rim cac phep th6 saw
a) a 11 2 3 4 5
2 3 5 4 1
b) 5=
r1 2 3 n n+1 n+2 211 2n+1
I
ll 4 7 ai-2 2 5
31
12

Lai gidi
a) Phan tick) a thanh tich cac chuydn tri:
=
(1 3 5 4 1
2 3 4 5)
= (1 2 3 5) = (1, 5) (1, 3) (1, 2)
2
(chi) 9 la pile)) nhtin cae chuydn tri dude thuc hign tii phai
sang trai nhu hap thanh cua the Anh xa).
Vay sgna = (-1)s = -1
Co the lam each khan: Cam nghich the cua a la (1, 5), (2, 5),
(3, 5), (4, 5), (3, 4).
Vay a có 5 nghich the nen sgna = -1.
b) Ta hay tinh sS nghich the cua boa)) vi (1, 4, 7... 3n-2, 2,
1 khong tham gia vao nghich the
4 tham gia yen 2 nghich the voi the s6 thing sau no.
7 tham gia van 4 nghich the.
3n - 2 tham gia vao 2(n - 1) nghich the voi the se dung sau no.
2 khong tham gia vao nghich the nao vdi the se dung sau no.
5 tham gia yen 1 nghich the voi the se dung sau n6.
S tham gia vao 2 nghich the vdi cae s6 dUng sau n6.
3n - 1 tham gia vim (n - 1) nghich the voi the s6 thing sau no.
Cae s6 3, 6, 9..., 3n khong tham gia vao nghich the nao voi
the s6 (hang sau
13

Vay co tat ca 2 + 4 + 2(n-1) + 1 + 2... + (n 1) - 3(n -1)n
2
(n-1 )n
nghich the trong hoot vi da neu va do d6 sgn S = (-1) 2
Khi n = 4k hoac n = 4k + 1 thi sgn 5 = 1
con neu n = 4k + 2 ho4 n = 4k + 3 thi sgn = -1.
Vi du 1.2
_ 1 2 3
Cho phep th'e' f - en dgu la (-1)1
312 fn
Hay gag dinh da"u dm:
a) 1-1
b) g = (In

2
fn-) • n )
Lift( gidi:
a) Vi sgn f. sgn = sgn (f.
= sgn(Id) = 1 non
sgn (e1)= sgn (f) = (-1)k
b) X4t phep the"a = 1 2
nj
n -1 . 1
thi g = f. a
Do gay sgn g = sgnf . sgn a.
n(n-1)
Nhung sgn a = (-1) 2 nen sgn g = (_])k+C;',
14

Vida 1.3
ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf
j) v6 ben trai Wring throng v6i viac dai cha car s6 i , j a clang
drah cna phep the.. Cling nhu vay, nhan mat phop th6 Nth ehuynn
trf (i, j) v6 been phai tunng during voi del eh?, I, j a dong tit 66a
phep th6.
Lo gidi
Gia sif a la
truong hop nhan
Gitisi2 a=
Theo d nh nghia
phop th6 cho j) la phep
ben trai tile la f = (i, j). a.
(3 n
1 2
(i, j) = (
9 2
_ (1 2a
ai
nj
n
)
chuy6n tri. Xet
Trunng hop nhan ben phai dude ant Wring fib
Vi dy 1.4. Cho f va g la hal phop th6cua n strtn nhien clAu tien.
a) Chung minh rang có the' cilia f va g bang khong qua (n- 1)
phop chuyan trf (nghia la ton tai k phop chuyan trf al , cr2, ak,
k 5.11 - 1 g = ak ak_,... a,. f).
b) Chling minh rang khOng tha giam bat s6 chuOn trf rah
trong cau a) titc la en the' chon f va g sao cho khong the dua f vd
g bang ft han n - 1 phop chuyan trf.
15

La gicii
a) Xet phep the g o f', phan tich g o e' thanh tfch cac vang
xich dOc lap T1 , Tp .
g o e' = ... T2 .Ti
Neu kf hiOu mi nt do, dai cart yang 'dell Ti thi

± m2 + =

rang mOt vong xfch (a l , a2, am) la mOt plop the
a cac s6 tv nhien Ui 1 den n sao cho a(a) = a 1+1 (i = m-1) va.
a(a„,) = a l, con a(l) = 1 nen 1 yen moi i = 1, .,., m. Vong xfch
(al , a2, u„) goi la ce do, dai m.
Ta da hiet rad yang xfch do, dai m deu phan tich duo
thanh m - 1 chuyen trf. Vi vay g o e' phan tfch duo. thanh 'Lich
caa i(mi -1)= n -p = k phep chuyen trf.
i=1
Nhungpa. 1 n-1:115n-1
Nhu vay g o f-1 = ak (s, - chuyen tri)
TV do g = ak 0 ak.,, ... 0 0 f, k n - 1
va cac a, la cac phep chuyen trf.
2 ...
njri
b) Cho g = la phep the ddng nhA
0.
( 1 2 3
nOa 1 2... n-1)
f ve g duo Wang it hdn n - 1 phep chuyen trf.
on f = . Ta se chUng to rang killing dua
16

'
WA met phep the" h = 1
1 2

ta not rang la
2 n
phan tit chinh quy ngu i. De" rang neat nhgn vao ben trai
cua h met chuygn Lri thi A:C. 1)111in tit chinh quy tang cling lAm la
met don vi. That way, ngu ngudc kg, cheng hen i, j la hai phan
tit kheng chinh quy cim h ma nob d6i dig hi voi hj ta Jai &tog
hai phign tu chinh guy (cum phep th6 m6i) th6 thi: h, < i. h, < j
nhang hj 2 i, h; j vti 1Y.
Do f chi co met phAn tei chinh quy. ye g co n phan tV chinh
quy, vi vgy khong thg clua f vg g bring it hon n - 1 phep chuygn tri.
Vi du 1.5
Chung minh rang vdi mei so k (0 <k < C;) t6n tai met
phep thg a e S„ co dung k nghich thg.
1231. giai
Ditch 1: Ta hay cluing minh met It& gug manh
Nigu a = (a,, la met hogn vi cda 1, 2, ... n va ace 1:
nghich thg, 0 < k < , thi có thg ct6i che hai phfin tri nao
do de thu ridge hog') vi c-3 co k + 1 nghich thg. That Nay, int&
hgt ta nhan thy rang negi > vdi moi i = 1, 2, ..., n-1 thi
coa nghich the'. Vi vay, do so' nghich th6 caa a la k < ,
nen ten tai i o dg oc ii) < ai0+1 ;
Xet hoan vi p = 0„) trong do 111 i = a, ngu i # i„, i„ + 1,
con p io = a;0+1 , pi+, =a,„ thi HI rang 13 co nhigu hon a met
nghich the". Nghia la s6 nghich th6 ciga p la k + 1.
17

Ta tha'y A= B. ca do
1
1 0
1
B= vi C=
11
t1
Nhan cot thu nhal ciia ma tran A vdi -k rdi cOng vac) cot
this k, ta dude:
1 -1 -2 ... -(n-1)
1 0 -1 - (n - 2)
1 0 0 .. - (n - 3)
1 0 0 0
det A =
Khai trio'n the() dung Ulu n, ta ea:
-1 -2 ... -(n -1)
-1
= (-1)"+1. (-1)"-'=1
Cdch 2.
ma detB =1, detC = 1 nen detA = detB. detC = 1.
Vi du 1.9. Hay tinh
cosa 1 0 0 0

1 2cosa 1

= 0 1 2cosa 0 0
2cosa 1

0 0 0 1 2cosa
21

Lai gidi:
Khai trim dinh thac Lheo cot cub" to co
D„ = 2cosa . - 17,1_2
De thay D, = cosa.
cosa 1
2cosa,
= 2coi2a - 1 = cos2a.
Gia sa D1 = cosia TM moi = 1, . k.
Ta có
Dk,I = 2cosa . Dk - Dk_ i
= 2.cosa . coska - cos(k -Da.
= (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a.
Nhn vay D„ = cosna
Vi do 1.10
Hay Dull
1
1 eP + e-(1)
a do the phan to tren &tang choo chinh bang nhau va band
eq) +e-9 ; the phan tit tren hai &tong xien Win nhat TM (Mang
the() chinh bang 1, con the phAn ta khac bang 0.
+ a-9 1 0
1 e" +e 1
A„ =
1
0
0
N x0
22

Khai trin theo cot tht nhEt, to c6:
An = (eP +e-P)A n_i:
e21' - e-2(P
Nlinn xet rang 4 1 = 69 +Cc =
A 2 = ((i
ero e 6
36 - -39
e(P -
e(1+1)6 - e -(lrv1),p
Girt sit AR -
e(-0
, k - 1, 2, , n - 1.
-e (P
Ta c6 An = (ec e-P)A n_i -A n_,
=(eP +e w)e
nip - e npe( n-1)n4)
- e e
(:+1)o - e-(n+1)4'
ew e q) - —
e (n+1)p
- e
-(n+1),p
Nhu v .6}.: A n =
Vi du 1.11
Tinh:
l l a 1 a, an
1 al: +hi a, an
D = dot 1 a1 a, +b., a n
a1 a.9 ... an +b n
-
e 1 -e
23

Lai gidi:
LAY ciOng dAu nhan vOi -1 r6i Ong vao cac thing con
1ai, to có ngay D = b, 1)2 ...b„.
Vi du 1.12
Cho da thric P(x)=x(x+1)...(x+n)
Hay tinh Binh thdc:
P(x)
P(x)
P(x +1)
P(x +1) ...
P(x + n)
P(x +n)
d =
P(n-1)(x) Pth-1) (x+1) Pth-1) (x + n)
Pthl (x) Pthl (x +1) P(n)(x +n)
gidi:
Ta b6 sung de' dude ma Han dip (n+2):
P(x) P(x +1) P(x +n) 0
P(x) P4x+1) P4x+n) 0
D =
P(n) (x) Pthd(x +1) P0P(x + n) 0
P(„+l) (x) 1301+1) (x +1) pg+0(x +n) 1
RO rang det D = d
Nhan dOng 1111 k cua ma trail D vdi
(x +n
dc-ix( 1) k-1 r6i
(k-1)!
Ong vao clang Hirt nhgt vgi tat ca k=2, .. n+2).
Khi do, phAn tii dung dau có clang:
poc + 0 + P(k)(x +0.(x +11.) k o k = n).
k!k=1
24

Pkx+1.) P(x+n)
PTUx +1) . P(n)(x+n)
PT+I kx+1) PT+Ikx+n)
d = del D -
(x+n)°+i
(n+1)!
(-1)1T1 (x +n)n+i
n; con phan tit a cot cuoi bang
nghia 13 clang thg nhaa c6 dang:
(n+1)!
(0, 0, , (+1)"+1(x+n)ni ).
(n+1)!
Do do
Ta ki hieu dinh thge a v6 phai bai C va ma tr5n Wong ring
hai (6-1 Vi da thge P(x) = n(x + i) Ken P'"0(x) (n+1) !, vi vay
i=0
the s6 hang a dOng cu6i &au bang (n+1) !. dgn gian ki higu
va each viek ta dirt xk= x+ k, k = 0, 1, n. d dOng thg hai tit
&leg len ciaa ( ta co:
(Pw(x0), P("ax,), PT)(x.,)) ((n+1) hco + al , , (n+1) tx„+
a do al la hang s6 &do do. Khi nhan (long cue"' ding voi
r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang
P(x0) P(x] ) P (xi) )
P(11-1) (x0 ) Pth-e (xl ) PT-1) (xn )
(n+1)!x0 (n+1)!x1 (n+1)!xn
(n+1)! (n+1)! (n+1)!
a l
(n+1)!
(*)
25

Deng this ba tit dUdi len caa ma Dan (*) co dang
(n +1)! 2
xo +ai xo +a,,..„
(n+1)!
x n
2
+ai xn +a.,
2
2
Cong vac) (long nay hai clang °Ma sau khi nhan voi cac s
a,
va a1 ta nhan dude clang
(n +1)! (n+1)!
(n+1)! 2 (n+1)! 2 (n +1)6 9
xo ,
2 2 2
Bang each bidn ct6i nhu vay vdi cac dong con lai, ta clan m
Dan ( '61) ve clang sau ma khang thay d6i clinh thfic caa no.
c = det = det
x on X n
n-1 x n1
X0
(n+1)!
= (n+vn
k=1 IC!
X0
n(n+1)
(-1) 2 .((n +1)!6'1
.Dn .
Jk!
k=1
26

6 do D„ la dinh thiic Vandermonde cua clic so" x„, x„
n-1
D6' thay D„= n(x k - xi )= n (c_i) = 11(n-o!= 1.
k>i>0 k>1 1=0 k=1
Vay
d = detD = (-1) 2 [(n +1)! in .(x +n)n+1 .
Vi du 1.13
Gia sU A e Mat(n, K), A= (au) a do ali = 0 vdi moil= 1, 2...,
con aij bang 1 hoac 2001 Ted i t j. Chung to rang ne"u n chan,
thi det A # 0.
Lbi gidi:
Nhan xet rang neh to them vim mit phan to ajj 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
hi; = 0 n6u i= j va = 1 ngu it j.
Ta có:
0 1
1 0
det B =
1 1 0

nhan Bong ddu veli -1 fee cling vao cac dOng con lei ta dttuc:
detB =
1 0 0 -1
Deng vao cot thu nhat ca car cot con lai ta cO:
detB = . (n-1).
Vey khi n than thi detB la sidle, do vey detA # 0.
Vi du 1.14
Tinh Binh theft:

' 1 1 1 1
0 C3 CI Ci

2 n+1

D = det q C., C2412 Cn+2
...
n-1 n-1c c C2-11C.!;1-11-.)

n n+1 2n-1 >
LtlL gill
Vi CI, +Clr = C74, nen hieu cUa moi phan tit vol phfin I
dUng b8n trai no thi bang phen td thing ngay tren n6. De tir
D, ta ldy cot thit n tr3 di del n-1, r6i ldy cot n-1 tie( di cot
2,..., ldy cot the] 2 trU di cot thU nhat, ta co:
28

1 0 0 0 0
C12 1 1 1 1
D = det Cy c?c, C 1 C1„1
" •
011 C112
cn-2
C1,1,12 2n-2
lai lam nhtt tren, to co
1 0 0 0
Cl2- 1 0 0
D = det C3 C13 1 1
c^-3 n-3
2n-4
sau n -1 budc nhit v'ay, to
1 1 0 0 0'
1 0 0
D = det
cn-1
CI3
Lin-2
1
cn-3
CI
0
1
= 1.
VEty D = 1.
1.71 du 1.15
Cho A=
cos9 - sin9
sin9 cow ,
fray tinh An .
Lai gidi:
cosy -saw cos9 -sine cos2p -sin29
Ta co A2
sine cost° sing cosy, sin2c cos29
29

Gia Ak
= cosky -sinkp vOi k = 1,
n-1.
sink cosk(p
Ta ttnh
cos(n - lap - sin(n cosy - situp
A" = A n-1 .A =
sin(n -1)9 cos(n -1)9 2 sing cosy ,
cos (n -1)9 cosy - sin(n -1)9 since -cos(n - lap since- sin(n -1 9 cosy
sin(n - lap cosy + cos(n - lap situp cos(n - lap cosy - sin(n -1)9 sirup
cosmp - sinmp
sinmp cosny
Vay
= cosny - sinny
vdi 11191. n E N.
sinny cosny ,
Vi (11..t 1.16
0 1
Cho A = hay tinh A 200"
0
Lbi gidi:
-1 0 0 -1 (1 0
Ta c6 A 2 =
0 -1 1 0 0 1
ma 2000 chia het cho 4
ray A200° =(A)500 =I, (I, la ma tran thin vi cep hai).
Vi du 1.17
Ma Iran vuong A e Mat(n, K), A = (ad throe goi la Ina tran
phan del xang netu aii + = 0 vdi moi i, j = 1, n.
30

Hay chang minh: 'rich caa hai ma tran phan doi xang A va
la mat ma tran phan doi 'ding khi va chi khi AB = -BA.
Lai gidi:
Gia sid A = (ad, B = (bid d do ait + ai, = 0, by = 0
di mm L j = 1, ..., n.
Dat C = A. B = (c,); =
=1
D = B. A = (c1,0 = Ibijajk
id]
Ta ca.: c,k = Zajibik - Eb kiaji - d ki
Nha vay AB phan del xang <=> c, = V i, k.
tt=> = vet mot i, k c> AB = -BA.
Vi chi 1.18
Cho A, B la hai ma tran vuong c9p n. Hay minh
let(A.B)= detA. detB.
gidi:
G1a. sit A = (a,,) , B = (It o ) yea i, j = 1.
Ta lap ma tram
an 0 0a1.,al n
a91 a22 a2n 0 0
ant a

n11 0
- 1 0 .. bth

0 b21 b2n
C=
0 0
31

Khai tri6n theo n clang dal) (thee dinh 12) Laplace), ta co
detC = detA. detB. (1)
Mat khac, bi6n &it ma Iran C bai phep bi6n ct6i sd ca) sau:
Nhan cot thu nhal vat b 1, cot thil hat Null cot thin n vet btl;
r6i Ong vac) Ot thu n + j (j = 1, 2, n), ta dude ma tran D
Bang sau ma dinh thiic cua D va cua C bang nhau:
al a a dta d
a21
am am ... ann dm d, , ... dnn
-1 0 0
0 -1 0
0
a do = riaikbki , nghia la (dij) = A. B.
k=1
Khai tri6n theo n cot cuoi (theo dinh 157 Laplace) ta có
detD = det(dij) = det(A. B) (2)
Tit (1) va (2) va do detC = detD nen ta co:
det(A0 B) = detA . detB.
Vi rtu 1.19
Cho X la ma tran vueng cap n. Chung minh rang X giao
hean vOi moi ma tran vueng Ong ca) = X co clang XI,,, a do I„
la ma trail dun vi cap n.
D=
32

Lari gidi:
Ni111u X = 1 thi re rang X giao hoan vdi moi ma Han
along cling cap.
Ngnoc lai, gia sit X = (X„) giao haul vdi moi ma tran
yang cdp n.
VOi i„ # j0, to chiing minh x inio = 0. Mudn stay chon A = (ad
trong do ajoio =1 con one phan tit khac ddu bAng Thong. Phan to
ding io cot j„ cim ma tran XA Wing x. •' can phan tit a thing io
cotj,cimAXla0.Tirdi6ukienAX=XAsuyra=O.Nhu
y X co dang:
k r 0
„
0 k„
Cho ma tran A = (a) a do a], = 1 vii moi i, j. Khi do phan tit
o dung i cat j cim ma Han XA la X , con phan to 0 ding i cot j
dia ma tran AX la 2. nen k, =
Vi vay:
X= 7. I,,.
Vi du 1.20
Cho ma Iran cap n:
a b
b a
A=
b
a) Chung minh detA = (a-b)° (a + (n-1)b).
b) Trong trudng hOp detA s 0. Hay Huh ma trdn nghich dao
A+ oda ma tran A.
X=
33

Lai giai:
a) Gang the clang vao dOng this nhat ro'i nit
a + (n-1)b a clang dau, ta (Woe
1 1 ... 1
b a ... b
detA = (a + (n-1)b) x
b b ... a
Lay dOng chin aria (Anil th.fie tit nhan voi -b r6i Ong vao
the dong sau, ta eó:
1 0 ... 0
0 a - b ... 0
detA = (a + (n-1)b) x
0 0 ... a - b
detA = (a + (n-1)b) . (a-b)"-i.
h) A kha nghieh <=> detA 0 .(=> a b va a + (n-Gb O.
Goi B = (bii) la ma tran nghich (lac) cila A = (a„).
1
Ta Wet rang bid = detA . A„ a do A.; la phAn Phu dai so
etia phan trong ma tran A.
a b b
a
Vol mdi i thi = la nh thiic cap (n-1).
b b a
Theo phan a) thi A i = (a-b)"-2 . (a + (n-2)b).
34

A a+(n-2)1(
Vay b„ -
det A (a 1- (n -1)1).(a - b)
Vol i # j = (-1)H , d do kl„ la chub thue e6p n-1,
co dupe ba' lig each x6a &Ong thu i va cat tha j eila ma tran A. Do
A dovi ming nen = Gia sU rang i < j, khi do cot thfi i va
dung thu j-1 cua Mo gain town nhang phan t& b. N6u d6i eh.6
Bong j - 1 len tren Ming dau (Mu nguyen cac dung khac), rei lai
MS( nit i len 6;4 thu nhai (va van gib nguyen the cat khac), thi
to (Moe:
m = = (-1)'34 b.(a-b)"-2 .
Nha v(6y b
dot A (a +(n -1)b)(a - b)
Do do
a +(n -2)b -b
1
(a +(n -1)b)(a -b)
-11 a +(n -2)b
C BAI TAP
1.1. Chung to rang mai phep chuyan tri dm (n 2 2) lh
mot phep the le.
1.2. Hay phan Lich eau pile') the! sau thanh tieh ene phep
chuy6n tri
-b
35

1 2 3 4 5 6 7 8 '
a)
8 1 3 6 7 5 4 2 )
( 1 2 3 4 5 6
b)
6 5 1 2 4 3
1.3 Tim s6 tat ca the phep the a e S„ sao oho a(i) x i vol
mm i = 1, 2, ...n. Cluing t8 rang khi n chan, so" one phep th6
clang tren la m54 3616.
1.4. Ki hi8u (n, k) la secac hoan vj rim 1, 2 n ce dung k
nghich the.. Chiing minh cong thile truy 146i sau:
(n+1, k) = (n, k) + (n. k-1) + + (n, k-n).
vdi guy vac (n, j) = 0 nevu j < 0 hoac j >
9
1.5. Ta goi 45 giam ena phdp the f Ia hi5u cna s6 the phan
tic khting bat dog (nghia la s6 the pilau tit i ma f(i) i) va s6
clic Yong xich <IQ dai ion him 1 trong phan tich cua f thanh tich
car yang xich 458 lap.
a) Chung minh f cotang tinh chat chan la vOi do giam cua no.
h) Chling minh rang s6 161 thidu one nhan tic trong phan
tich cna f thanh tich the chuydri tri bgng 45 giAm cua f.
1.6. DM h hai s6 x va n nguyen, n x 0, to ki hi5u r(x, n) IA
s6 chi khi chia x cho n : 0 r(x, n) < n. Chiang minh rang nal
n > 2 va a la so? nguyen, nguytin to' d61 vdi n, thi tthing Ung
ki—>r(ak, n) la mot phan tit cua S„,, k e 11, , n-1}.
1.7. Cluing minh rang moi phep th6 cap k > 1, dou phan
tich dupe thanh tich nhiing chuydn tri dang (1, i) vdi i = 1, k.
36

1.8. Xac dinh davu cua cac phop the' sau:
a) I 1
2 n n +1 n + 2 ... 2n 2n +1 3n
43 6 ... 3n 2 5 1 3n-2
b)
1 2 n+1 n+2 2n
2 1 ... 2n 1 2n-1,
1.9. Chgng mink rang:
a) Dinh thiec cap ha ma cac Phan to Wang +1 hoac -1 le mot
s6
b) Dinh attic ay Ion nhig la 1.
c) Dinh thew c)41) ba ma cac pga'n tit la 1 hoac 0 dot gia tri
IOn nheit biing 2.
1.10. Tinh dinh thew cap 2n D = det(cM),
a vai i= j
dO di; = b yen i + j= 2n +1
0 i=j vet i+j 42n+1
1<i, j S 2n.
1.11. Cho A = (an) la ma trn cap n, a1 e R. Chung minh
rang detA khong thay dei, neit cac phein tei a i; ma i +j 15 dooc
thay bai so' dal cga no.
1.12. Chung to cac dinh thew sau day bang khong:
1 cosa costa cos3a
cosa costa cos3a cos4a
a)
costa cos3a cos4a cos5a
costa cos4a cos5a cos6a

a)
c+-
1
b+-
1
c b
2 a + —
1
a
2
1
c+-
c
b + 1 a+ 1 2
b a
(abc -1)
b) =
1 1 1
-1 0 1
a) Dn = -1 -1 0
-1 -1 -1
0
-1
-1
1
0
-1
1
1
0
1

-b1 a1 - b2 al -bn

a, -b1 a2-b2 a, -ba
h) ; n > 2.
an -b1 an -b.,-bn n
1.13. Cho hai ma tran A. B sao cho:

5 11 1 14
A B = I, B. A -

J1 25 ) 4 y
Hay tam x, y va A, B.
1.14. Hay khai tridn dinh thiic va chilng minh:
b)
(a+b)2 b 2
a2
b 2 (b+c)29c-
a = c= (a + c)2
= 2(ab be + ca)2.
1.15. Way tinh cox: dinh thric cap n saw
38

1.16. Hay tinh dinh thdc sau:
1+x'2 x 0 0
x 1+x2 x 0
a) D„= 0 x 1+x2
0 0 1+x2
nghla Dn la dinh thiic cap n ma cac phan tit tran during cheo
chinh bang 1 + x2, the plidn tit thuOc hai [Wang chat) grin during
cheo chinh bang x, the phOn tit con lai bang 0.
2 1 0 .. 0
1 2 1 .. 0
0 1 2 0
1
0 0 0 1 2
1.17. Cho da thite P x) = (x - al)(x - (x - a„).
a d6 a, la cac so thvc dot mot phan biet. Hay tinh climb thde sau:
P(x) P(x) P(x)
x-a1 x-a, x-a n
1 1 1
a l a, a n
a;
a
an-2
n-2
an -2
1.18. Xet hai ma tram ph& cap ha:

'a b c ' 1 1 1

A= c a b va J= 1 j
b c a, v 1 j-
2n . 1
dO j = e = cos— +. sm-
2n
= .
3 3 2 2
b)
39

Way chang minh det J # 0. Hay tinh ma tr'nn AJ. tit do suv
ra gin tri detA. Hay nen hal Loan Wong to kin A ya J cite ma
1.rn cap n.
1.19. a) Hay tinh dinh thing cap n san:
a+13 (A.13 0 0 0
1 a+11 a.{3 0 ... 0
14„ = 0 1 a+ ^ 0
0 0 0 0 1 a+ 0
1)) Chung to rang dinh tithe sal' khong plat thuhe vao y,, )
••• Ynt
1
g1+ Yl
(x1 11111)(x1 11 379)
1
x2 + y1
(x9 + N.1)(x9 + y2)
(x n +y 1 )
(x„ +y 1 )(x 1 +y2)
D„=
n-1
11(X1 + Yi )
1.20. Cho ma 14.'0
Hay tinh A u1°.
n-I
Fl(x.) +yi ) n(xn +)'i)
1=1 i=1
( 1 -2 1
A= -1 1 0
-2 0 1
40

1.21. (ha si:t ma trgn A e Mat(m, n, va rang A = 1.
Jhisng minh rang cac ma trgn B e M(m, 1, K) va C e Mat(1,
1g A= B. C.
1.22. Chung minh rang ma trap A = a b )d thOa man(
pIntong trinh:
X2 - (a + d)X + (ad - bc) = 0,
0 do 12 la ma Iran don vi cap hai.
1.23. Cho ma trgn
9. 1 0 v
4 1
A=
90
vOi 4 x 0. Hay tinh A-'.
1.24. Cho ma tran vuong c5p 4.
cosa sincx cosa sina
cns2a :;in2a 2cos2a 2sin2a
cos3o. sin3a 3cos3a 3sin3a
cos4a sin1a 4cos4a 4sin4a
Chung minh rang A khg nghich khi va chi khi a ♦ kg (Ice Z).
1.25. Cho ma trgn A = (a) e Mat (n, 1H) ma the phan tit
doge cho bai tong that:
(-1)H- CV yin
vOi i = j
0 vai i>j
6 do Chi.,'
kHrn-k)!
- . Chung minh rang A2 =1,1.
(9, la ma Iran chin vi).
91

1.26. Gth sa X = ())) e Mat(n, R),
1-1
a do xii 4- (-1)")! Chitng minh X' = I.
01- 4,
Chit Se voi aeR k e N, ki hiqu
)a,
a(a - 1)... (a - k +1)
-
a
())) )0
)- 1
0k!
;
1.27. Gia = yXx„ x.„„ x„) vdi (i = 1, 2, ..., n) la ham
[cua cac bi8n doe lap x1 , x2, .. x„. Ma tran J = J(Y, X) = aYi
dx•1 i
dliciC goi ]a ma tran Jacobi cua phdp bi6n din, con Binh thud dm
no duck goi la Jacobien dm phep biers den do.
Bay gin /cot m6i quan hq gilla n2 ham vii va n2 bien xii dune
cho Isdi ding thiic:
Y= A.X.B, a do Y= (yi) , X = (xi),
A . B e Mat(n, R) la hai ma tran the trn6c.
Chung minh rAng det(J(Y, X)) = (detA)" . (detB)".
1.28. Cho X = (x„) e Mat(n, R) la ma tran tam gide dudi;
va Y = X. X.
Chung minh rAng det(J(Y, X)) =
1.29. Cho Z lA tap cac sqinguyen; A, S hai ma tran vu8ng
cap n, cac phAn t> la nhilng s6nguyen (ta vie) A, S e Mat(n, Z)).
Hun nua detA = 1, det S x 0.
Dal B = . A. S; Chung minh rAng cOs6m nguyen dding
de Bm e Mat(n, Z).
42

1.30. GM sit trong ma tran A = (a u) c Mat(n, R) da cho
rude tat ea the phan ti ad (i # j). Chang minh rang có thk dien
Mo &rang char) chinh the s60 hoc 1 de ma tran A kheng suy Bien.
1.31. Tim tat ca cac ma tran A e Mat(n, K), A= (dj), 0
na tan toi ma tran nghich dao A-' cling có the phiin t5 khOng am.
1.32. Cho ma tran vuong A co the phan to la s6 nguyen.
rim diet' kien can va du de ma train nghich dao cung c6 the
-Men to la s6 nguyen.
D - HDONG DAN HOAC DAP S6

Xet T e S„ (n 2 2) gie sii i < j va T = j, T (j) =
T (k) = k vdi moi k s i, j. Khi do the nglach the eaa
ji, k} Nob < k < j
j/, j} Nob / i + 1, j - 1
With vey co tat ek la U - i) + (j - i - 1) = 2U - I) - 1 nghich
the Vi so nghich the le nen t la phop the le.
1.2. a) Phop th6 da cho phan Deb thanh hai yang xich chic
lap (1 8 2) (4 6 5 7) = (1, 2) . (1, 8) (4, 7) (4, 5) (4, 6).
b) (1, 6, 3) (2 5 4) -= (1, 3) (1, 6) (2, 4) (2, 5).
1.3. DS. S6 Dat ca cac phep the a e a(i) # i vai moi
f n
(- 0
n +1 . NMI yky A, (i =1, 2,..., n-F1)
i =1,n la n! E
k0 kl=
1.4. Xet tap A gam at ce eke Minn vj (Ma 1, 2, ..., 14, n+1 co
thing k nghich the. Ki hieu A i la b° phkn eaa A gdm cac imam NO
an , a n+1 ) ai =ma
la nhang tap con rot nhau caa A va A = A, U A2 U U... U AnA.
43

Xet Anki = {a = (a p a2 ,..., an+1)1 = n +1}
NMI 0y so cac nghich the cna a bAng se cac nghich the cue
1 9
nghia la bling 1+ Dieu do cheng to so ea(
U Ch 2
Jhfin to cria A„,, bang (n, k).
Xet tap Ai (i = 1, ..., n). cis su a e Ai, a = (a, aj, a„4 ,)
Theo dinh nghia a; = n+1. Nhti vay (aj, khong la nghich th(
vdi j < i va (x„ ai) la nghich the vdi j > i. Do do a, tham gia vac
n+1 -i nghich the.. Xet hoan vi a' = (a l .— aa,_„ (>61, cCia S2
S5 nghich the maa a' bang sernghich the cna a trii NM;
vay ta có met song anh tit A; len tap cac hofin vi cua S„ co thing
k-n-l+i nghich the., do do so' phan t& cera A; IA (n, k-n-1+i). Td
do, sephan to cUa A la:
(n + 1, k) = (n, k) + (n, k-1) + (n, k-2)+... + (n, k-n).
1.5. a) Xet phan tich f = a l o a, o ... a,„ thanh tich cac veng
xich dec lap di) dai > 1. Gia sa do dai al, la dk; ta they f(i) # i khi
va chi khi i thuoc met trong cac yang xich do. Vay do giam coal
IA d,+€1.2 M6i vOng xich dai dk phan tich dude thanh
dk-1 chuydn trf. Do vay f phan tick deck thhnh d 1+... + (.16-1n
chuyen trf. Do do kha'ng dinh a) driec chting mink.
b) Goi / a di) giam can. f. Theo a) f phan tich (kw thanh /
chuydn trf. N5u có phfin tich f thanh h chuyen tri nao do, ta
phai chring minh h 2 1. Ta có bd de sau: Neu a, 13 tham gia van
met yang ;rich cas phep the f, thi khi nhAn f vdi chuydn trf (a, f3)
(ve ben trai hay ben phai) yang xich dji not phan thimh hai
Tong xich d'ec lap. Con neu a, tham gia vao hai vOng xich cua
44

Thep the f, thi khi nhan viii chuyen tri (a, (i), hai yang xich se
Thep lai lam met (136 de nay a thing chfing minh). Tr/ do neu g
a phop the. va T la met chuyen tri tin dp Mem cua g o T khong
vire); qua do giam cfia g ceng them 1. Vi the nen f phAn tich
luec thanh h chuyen tri thi do giam cUa f khong wet qua h.
1.6. H.D. Do a va n nguyen t6 vfii nhau, nen a k khong chic
het cho n vdi mm k = 1, 2.... n-1. do de r(ak, n) la nhfing so
phan hied.
1.7. Vi moi phep the phan tich due() thanh tich cat ghee
chuyOn tri, nen chi can chfing minh bai town cho phep chuyen
tri(1,j)e Sk vei 1, j # 1. Ta ce (i. j) = (1, i) . (1, ) . (1, i).
1.8. Xem vi du 1.1
- 1-1(n+D
a) DS: (-1) 2 .
r(1121)
h) DS: (-1) 2
1.10. Iasi trien cot dau, to ca. D ykk =(a2 _
" " --1 2n-2 •
Do D2 = -13.2 nen D 2n = (a2
1.11. A = (aid , detA = E sgna ai,(1) ana(N
E S..
Trong moi tich ako(k) angfrik , tong E(k +cr(k) )= n(n +1)
k=1
la sir than; nen re met se cliSn the thfia so ak.,0.;) ma tong k+G(k)

le. VI Vey khi thay ha; nen i + j 1e, thi cfic tich
anc(k) khong dei, do do detA khOng del
45

1.12. a) COng dOng thu nhat vdi (long the./ ba, ta dude don
tY le vdi (long thu hai.
b) Nhan dong thd nhat vdi (-1), rdi cUng vao die (long th
hai va Ulf( ba, ta dupe hai dOng CY 15.
1.13. Ma tran A, B e Mat(2, K). Ta có hai bat bian detAB
detBA va tr(AB) = tr(BA). Vi vay:

fx+y=30 {x = 20 {x =10
hoar

x.y = 200 {y=10 {y=20
20 14
a) x = 20 va y = 10 o BA=
14 10
20 14' 5 11
Ta co ABA = A A.
14 10 , 11 25 2
Tit do ta tinh dude A =
5k -14X
, A E R
-11k 0
va B = A-
5 11
11 25,
10 14
b) x = 10 va y = 20 BA=
:14 20,
10 14 5 11 -3 5 l
ABA = A X A= P.
14 20 4 11 25, X 55 32
1.15. a) Di, = 1.
b) An = -A„. 1 + 4,1 = 1.
Tit do suy ra A2p = 1, A2p+, = 0
46

1.16. a) Khai tri4n D„ theo cot thu nhat, to cee
DT, = (1+ x2 )Dn_i -,4211)„,, , D, = 1 + x2,
D2 = (1 + x')2 - x' = 1 + x1 + x4 .
Taco: Da -D„-1= x2 (D„_i -Dn_2 )= x1 (131-2 - Dn_3 )
x 201-2) (D9 _ D1). x21
Tct do Dn = Dn _1 + x2”
Do vay Dr, =1+ x2 + x3 +...+ x2"
b) Ap clung eau a) vdi x = 1. DS (n+1).
1.17. Khai tri .on A(x) then (long tha nhat, to co:
P(x) P(x)
A(x) = D(a9 ,... an )
x - a l -
+ ( 1)"7
I3(x)
; a 46 D(a2,
x -an
la dInh Dane Vandermoncle eaa cac s6 a2, , an;
Cho x = a,, to co A(a l) = (a, - a2)... (a, - a„) n(a
j>i22
A(a,) = (-1)" n(ai -a1 ).
j>141
Thong ta A(a2) = = A(a„) = A(a 1). Da tithe A(x) bac nhO
hOn hoar bang (n-1), co the gia Da hang nhau tai n diem phan
biat, vi vay A(x) la hIing.
Tit do: A(x) = (-1)11' Fr(a1 -a.).x
ISIS“
47

1.18. J In ma tran Vandormonde vat (Me s6
biet nen dot J * 0.
Ta
a+b+c a+ bj+ cj2 a + bj 2 + ej
A . J = a+b+c c+aj+bj 2 c+aj2 +bj
•
a+b+c b+cj+aj2 b+cj2 +ttj
dotAJ = (a + b + c) (a + bj + cj 2) (a + bj 2+ cj)det
1.19. a) Khai trien theo cot the] ula. La

D„ = (a + a0D«-s.
Ta co: D, = a + 13, 1/2 = az + an + 132,
k

Gia sit D k = Ea i f3k-i (k = 1, 2, n-1).
=o
n-1
Ta co = (a +p) a ir
-1-i
- an Ia'0 2-2-I
i=o i=o
n-1 n-1 n-2
= ai+lpn-i- orkira .2 a al n n-i-1
1.0 i=0 i=0

n-1 n-1
= r3n ct i v_i
-
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 = = x2 - x, kheng phi thuec y,.
+ yi x, +yi
n = 3 D„= (x, - x,)(x,, - xi)(119 - x1)-
Gia stir Dk =

11(Xj -x i ) vdi moi k = 2, n-1.
Sii <MX
Ta chting minh ding thim tren dung voi
Khai triL 1)1, theo clang flu./ nhal, r6i rut cac thira soi chung
6 die cot ye sit (King gia thiel qui nap, La cc):

Di, = (-1)" n( Yi) n(X j - X1) (1);
k=1 /=k 151<“n
i,j #1:
Neu ee i x j ma x, = xp thID„ = 0; ve ming theft quy nap
Van dung.
Neu x; x x vol moi i #1; thi to Ming thee (1), to co:
DT, =
1 i<g nI
(4) k-1 n (xi +
44
11(x, -xk ) Fl(xk -x j )
j>k
Net 14(y 1 )
• (-D kjjl 11(x2 +y 1 )
lxk
k=1
n _ x, n (xk k=1
(x/ +y 1 )
ixk
(xj -xk )
jxk
P(y,) 13 da thvc bUc < n - 1 doi vdi y,. Vbi y, = -x; win moi
i = 1, 2, Ta co P(-x,) = 1, do de P(y,) = 1. Tiido ding thirc
quy nap three chiing minh.
49

1.20. agA=B+I3 L la ma tran don vi,
0 - 2 1' 0 0 0
B= -1 0 0 , t3do B2 = 0 2 -1 ; W = 0
-2 0 0 0 4 -2
Ding khai trie'n nhi thric Niu-ton to co:
A100 = (0 , 13 ).100
13 +100.B+
100;99
u 1-
2
1 -200 100
Suy ra A100 = -100 9901 -4950
-200 19800 -9899
1.21. Vi rang A = 1, nen tat ca car (long tS, le vb7 mot dong.
Wiy ma tran C E Mat(1, n, K) la ma trqn dong do. Giq sv clang
thd icim ma trqn A bang b,. C, lay B = (b) la ma Lean cqt. Ta co
A = BC.
1.23.
A-1 =
1.24. Tinh detA = Tit db suy ra ket qua.
1.25. Bat A2 = (b(), khi bi; = iaikak;
k=1
+ Vdi i > j, 13;; = 0 vi = 0 ne'u k j con k > j thi ak; = O.
50

+ VOi i <j bli = Zaikaki =
= E(_1)k_i cry-1)Hc i(
k=i k=i
. •
0-0!_ z ( 1) )+1; (J -1 )•
k=i (i -1)!(j - i)! (k - i)!(j- k)!
(j-1)!
(j-k)!(i-1)!(k-i)!
= (-1)1+1c1
- 1
1 J-.(-1)kIck• 1 : fiat k - = /
J
= (- 1)1+3 cl-1• Z_.r J-(- 1/ c i 3 = 0 , vi
1-0

(- 01 ,c = 0 - =-0 .
1-0
+ Vert = j, ta co b ii = Ea ik =1ki •
k=i
NhLt vi,ty A7 = I„.
1.26. Triloc h6t to có nhan xet sau: Cho a, b, C e N, c 2 b
Khi do:
b 7(2.-1,'
E(-1)k

kzo - ‘a., - a,
Cong thtic tam co the thong minh guy nal theo a = 0, 1, 2, ...
va hiu y la = 0 voi k > a.
flat X- =(3.7 .5 ) th y], = [_,x lk xkj
k=1
••
=
•
=
k=r n-k, 3 - i ,
xot tren).
(do dp dung nhan
51

Gia sit = (20, to ed za = Zyikxkj
kst

kbl
in-i‘ -
Ora
k=1 Tr -1, „n - j,
k„ iEnn+1+1+k n- i
ki=1 „k -1 n
Neui<j zu =0
Neu i= j = 1
Ngu i > j z,i = E Enh-Ft

aXt<l) t b
day t = k -1. a = n - b = n - j).
Nha vSy zij = D_ob+t t! b!

a<t<b a!(t-a)! t!(b-t)!
b

b ! x
(_ i )1)-0

a 1)+1 t-a
Cb 1) C b_ik

a<t<b (b - t)! (t -a)! t=a
Ixa
= (-1)am
s=0
Nha vky = 8,j nen Xx = I„.
1.27. Trade het to Chang mink rang ne!u Z = AX,
a (Id Z, A, X e Mat(n, R.) thi det(J(Z, X)) = (detAr
That vay, gia sit Z= (20), 21; = aik xkj .
k=1
52

Tif do
aik n6u / = j
0 nau / j •
(1)
Ta coi ma tran J(Z, X) E Mat(n2. 1K) vdi can dOng va can cot
duct danh s6bai can cap c6 thit Lu (i, j) , 1 < i , j n. Thii ti cac
thing clang nhu cac ctit la (1, 1), (2, 1)... , (n, 1), (1, 2), (2, 2),...
azi
(n, 2), (1, 3)... (n, n). Phgn t& a clang (i, j), cot (k, /) la 3 . Tn
ax k/
do, do (1) ta en:
A
A
J(Z, X) =
0
va nhn vay det(J(Z, X)) = (detA)".
Turing to n6u Y = Z. B thi ta clang ce):
' det J(Y, Z) = (detB) ° . Xet Y = A .X. B = Z. B.
Ta co: J(Y, X) = J(Y, Z) . J(Z, X). Mut vay
det(J(Y, X)) = det(J(Y, Z)) . det (J(Z, X))
= (detB)n . (det(A))n.
1.28. Gin s>1 X = (xi) e Mat(n, K); x,, = 0
vdi i < j. Ta có Y = X. Xt = (Y,)
Y;i = xi (xk Exik jk
k=1
Nhv vay Yii =
53

OY •Xet
ax
13 . Do (*) to có
,j aX
2X 11vat i = j
X JJ

j
x is ngu i=j,r=j
Ta có ay'J ngu r = = j, j
(*)
42/ci„, neat r = i = j
0 trong the twang hop con loi.
Ma trOn (1(Y, X) e Mat(n 2, R), the (Ping nä cac cot dttoc
darth s6 nhu trong bai 1.27. Phan tn a hang (i, j), cot (r, s) Pt 13
ors
Ta nhan tha'y vdi (i, j) < (r, s) (nghia Itt i < r ngu j = s hotic
dy
j < s) thi - 0.
Mt n;
Nhu vay J(Y, X) la ma tran tam gitic &RR
Nhu ray det(J(Y, X)) =
aXij _
-
'J=1 jX j1
=1l fl - n(2xn)=-JJ j1
J
1.29. Goi A = (AO la ma tram phu hOp cat) A.
Ta co A41 = 1 At - Al e Mat(n, Z).
det A
Ta co B = SI1 . A. S Brn = S4 . S
VI ma tran S g6m cac so nguyen nen k = Ida S e N* (do
gia thigt dotS = 0). Vol m6i se; nguyen dming p, )(et ma tran
AP e Mat(n, Z), gia sit AP (ap ; goi R la set du khi chia a!; cho
54

k = Idet S. Idat = 0'0 vi 0 < ri; < k, nguyen nen so' dile ma
tran dang Rp la huu hen. VI Oy t&n Lai p, q c N*, p > q sao cho
the ph&n tii taring ung cad AP va AP bang nhau thee modk.
Do do AP = +(detS).0 , C e Mat(n, Z).
Tit do: = +(detS).C.(Aql . that m = p - q.
Rhi do: I - 1 A"'S=1„ +detS.S-1 .C.A-4 S;
Vi S e Mat(n, Z), (detS). Sl' = S i la ma Iran phu hop cua S,
nen S e Mat(n, Z) va vi vity: B" c Mat(n, Z).
1.30. Ta chfing minh quy nap then n.
n = 1 : hien nhion,
all alp
a21 22
Neh cho track a2 , . a10 t 0, to chon a, 1 = a22 = 0.
Con nein a21 . a u, = 0, La chon a„ = a22 = 1.
Nha vay n = 1, 2 thi bad Wan dung.
Chi sit bai toan dung laid moi dinh tithc cap < n - 1.
Ta chiing minh no dung ved dinh at& cap n.
Gia silt A = (ap) e Mat(n Xet A„ la phan phu dai so' caa
ph.dn tit an . Theo gia thi6t quy nap, to da Hein dude a22, a„„ de?
A, 1 0.
n=2, A= =an a22 -a 21 at2 .
55

DAt all = x, khai trien theo clang thu nhAt ta co:
detA = x. A, 1 + ZaliAli
Ta co I:A uAli 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 i2. Chon clang k vdi k # j cila ma tran A'. VI A-1A = I„ non
tich cua dOng k cua A' vdi cot j CEla A bang 0. Nhu vat), cac phzin
t0 cfm cot thil i, va i2 nam tren clang k deli bang ki:Mg (k x j). Do
do hai cot i2 va i2 cua A-1 tY le vdi nhau. Diu do clan deIn mau
thuan.
NMI vAy m61 cot cim ma tram A co dung met s6 clueing (con
lai dgu bang 0). Dicing te, m6i deng cila ma tran A clang có
dung met s6 throng. Giao hoan cac demg (hoac cac get) ta duoc
ma tran chg.°.
Nguoc lai, tat ca the ma tran 'Then dude tt ma tran cheo
(ali > 0) bang each chuygn clang host cot deu kha nghich. Ma
trAn A-I nhAn dude to ma tran A bang each Iay nghich dao cac
phan to khac kitting rdi chuyAn vi.
1.32. DS det(A) = ± 1.
56

Chuang 2
KHONG GIAN VECTO - ANH XA TUYE-N TINH
- HE PHUONG TRINH TUYEN TINH
A - TOM TAT Lt THUlt
§1 KHONG WAN VECTCS
1. Dinh nghin: Gin sit K la mot traong. Mt tap hop V
kirk rung cimg vdi hai phep town "+" : V x V —> V
(a , p)1—* +
va phep than " •" : K x V —> V
(k, a) 1--> k. a
dime goi la mot kheng gian vec to tren twang K n6u no thOa
man cac tinh chgt sau. mot k, /, c K va moi a, p, y, c V,
to CO:
a) a +p=p+ a
b) (a+ p)+ y= a+(p+ y)
c) CO phlin tit 0 e V sao cho:
a + 0 = 0 + a = a vin moi a c V.
d) Vdi a e V, ton tai (-a) e V sao cho:
a + (-a) = 0
e) k(a + is) = ka +103.
g) (k + /)a = ka + la.
h) (k. / )a = It (/ a)
k) 1. a = a, 1 la phan ta don vi cita truang K.
57

Mot kh8ng gian vec CO tren throng 1K con &foe goi K -
kKong gian ecto.
Khi 1K = R, kh8ng gian vac to &loc goi la kh8ng gian vec to
thee. Khi K = C, kh8ng gian vac to dude goi la kh8ng gian vecto
phiie.
2 - Sit crOc 15p tuygn tinh vit phu thqe tuye'n tinh
Vec to 13= k,a, + + k„,a„„ (a, c V, k e K)
goi Ht met to' hop tuyo'n tinh cua cac vec to 4 0 = 1, Ta
ding not p bieu thi tuy6n tinh qua cac vac to a t, ..., an„
MOt he vac to la,, ..•, caa V dude goi la he phu thmic
tuyeIn tinh nen c6 cac s6 = 1, m) kh8ng deng thoi bang
kheng cna ]K sad cho = 0 . Met Oat hkeu along &king:
1=1
he (st„ am) ka he pha thuec tuy6n tinh nen c6 met vec Lo nac
de cim he bleu thi tuyen tinh qua cac vec to can lai oda he.
Met he cac vec to kh8ng phu thueic tuy6n tinh doge goi
he doe lap tuyen tinh. Nhu ray, he {a„ am} la he dOc lar
tuyen tinh ned.1 moi t6 hop tuy6n tinh k iai =0 to suy ra
k, = = km = 0.
Cho he 14„ aid cac vec to doe lap tuygn tinh caa kh8ng
gian vec to V ma m6i vec to caa no la t6 hop tuyeIn tinh caa ca(
vec td cim he ([3„ thi k < 1.
. Cho he vec to {di } ; e trong kheng gian vac to V, I la tap chi
so gfim huu han phan tit He con (a3) jedcI goi la he con de(
58

lap tuyen tinh t6i dai cim he da cho neu n6 la he dec lap tuyen
tinh, va nen them bet kST vec td ak nao (k E I J) thi ta dude met
he phu thuec tuyen tinh.
Cho he hau han vec td {a, , , am} trong khong gian vec td
V, thi s6 phain tit cim moi he con dee lap tuyen tinh t6i dal ciia
he tren deal bang nhau, so/ do duoc goi la hang cim he vec td
{ao ...,
3 - Cd sd va so clueu cua khong gian vec td
Met he {e„ , en} the vec td doe lap tuyen tinh am killing
gian vec td V duos goi la met co sa oda V nen mm vec to cim V
deli la t8 hop tuygn tinh cim vec to {c„ , en}. Khi V c6 ed sa
g6m n vec td thi moi co sa cim V deu c6 dung n vac td. S6 n goi
la so" chigu cim V, Id hieu dimV. Neu kleing tan tai mat cd sa
Om him han vec to, thi V goi la khong gian vec td v8 han
chieu.
Cho co sa e = le i, , ej, yen vec tO bet kS7 a e V, ta co
a , , x„) dupe goi la cac toa doo cim vec td a del vOi
ed sa {e„ , en}, ; la toa de thu i cim a doa vdi cd 56 do.
Gia sit co ed sa khac c = {c,, , E„} ciaa V, ma si =
]=1
= 1, n). Nen vec td a có toa di) (xi) trong cd s6 va c6 toe
de' (x11) trong ed sa thi ta c6:
= Ec oxli , , n.
Neu ta ki hieu ma tran C = va ma trail X = (x), X' = (x')
la cac ma tran cot, thi X = C X'.
59

4 - Klaang gian vec tei con va klaing gian vec td thacing
Tap con khong r6ng W ciaa khong gian vec to V duoc goi IA
khong gian vec to con ciaa V nen W la khOng gian vec to, voi cac
phep toan ciaa V han dig tren W.
Tap con khong rang W dia. V la khong gian vec to con ciaa V
khi va chi khi W on climb dna vdi hai phep toan dm V, nghia la
vdi a, 13 E W va k E K, thi +p E W va k a E W.
Cho W,.... WEI la cac khong gian vec to con cria khong gian
11
vec to V, khi do nW, IA mOt, khong gian vec td con Gem V, IA
khong gian con ldn nhat nam trong moi W i, i = 1, 2, ... , n,,.
Cho tap hop X c V, khong gian vec la con be nhat cna V
chga X duck goi 1a bao tuygn tinh cila tap hop X, ki MO <X>
hay Vect(X). Neu X = , bao tuygn tinh ciaa X thick
ki hieu la <at, , a„,>.
Cho W„ , ho nhUng khong gian con ciaa V. Khi de
bao tuygn tinh ciao, tap hop W,U... UWE, dude goi la tang cim cac
khong gian con W, ,W„, ki hien Ia W, + + W„ hay ZW; .
Ta thay rang a e EW; khi va chi khi a = Za i , a, e
i=1
Neu moi e /113/4 , a vigt chicle mat each duy nhat a
i=1
dang a = a, + + a„ , a, e thi t8ng W, ch.toe goi la tang
I=t
true tigp cua n khong gian vec td con (i = 1, 2, ..., n) va ki
hieu la W, e W2 ED ... ®W„ hay S W,.
=1
60

Gia sit V la klafing gian hitu han chikt,W, va W2 la hai
khfing gian vec to con dm V, khi do:
dim(W,+ W2) = dimW, + dimW2 - dim(W, fl W7).
Cho W la khong gian vac to con cim khong gian vec to V.
Ket quan he Wong throng tran V: a-Paa-Pe W. Lop Wong
:lining cam vec to a dirge ki hiau la [a].
Tap thudng V/W vdi hai [top toan:
[a] + [in = [a + [I] va k[cx]= [kal
vdi moi a, 13 E V va k e K, lam thanh mat klihng gian vac td,
durie got la kitting gian vec to thtfong (ciia V chia cho W):
dimV = n, dimW = in (0 m n), thi dim V/W = n - m.
Anh xa it : V -> V/W ma a(a) = [al (Inge goi la phdp chi6u
tdc.
§2. ANH XA TUYEN TINH
1. Dinh nghia. Cho V va W lh cac khong gian vec to tram
twang K; Anh xa f: V -> W duo° goi la anh xa tuye"n tinh (hay
itIng ca'u tuy6n tinh, hay toan tit tuyan tinh) n6u no bao
phop toan cua khong gian vec to, cu the' la: veil moi a. p e V.
1119i k e IK, to ce:
P) = f(a) f(P)
f(k a) = k . f(a).
Anh xa tuy6n tinh dude goi la don din nett ne la don anh,
:oan can n6u n6 la than anh, va dang 6.'11 n6u n6 la song anh.
61

Hai khong gian vac td V va W chive goi la clang cdu vol nhau
nAu ce met ding cau f tla V len W.
2 - Cac phep than tren cac anh xa tuy6n tinh
Ta ki hieu tap cac anh xa tuy6n tinh tit khong gian vac to V
dAn khong gian vac to W la Hom(V, W) hay HomK(V,W) de chi
rd K la tniang co sir.
HomK(V,W) la met khong gian vec to tit truong K vbi hai
phep town nhtt sau:
Vai f, g e HomK(V,W); anh xa f + g e HomK(V,W) the dinh
ben (f + g) (a) = f(a) + g(a) vdi moi a e V.
Vol k e K, f e HomK(V,W), thi kf: V —> W xac dinh boi
(kf)(a) = kf(a) vdi moi a e V.
Anh xa f + g dupe goi la tong Kin hai anh xa f va g.
Anh xa k. f ddoc goi la tich eda anh xa f vdi vo hdong k.
3 - Di6u ki6n xac dinh anh xa tuy6n tinh
Anh xa tuydn tinh f: V —> W hohn toan chidc :Mc dinh khi
Mei anh cOm met co so.
NMI le„ , e„) la co se, cOta khong gian vac to V va a,, a„
la n vec to cent MI:Ong gian vec to W (V, W la cac khong gian vec
to tren clang rant trnong K), thi Kin tai duy nhdt met anh xa f e
flomK(V, W) de f(ei) = a, yea j = I, ... f la don cdu khi va chi
khi he , a„} doe lap tuyetn tinh, f lA clang cdu khi va chi
khi he M I , , a„) la co sa oda. W. Gia = EJ la cd
in
ciaa W, thi f(e,) = E, = vaaha tran A = (ad goi la ma
62

tan dm anh xa tuy6n tinh f dOi vOi co sa {e,) va {EJ. Nhg vay
16u cho cd sa E cua W va co sa e cua V, thi dinh 19 tren chring to
:Ang co mOt song anh girla tap Hom K(V, W) va tap 1),E#t(m, n, K).
FlOp thanh cua hai anh xa tuy6n tinh la mot anh xa tuy6n
rhh, nghia la nOu f: V —> W va g: W —> Z la the anh 'ea tuygn
thi g f: V --> Z cling la mot anh xa tuygn tinh.
4 - Anh ca hat nhan cua anh xa tuygn tinh
Cho f: V —> W la thing thu tuy6n tinh gifla cac khong gian
7ec to, neh X la khong gian vec to con cua V, thi f(X) = { f(a) I a E X)
a khong gian con cua W, va n6u Y c W, Y la khong gian eon
W thi f-'(Y) = e VI f(a) e Y} la mot kWh:1g gian con cua V.
Ta goi Kerf = f-1 101 la hat nhan cda anh xa f va Imf = f(V) la
inh cua anh xa f. S6 dim Imf doge goi la hang cua anh xa f, kf
lieu rang E
Gia sit f e HomK(V, W),
la don eau khi va chi khi Kerf = {0}. Nth dimV Fa hfiu han thi
IimV = dim Kerf + dim Imf.
Cho ma Wan A e Mat(m. n, 1K), xem A nInt ma tran cua
inh xa tuy6n Unh f: K n —> Km trong thc od sei chinh tdc. Khi do
tang cua ma trail A (da dude dinh nghia trong chudng I) hang
rang cua f va chinh la hang cua he vec td cot cua ma tran A.
5 - Ma tran aim t‘i thing cgu trong the cd ad khac nhau
Cho f E HomK(V, W), trong co se: e = (e„ , en) f cO ma Wan
= (ad), nghia la f(e1) = Za ijei .
i=1
63

Gie sii & = (6o••ogs) le mot cd sa &bac, ma E, = /Cijej , tron€
i=1
cd s6 6, f co ma tren B = (b ii) ughia = Ebii Ma trar
i=1
C = (co) chide goi la ma tren chuyen co sa. Ta ca:
B=C' AC
Hai ma tran A ya B dude goi la deng deng neiu có ma trey
khong suy bign C de B = C-' A. C. Nhu yey hai ma Win cue
ding mot phep bign (lei tuygn tinh trong hai cd sd khic nhau
&dug clang.
Ta goi vet min ma tren vueng A la t6ng cac phen to trer
• &tang Oleo chinh. Hai ma tran tieing deng co vet bang nhau
Vet cua mot to deng cdu tuygn tinh la vet cila ma trail cfla ni
trong mot cd sd nal] do. Vet clad ma tran A dude ki hieu la trace
A hay trA. V6t cua td d6ng cliu f thick ki hieu la tracef hay trf.
§ 3 HE PHUtiNd TRINH TUYEN TINH
1 - He pinking trinh
He Ea kixi = bk (k = 1: 2, ... ,
i=1
dO aki , bk E K cho trUoc, k = 1, , m; it= 1, , n.
xi la cac en, dude goi la h0 phudng trinh tuyen tinh (hay 14(
phudng trinh dai s6 tuygn tinh) g6m m phudng trinh, n an so
Khi K la truang s6 (nhu R hoac C), thi cac a to goi la the he 66,
hQ se" to do.
Ma tram A = (ak;) goi la ma tram cac he sg.
64

a ll a12 aln
a21 a 22 a2n
b
b2
Ma trail Abs = goi la ma tran
b6 sung, no có ducic tii ma tran A bang each them cot cac he so'
tti do vac, cat thu (n + 1).
b
Neu ki hieu X = va B = la ma tran cot,

xni
bini
thi he phtiOng trinh (1) co the vieit duoi clung:
AX =B.
2 - HQ Cramer
He n phudng trinh tuyein tfnh n an s6, ma ma Iran cac he
s6khong suy bi6n goi la he Cramer.
HO Cramer c6 nghiem duy nhdt. each tam nghiem nhu sau:
Cdch 1: Xet phasing trinh ma trdn AX = B, vi detA # 0 nen
tan tai ma trdn nghich dao Al', va ta co:
X = B

Cdch 2: Xet he vec td cat an}, ma ai = (aii ,asi ,...,amj)
va b(b„ , b0,) la vac to cat td do, the thi he viat dude dU6i
clang =b. N6u ta goi Di la dinh thiic cda ma trail nhain
dude bang each thay cat thit i cda ma trdn A bat cot cac he si6 tp
do, thi x
I
=
D
ado D la Binh thiic cim ma tran A.
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 = (ao) nhu ma Wm cua anh xa tuyen tinh f:
K" —>. Km trong cAc co s6 chinh tac cua 1K'l va Km, thi tap hop
nghiem cilia (2) chinh la Kerf. Mei cci sa caa Kerf goi la met hee,
nghiem co ban cilia (2). HO (2) ludo có nghiem x, = = x„ = 0,
nghiem nay goi la nghiem tam thuong. KM rang A = n, thi he
chi co nghiem tam thuang. Khi rang A = p < n, thi tap hop
nghiem la kh8ng gian vec to n - p chieu (n la s6 an ciia phudng
trinh).
4 - He phudng trinh tuye'n tinh tong gnat
a) Dinh b./ (Gauss hay Kronecker - Cape11i)
He phudng trinh Za iixj = hi (i= 1,..., m) (1)
i=1
ce nghiem khi va chi khi rangA = rangAhs.
b) Phining phap khet nem Gauss
Cho he pinking trinh (1), n6u dung cac pilau Men dpi sau
day thi La van nhan dude met he phudng trinh thong doing TM.
he (1), nghla la he cO ding tap hop nghiem nhit he (1).
+ :Than hai ye cUa met phudng trinh nao do cem hee, vat s6 k #0.
+ Geng vao met phudng trinh cua he sau khi da nhan met
s6 bat ky vao hai ye cem phudng trinh khac.-
+ DAi the to cua phudng trinh cem he.
66

Cap Oen bi6n ling during cl6i vOi he phuong trinh
chinh la cac Olen bin d6i so cap thy(' hien Den cac clang dm
ma trail 1)6 sung Abs cUa he.
Dung phuong phap khil Gauss la Dille hien cac phen bign
ckii Liking during de chia he phuong trinh (1) v6 he phuong trinh
ma ma trail 06 dung:
P
0 1
I 131
hip
p+i
m-p
0 0 b'm
p n-13
phan t>i a phan goch (*) co the khac 0.
Khi rid n6u 13,;+1 + + > 0 thi he ye nghiem,
n6u = b'„,= 0 thi he có nghiem phn thuOc n-p tham s6.
§4. CAU TRUC CUA T11 DoNG CAU
1. Khong gian rieng - da thitc dac thing
Cho V la khong gian vec to tren truong K (1K bang K hoac C).
Wit anh xa tuy6n tfnh to V d6n chinh no dude goi la mat to
dOng calu tuy6n tinh. Tap ode tv ding cOu tuyeyn tinh eau V kY
hieu la HomK(V, V) hay EndK(V). Gia f e EndK(V), W la
khOng gian con ena V; W chick goi la khOng gian con bkt bi6n
elia V n6u f(W) c W.
67

Vec to a # 0 thuec V dude goi la vec to rieng cim f e EndK(V)
ling vat gia tri rieng X ngu f(a) = Ira, A e K. Khi do khong gian
met chigu sinh bai vec to a Et met kh8ng gian vec con bgt bign
cim f.
Val A E 1K, tap ker(f - AId) khi no khac {(5} la khong gian
con cim V, gam vec to khong va tat ca cac vec to rieng caa f ling
vdi gia tri rieng X. Kitting gian nay goi la khong gian rieng cim f
v6i gia tri rieng A, ki hieu
Gia sit ta cl6ng eau f e End(V) trong met co sa Mao do cim V
co ma tran A, thi det(A - XIn) 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 In I . Nha vay A la met gia
tri rieng cim f khi va chi khi X11 nghiem &la da that dac trang
eim f.
GM sit , Ak la cac gia tri rieng cigi met phan biet caa I
Px Pik la cac khong gian rieng Wang ling via ale gia tri
rieng do, thi t6ng Pr + Px2 Pxk la tong true tigp.
2 -Ong gian rieng suy Ong
GM sit V la khong gian vec to tren (Wong K, f e End K(V).
Vdi mei A e K, xot tap {a e VI co s6 nguyen m > 0 de1
(f-XIcl,)m(a) = 6). DO la met khong gian con caa V, khi khac {0}
no &arc goi IA khong gian Hong suy rang cua f ling vdi A va ki
hieu IA Ta thgy rang:
68

+ Vdi moi khong gian rieng suy rang 3 k 7r IA met gia tri
rieng cim f va Y, c Rs.
+ V6i X IA gia tri rieng cua f, dim2hx bAng bei cern nghiem 7.
cua da thitc dac trung /. f(X).
+ Mot g?",, la met khring gian con cila V bait bi6n qua anh xa f.
+ Gia sf1 , "f.k IA cac gia tri rieng phan biet tiing cap
va u, e {0} = 1, 2, ..., k) thi he cac vec td {u„ uk} doc
lap tuy6n tfnh.
3 - Tit citing clu luy linh
a) Ta not ring f e EndK(V) IA tu ding caiu luy linh nau tan
tai s6nguy'en k > 0 de'f = f o .. of= 0, hdn nua, nau # 0 va
k hin
0' = 0 thi k goi la bac luy linh cua E
Tu dang udu f e EndK(V) ma co cd sa te„ , en} sao cho
Re) = (i = 1, , n-1) va Rep) = 0, thi hay linh bOg n va ed sa
{ei, ..., en} dude goi 11 cd so xyclic d6i v6i f. Trong cd SO xyclic ma
trail cua f co clang:
0 0 0'
1 0
0 1 0
1
0 0
.. 1 0
69

b) f e EndK(V). U la khong gian the to eon cem V, U chide gui
IA khong gian vec to con xyclic chid vdi f nEu U IA f- bas biEn va
trong co mot ca so xyclic dee vdi f/U: U -> U.
c) NEu f e EndK(V), dimV = n, thi V phAn tich dude thanh
tang trip tiEp cua cac khong gian vec to con xyclic doi vdi f. Vdi
mdi so nguyen s 2 1, sE cac khong gian vac to con s chiEu xyclic
doi vdi f trong moi each phan tich dEu bdng nhau va bAng:
ran g(f 8') - 2rang(f s) rang(f
d) N'Eu 9 ) IA khong gian rung suy rang cua f dng vdi A, thi
(f - A.Ick) la td d6ng eau Iuy linh cern V.
4 - Ma tran clang chutha lac Jordan caa tat d6ng eau
GiA sei V IA khong gian vec td hisiu han chigu tren truang K,
f e Endx(V) ma da that ddc trung 94(X) cd dang:
i(X) = (>1 (X,
-X)`2 ... -X)sk
(cac &Ai mat phAn biet, i = 1, 2, ..., k), khi do V la King true tiEp
tha cac khong gian riAng suy rang V = x G3.? X2 e Xk
va trong V co mot co sa (ei) = 1, n; (n = dimV) dA trong co ad
do ma trdn din f tao bdi the khung Jordan
0
70

nam doe (Wang cheo chfnh so khung jordan cAp s vai phan
eh& X, bang:
rang(f - - 2rang(f - kildv)s + rang(f -
Ma tram clang tren cua f xac dinh duy nhAt sai khae each
sap xap cac khung Jordan. Ma tran do dude goi la ma trail dang
chuan tac Jordan cila tp (tang eau f.
B- VI DV
Vi d4 2.1: Xet R", R!" la the khOng gian vec td tren R. Cho
x„ x2, , x„, la m vec td thuee R". ]E la met khong gian vac td
con eila ChUng minh rang tap IF the vec td ena K" dang
Ztixi (t1, , e E Fa khong gian vec to con ena K', dimF =
dimE - dim(E nN), trong de N la khbng gian cac nghiem tha
phtlong trinh:
E ti., =0 (a do ti, tn, la cac An).
i=1
Lai gidi:
Xet anh xa tuyan tinh u c Hom(Rm, R") a do u(t,, t„,) =
EtiXj . Khi do F = u(E), vi vay F la khong gian vec to con cna
R'. dimF = dimE - dim(Keru'), u' = uI E.
Kern' = (KerU) fl E = E (1 N.
Nhtt vat dimF = dimE - dim(E n N).
71

Vi du 2.2. Cho xi , , xi, la nhUng vec to khac khong trong
khong gian tuy6n tinh V. Gia sit c6 Oleg bi6n d6i f e End V sao
cho f(x1) = x1, f(xk) = xk + vdi moi k = 2, ... , n.
Chung L6 rang he {x1, , xj la doe lap tuy6n tinh.
Lb gidi:
Ta chiing minh quy nap theo n.
Via n = 1, thi {x1} dee lap tuy6n tinh do gia thiat x 1 # 0.
Gia sit menh de dung vdi moi he fx„ xki; k < n - 1. Ta
chimg minh menh d6 dung vdi k = n.
Xet t6 hqp tuyen tinh
ECiXi = 0 . (1)
i=1
Khi do:
) = CIX ] Zcif(xi) = c,x, + Eci (xi +xi_1 ) =
i=1 i=2 1=2

= ZeiXi . (2)

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

i=2 i=1
Do gin thik quy nap hee, fx,, , doe lap tuyen tinh nen
C2 =e2 = = cn = 0.
Tit do suy ra c,x, = 0 ci = = = c„ = 0 Nhu vSy he
{x1, , xj doe lap tuy6n tinh.
72

Vi du 2.3. Trong khong gian vec to V cho he ak} cac
vec to doe lap tuygn tinh ma m8i vec to dm no la t6 hop tuyen

tinh cua cac vec to cim he 113„ , Hay ehung minh k < 1.
Ta eó the' gia thigt ••• • la doe lap tuy6n tinh, neu
khong to se lay he con doe lap tuyen tinh t6i dal eim {p,, , pi}
g6m m vec to va se elnYng minh k s m < 1.
Theo gia thigt a„ , ak bieu thi tuyern tinh qua Di , p, nen
ten tai cac vS hudng a.„ e K sao cho:
a; ; 0 =1 k) (1)
Ta gia sit k > 1. Ta se cluing minh
tuygn tinh. Xet t6 hop euyen tinh
k 1
ak phty thuec
/Xiai
1=1
I
j=1
<=>
i=1
( k
.i=1
j=1
pi =o
=0
<=> EauXi =0
j=1
(2)
vdi j = 1 do he pi leriic lap tuyen tinh.
Vi he phuong trinh (2) la he phuong trinh thuan nhat, có s$
an nhigu hon sg phuong trinh, nen no co ve s6 nghiem, nhu vay
73

ton tai nghigm (x„..., xk) 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/E2 IA khong gian thacing va h: E, —> E/E2 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,)/E2 (fang eau v6i kholu
gian Ei/EinE2.
L of Rich:
1)a) X& E —> E/E2 la anh xa chidu chinh cac, h =
h toan anh a ME,/ =11(E).
E, + Kern = E + Ker n = E
E, + E2 = E (vi kerb = E2)
b) Ta en Kerh = E, r1 Kern = E, n E2
Nhtt vey h don anh a Kerh = 0 a E, E, = {O}
2) GiA. se F = E, + E2;
Xet F F/E2 va k = 1-1/E, •
Do phAn 1) k la toan anh va kerk = E l C Kern = E l 11 E2.
Do do to co El/El fl E2 ding can not E 1 2>E2/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 p2 = 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
<=> p2 + p.q + q.p + = p+q <=> p.q + q.p = 0
2) p.q la phep chigu <=> (p.q)2 = p.q.p.q = p.q = p2 q2
Er>p(qp—pq).q q (pq — pq) (Imq) = 0
.(=> (qp — pq) (Imq) c Kerp.
3) Vi mai phep chigu p co rang p = trace p,
va vet cim tong hai anh xa tuygn tinh bang tong cac vet cim no,
cho nen ngu p + q la Agri chigu thi:
trace (p + q) = trace p + trace q.
va rang (p + q) = rang p + rang q.
Tit do dim Im (p + q) = dim Imp + dim Imq, suy ra
Im (p + q) = Imp ED Imq.
D6 °hang minh Ker (p+q) = Kerp n Kerq
to nhan thay kerp nKerq c Ker(p+q).
75

Nguuc lai vdi x e Ker (p + q) thi p(x) + q(x) = 0.
Do Im (p.+ q) = Imp $ Imq nen MI p(x) + q(x) = 0
suy ra p(x) = q(x) = 0, hay x E Kerp n Kerq.
Vi du as. Gis sit E va Fla hai khong gian vec td tren
trulang K; Horn (E, F) IA tap cac anh xa tuyeal tinh tit F Mn ]F;
Ft la khong gian vec to con cna F.
a) Chiang to rang £ = e Horn (E, F) / Imf c F 1) la khong
gian con cria Hom (E, F).
b) Gia. s5 F= K la mot phAn tich cua F thanh t"o"ng tryc tip
i=1
ciaa khong gian F. Vbi mdi F, xet 2, = if E Horn (E, F) I Imf c Fd.
Chung to rang Horn (E, F) = 2.
WL gia'i:
a) Do g ding kin vai phep town tong anh xa va nhan anh
xA vdi mat vet hriong thuac K, nen hieln nhien la khdng gian
vec to con crIm Horn (E, F).
b) Vdi h E Horn (E, F), vbi m6i x e E, to phan tich h(x) = y
theo eec thinh ph'An y = h(x) = Eyi , yi e F1
i d
Xet E -, K, e Horn (E, Fi) = 2 hi(x) = yi
76

Va vdi moi x e E, thi h(x) = Eh; (x)
i=1
Do 05 h= , e
Gia six Eh; = 0 nghia la vial moi x e E,
( ) (x) = hi (x) = 0, h,(x) e F,
1=1 1=1
to do suy ra 11,(x) = 0 voi moi i vi F = e K.
I=1
Nhu ray Horn (E, F) = e g.
=1
Vi du a 7. GO Q la Huang s6 h> u ti va WI la mot tkp hap
khong rkng. Ki hiku E la tap cac anh xa tif c32 vao Q. E la
Q - khong gian vac to viii hai phep than: f, g e E, a e cc9, k E Q the
(f + g) (a) = f(a) + g (a)
K. f) (a) = k. f (a).
Gia sif V la khong gian vac to con ciaa E.
1) Chang to rang: nku f e V va co n dim x„ x„E Ca
sao cho
ingui=j . . —
k(x,) = = ej=1,n
Ongui#j
thi he 10, •••, f,,} dOe lkp tnyen tinh.
77

Goi W la khong gian con cea V sinh bed f„), khi do
m81 g E V 11611 viet dude met each duy nhat dudi clang: g = h, +112,
a do h, e W, h2 E V va. h = 2(x)= 0 yea mai i = 1,n.
2) Chung CO rang hai tinh chat eau day la Wong clueing:
a) dim V n
b) Ten tai n ham g„ thuec V va n diem x„ x„ ciaa
W( sao cho g, (x,) = 8,; vet moi i, j = I,n .
Lai gidi:
1) Xet to hop tuyen tfnh EXif, = 0, c Q.
1-1
Khi do vdi moi x E cam, LW; (x) = 0;
Chon x= x„ thi (xi) = 7.j = 0 ( = 1,2, n)
suy ra he {f„ , f„) doe lap tuyen tinh.
Val g e V, ham h, e W cAn tim phai thaa man
g(x) = (h, + h2) (xi) = h r (x,) vdi moi
Do If„ f„) la co sa cem W nen h, :11 1 (x, = EgfrA .
lei 1=1
Dat ha =g - h, eV
thi h2(xj) = g(xj)-112(xj) = 0 vdi moi j = 1 n.
78

2) Theo phan 1) neu b) clime th6a man, thi he (g„ g„) doe lap
tuyen tinh, do ay dim V 2 n; nglila la menh de b) a) thing.
Ta chung minh a) suy ra b) bang phydng phap quy nap
theo n.
Vdi n = 1; n6u dim V 2 1, tan tai fi x 0 va vi vAy có x, e c14
de fi(x,) = A x 0. Chon g, = thi (x,) = 1.
Gia sii menh de dung vdi moi k = 1,2, n.
Ta chiing minh nd dung vdi n 1.
Theo gia thi6t quy nap dim V n + 1 dim V ?. n nen tan
tai n ham g, E V va n diem e A de gi(x,) = 8,, (i, j = 1,2, ..., n).
Goi W la khong gian con sinh bid {g„ gn}, to cd. dim
W< dim V.
Theo cau 1), vdi f e V W, to cd f = h, + h 2, vi h, e W nen
h2 0; vi th6 c6 x,,, d'e' h1 0; (x,„., x x, vdi i = 1, 2, ..., n).
h2
hi-
) t
bin+, vdi MOt i = 1,2, ...,
n+1. Bay gia = g - = 1, ..., n),
thi vdi j = 1, n to co gi (xj )= (xJ )= si; va gi (xn+1 )
=g. (x„.1) - x; = 0 n6u 2,1 = gi (x„,)_
Nhu Nty co n4-1 ham {g-i } th6a man digu kien bai town.
Vi du 2.8. Gia sit 98 la ho dem doe cac anh xa tuy6n tinh
tit R" (16n R"I; vdi mai a e R" xet 0(a) = {f(a) / f e :43}. GM sit
g la Anh xa tuyein tinh to den âR" sao cho g(a) E :0304 vdi
moi a e K. Chung minh rang g c PC.
79

Lo gidi:
Ccieh 1. TM m6i x e R, xet vec to dx (1, x, x2,
H9 eac vee to dx c6 tinh chat:
a) vdi x„ doi mat phan biet, thi taxi dPe ld
tuy6n tinh, vi Binh
# 0 (Dinh thite Vandermonde)
b) V6i m6i dx e R', tan tai f E sao cho g(Ci;( ) = f( ).
Ta phai chiing minh g c 93, nghia la c6 f e d f
g@t„i ) tren bb la co so cua R".
Gia sii node Lai, vdi m6i f E c6 khong qui (n - 1) sd the
phan biet xi cl6 g(dxi ) = *xi ).
Do hop &dm Mtge eac tap bop hfiu han phan to 1a mat to
hap kh8ng qua deM dude, nen sod cle vee to d x e ma g(dx
e g3(ax ) le killing qua d6m dude. Digo nay trai vdi gia thie
Nhu vay phai co co sa Vt xi ,...,ax ) oda an vti 06 f E tgi d
= f(dixi ) vOi moi i= 1,n; nhu vay g=f e
80

(rich 2. (Mtn}, cho doe der hied mot sa sri kien cilia khong
gnin metric).
Theo gia thiet vdi moi a e der f e de" f(a) = g(a). nhu
vay — g) rad= 0 => a e Ker (rig).
NInt vay U (Ker (f — g) f E gq} = R.
Hia sir g vSy vdi mai f E thi f — g x O. do vay
dim Key (1— g) n — i. Vi vay Ker (1 - g) lie met kliong gian con
thing, khbng dau tra mat cent R". Dieu nay gay nen man thuan
do RS la khong gian metric day vdi metric dieing Hwang vii
dinh 19 Haire aid rang: met khong gian metric day kheng the
bang hop dem dude mita nheing rap hop khong dim Hu mat.
III. du 2.9. Cho hai so nguyen dining r, n ma r < n:
(it c Mat (Th R). rg." = rank rang cl = r m& (CP =
Hay chdng minh vet ()f = a,, + a i„ + + a„,, = r. (Vet can
ma Man red thudng duo( ki Heti bdi trace cel).
Xet f e End (IR") co ma trhn trong cd ed tly &nen
e = le,, e„). i(e,)= apie
Theo gia Hired to cd f2 = f.
Bat Y = - f(a) I a e khi de Y c Kerf, nen Yea a e
thi x = a - 1(a) e Keil, do vay a = f(a) + x e IMF+ Karl
8 1

Ta co Imf n Kerf = {0), that Gay, gia sit p e Imf Kerf, thi
co a e R" d f(a) = p va do 13 E Kerf nen f(0)= 0 suy ra:
f(p) = 12(a) = f(a) = p = 0, vay Imf n Kerf = {0). Nhu vay R"
Imf O Kern dim Imf = r va f I Imf = id, Chon ed sa s Rua R" sac)
cho {E,, £,.} c Imf, {E„,„ E„) e Kerf, thi trace f= trace al= r.
Cho Nhaat lai rang n6u f la anh toa tuyeat tinh cie'n
Jrco ma Ran oat = (ad trong cd sa e = (e,, nao do ciaa Tthi
so' a„ +,...,+ a„„ khong phu thuoc vita vice chop cd so oda °L va
clack goi la vat ciaa anh xa tuyan tinh f va chiac kf hieu la trace f.
Vi du 2d0. Gia sa °W la hai khbng gian vac td treat
twang K, f c HomK CP; °Tf).
Xet anh xa f: Kerr Gil"
[a]-a f [al = f (a),
Hay chttng to f la dun eau tuyen tinh va Imf = Imf , ta do
f la dang cau to /Kerf len Imf.
ianh xa f lh sac dinh, khong phu thuR vao dai dien. That way,
vdi [al = thi a - a' e Kerf, W do f(a) = f(a") hay II-al = f
thd thay f la Maya tinh va Imf =Imf.Ta chung to f la ddn cau.
That Ray RR [a] # [a'] thi a - a Kerf f(a - = f(a) - f(a) # 0
do do f(a) # f(a) suy ra f [a] # f Nhu way f don cau va do
do f la clang cau tit cliKerf len Imf.
82

Chu" y: neat so chieu Irau hen, thi tfx vi du Hen to
dim tillierf = dim Imf say ra dim "P= dim Kerf + dim Imf.
du 2.11. Giei he phudng trtnh
+2x2 +3x 3 +4x4 =30
-xi +2x., - 3x 3 + 4x 4 -10
x, - X3 +x 4 = 3

x i + +x3 + X 4 =10
1 2 3 4
-1 2 -3 4
D =
0 1 -1 1
1 1 1 1
Day 1a he phudng Huh Cramer.
gthi:
= -4
80 2 3 4

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

1), =

D,=
= -8
3 1 -1 1

0 3 -1 1
10 1 1 I

1 10 1 1
1 2 30 4

1 2 3 30
- -1 2 10 4

-1 2 -3 10

Da = -12 D'= = -16
0 1 3 1

0 1 - 1 3
1 1 10 1

1 1 1 10
NMI vey , = 1, x2 = 2, x3 = 3, x 4 = 4.
83

Vi dii 2.12. Gihi va bran luhn theo thaw s6
A.X1 x, + x• 1
Xx, + x. 2,
+ 4x3
I) =
a) Veil X s 1 vic a x -2. Day la he Cramer.
it +1 1
va ta ce, - , x -
X +2 2, + 2 +2
b) vdi A = 1, to c6 he Wring during vdi:
x i + x2 + x,, =1 hay x3 = I - x, -x.,
do x i , x, lay trtyl,
c) Vol X = -2. He co dang:

-2x1 + x., + x. = 1
xi 2x, + x3 g - 2
xi + x, - 2x 3 1
GUng vg still ve curt ba planing trinh Ln c6 0 = 3, nhn vhy he
voi nghiam.
84

Vi du 2.13. Dung phitong phap khn. hay giai he phuang trinh:
xt + 3x, + x 3 + 2x 1 )(5 = 2
3x 1 + 10x9 + 5x3 + 7)( 4 + 5)(5 = 6

2x 1 + 8x, + 6x 3 + 8x 1 + 10x5 = 6 (I)
2x1 + 9x„ + Sx 3 + 8)41 + 10)(5 = 2
2x 1 + 8x, + 6x 3 + 9x + 12x5 = 1
Lm giai:
Nhan hai ye aim phtiong trinh (tau vdi Inning so/ thich hop,
I.()) acing vac) eac phtaing trinh khae, ta clia; he pinning trinh
()rang throng vdi he (I):
x + :3x2 + x,, + 2x + X5 - 2
X9 + 2X3 x4 + 2x5 = 0

2x, + 4x3 + 4x4 + 8x5 = -2 (II)
3x, + 6x3 + 4)( 1 + 8x5 = -1
2x, + 4x3 + 5x 1 + 10x5 = -3
Nhan pIntong trinh thit hai cna he (II) vdi gag s6 thich hop
roi tong viva du; phuong trinh kink cita he, ta &toe he Wong
dyeing-
x 1 + 3x2 + x3 + 2x 1 + x5 = 2 (I)
x, + 2x1 + x, + 2x5 = 0 (2)
2x4 + 4x5 =-2 (3) (Iii)
x 1 + 2x5 = -1 (4)
(
3x1 + 6x4 -3 (5)
85

Car phtfting trinh thit (3), (4). (5) trong he (III) la tudng
during. Vi 04 he (III) tudng during vol. he
Giai
x i + 3x2
y 2 +
he (IV), to (little
xn = —1 —
X2 = 1 —
xi = 1 +
2x 3
2x 5
2x.
5xi
+
+
+
2x 4
x- I
x 4
3x5
+ 2x5 =
+ 2x3 =
a do x4. x
2
0
-1
Inv 31
(IV)
Vi dv 2.14. Cho hai ma lien A, B thuOr Mat (n, K),
A = (ad, B = 09.
Khi do A + B =(a i +1)0 (bloc goi la tang hai ma tren A va B.
Chang minh rang:
1) I rang A -rang B l < rang (A + B) a rang A + rang B.
2) rang A + rang B -n a rang (A B) a min (rang A, rang B)
3) Nan A' = E, tin rang (E + A) + rang (E - A) = n. (3 do E
la ma tran don vi cap n).
1) Gia stir f, g la hai ph&n tit cilia End (â "), co ma 'Iran A, B
Wong fing trong m(llt cc; se( s = (s 1) di( cho. Khi do f + g c6 ma
tren A + B. VI Im(f + g) c Imf Img, nen:
.dim(lm(f+g)) < dim(imf + Img) dimlmg.
86

Tit de suy ra:
rang(A + B) < rang A + rangB.
Mat khdc:
rangA = rang(A+B-B) < rang(A+B) + rang(-B)
suy ra: rangA < rang(A + B) + rangB
Tit do:
ningA - rangB < rang (A +
ta: rangB - rangA < rang(A + B). VI yay
rangA - rangB 15 rang (A + B)
2) Ta co f: K" , g: K" la hai anh xa tuy6n
tinh, Im(f o g) = Im(f img) c Imf nen rang(AB) < rangA.
Mat khan: rang(AB) = dim(lm fog) < dimlm f = rangB.
Do yay rang(A o B) < min(rangA, rangB).
Bay gio ta churig minh
rangA rangB - n < rang(AB).
Ta co dim lle = n = dimIm(f o g) + dimKer(f o g).
Mat khde 26t anh xa Kor(f g)/Kerg Kerf (1 Img
a do tI/4xl = g(2), yin X E Ker(f 0 g). De they (To la (Tang au tuyeal
tinh. Vi
dimKer(f o g) = dimKerg + dim(Kerf fl Img).
Tit do dim Im (f 0 g) = n - dim Ker(f o g)
= dim Img - dim(Kerf Img).
87

Nhu vay:
rang(AB) = rangB - dim(Kerf (I Iing) je rangB - dimKerf
rang(AB) -e rangB+rangA- n.
3) Vi A2 = E (E la ma (ran den vi), nen:
(A - E) (A + = 0
Vt vay, then phan 2), to co:
rangA -E) + rang(A + E) - n <0
hay rang(A - E) t- rang(A + E) n.
M3t khac, then phfin 1)
rang(A+E) + rang(A-E) 2 rang(A = rangA = n
Do vay rang(A + + rang(A - E) = n.
V( du 2.15
Cho ma tran A e Mat(n, R), cac phan to (ran duang cheo
chinh bang 0, con cac phan to khac hang 1 hoac bang p, d (16 p
la mot eel nguyen lan hdn 2. Chung to rang A e n-1.
142i
)(et ma tran II = (h ij ) e Mat(n, 1R),
.3 do b ij = -1 vol moi i, j. Khi do rangB =1.
va A + B = (CO, vdi C,, = -1, con thc phfin to khOng thuOc during
cheo chinh bang 0 hofic bang p - 1.
Nhu vay det(A+B) = (-1)" (mod(p-1)).
88

Do p-1 > 1 nen det(A +B) # 0 hay rang(A+13)= n.
NhUng Chao vi du 14, to co:
rang(A+B)< rangA + rangB = rangA + 1
Do vay rangA 2 n-1.
Vi du 2.16
Gui sit (p la 1-1101, L1.1 long cau run khang gian vac td phis n
chieh V. Ching (Dinh rang tan tai (ig khong gian vec td con 9 -
hat bin k chieu Vk:
0= V„ V, c c V„ = V.
gidi:
9: V —> V la Di dOng cau clic( C - khon g gian vec (5 V, 11611
ce mot vec td Hong e t
Bat V, = Vect< el >, thlkhong gian v@c td con 1 chi6u
(p - brilt biOn.
Gia sv (ili xily dung duo() car khong gian con (4) - bat Heal V,.
V2, ... Vk ma V, = Vect <el >,
V2 = VITA. <ei , >.... Vk = Vect e h 8: dimVk =k.
Tel get long cau Wk : y wk.vk
[U. I H> [(P( )]
89

cam sinh bai clang cau p. Do tirk IA ta clang can caa khong gian
vac to V/Vk tren truang se" phtic C, nen Co vac td rieng rea +i
ling vdi gil tri rieng can chfing minh:
Ubl Vect le, ....,e k , ek*, I
khong gian con k+1 chieu dm V va la - bat Man.
kik
Xet re hop tuyan tinh ei = 0 .
i=1
Neu ?Lk+, 0 thi ek,i e Vk [ ek+1 = [6], trai vei vice
k
chon{ ak,, I la vac td rieng, vay 2 k,l = 0 tit do =0 nen
iat
= 12 0 do he {a; ...6k doe lap tuyan tinh. Nha way he
lei , eki doe lap tuyan tinh = k + 1.
Ta chUng minh Vkk i lap - bat Bien. Chi can chang minh:
{14 ) e Vik*I-
Ta cif) pk k im(rp )1 ) 1k- k+lk = ` T‘ -k+1 kk = •`k-hlk k-kk = `- k+1 k
p (e +, ) - X e k+, e Vk C Vk+l, tit do suy ra
Bang each do, ta say dung clam cat khbng gian:
Vo = 0 c V, c c Vn = V p - bat hien
Nhain xet: Ma tran tha ta clang cgu p trong co sa 1 , ... e „}
)(ay clang a tran c6 clang tam giac tren. Dung ngan ngil ma [ran,
ta c6 the phat hie: moi ma tran phew vuong cap n dgu d6ng
Bang vOl mat ma tran tam gile tren.
6k+1 ) e Vk-kik
90

Vi du 2.17. (Dinh It Hamilton - Cayley)
Cho V la mot K - khong gian vac to; cp e End(V); 9 c6 ma
tran A trong cd so e = (ed i = 1, ... n; 1,, - la ma trail dun vi, khi
do det(A - xI„) IA ma da thdc bac n, doe gut la da thitc da, c
trung cUa tii (ItIng cau cp. D6- thay da thuc nay khong phu thuc
van ed so e = (u) da chon. Ta cling goi da tilde tron la da HI&
dac trung dm ma trail A.
Gia sit det(A - xI„) = aox" + alx"-'+... +a„_,x +a„.
Khi do to co: a0A" + a,A"-' + +an_,A +a„ = 0
hay 42'1 + a,""-1 + + an.Id = 0.
Lai gidi:
Xef ma luau B la ma trail chuyi1n vi ci'm ma tran phil hop
am A - xI„. Cac phan ;I'M ma Iran B IA nhfing da thilc cua x
v6i he tiI trong truong K, c6 hac khong qua n-1. Ta
B = B0 +13,x + .„ + x"-2 + Bn_1 . x"-1
trong do Bo, Bi, B,1 la nhung ma trait vuong cap n, khong
phu thuiic x. Theo tinh chat dm ma tran phu 110p, to co:
(B0 + 11,x + +BR,. x"1') . (A - xI„) = det(A - x1,). =
= (aox" + + + an_ix + a„). I„.
Ta co mot he cac clang thitc sau:
B0 A = an. I.
- Bo = and • In •
B2A - B, = an-2 • In.
91

B„-,A - 13,12, = a, 1„.
-B„., = a,,T„.
NhAn vg voi vacua the clang their trcn Bin Met vdi I n, A. X-
A" rOi enng Jai, to (kith
an A n + + a, ^ A+an I,= 0.

Chu Dinh 19 Hamilton-Cayloy thadng dude ghat binu:
MO ma tran vuting dela la nghiem maa da tithe dar trnn,
ua ne.
Vi rig 2.18. GiA s t B la ma trap ley linh, la ma tran giro
halm vdi B. Hay chtIng mink
det(A + B) detA.
Leh gidi:
a) Xet twang hen detA = 0.
cia stY n la bac lily linh rem B, nghia la Bn = 0. Ta re:
= zcnk A kBn-k = Ecnk A k w _k =

(A + B)" A Ick Ak-1 n--k
k=0 k=1 k=1
Vi detA = 0 nen det(A+B)n = 0. Vi way
det(A + B) = 0 = detA.
b) Ttheng Mk) detA = 0.
Do AB = BA non BA-' = A-1 B.
92

Theo gii thiat B lay linh, nghla Isl co se n de 13" = O. tit do
-1 B)" = 0 vi vay A-1 B lay linh. NM( vay tan tai ma tran kheng
ay Men C do C- 911-1 B).0 la ma tran dime tan hal nhning ma
An inking Bang:

' 0 o
1 0
1

0 1 0
am riot thco during cher) chinh. tr.( do ta co det(1 + A -1 B) = 1, a do
la ma Ban don vi (tang dip vdi B va A.
Nhu vay det(A+ B) = det[A(I + A-113)I= detA. (-Iot(I +2.111 B)= detA.
Vi dy 2.19. GiA six V la met khong gian vec td tren truang
va f e End,(V). Chiang minh rang Kerf = Imf khi va chi khi
= 0 vh ten tai h e Enci,(V) do h o f+ foh= Hy.
Didu ki(es chi: Tu f = 0 say ra Imf c Kerr.
Bay gia ta chUng minh Kerf c Imf. VOi IV la Itheng gian
ni bat 14 cna V, to co:
(1) 0 f + f 0 II) (NV) c hof (W) + f 0 h(W).
Do ta co of + f oh= Id, nen
W c ho 1(W) + foCuh(W) vdi mei W c V.
Lay W = Kerf, thi f(W) = 0 WI ta
Kerf c f o h(Kerf) c Imf.
Nhu vay Karl= Imf.
93

Digit ki•en din
Gia si7 Imf = Kerf = V,. Goi V2 la phein bet tuygn tinh ego

V, trong V nghia la V = V, $ V2; pl : V VI la phep chi&
chinh tac, nghia la ven x e V, x = x, + x 1, x, e Vi thi p,(x) = xl .
Xot anh xa ?: V2 -)
x2 1—› 1 (x2) = nx2).
De thgy If la Bang ego. Dat h = I -1 o P I , to en:
f 0 h(x)= f -'. pi(x) = pi(x) = xi.
Thong tki h f(x) = ho f(x, x2) = ho f(x2)
= -I 0 Pi 0 f(g2)= f((2)= -1 • ?(x) = x.2-
Do vgy (10 h + h (x) = xl + x2 = x.
Do do foh+hoN Ido.
Tu gia thing Imf = Kerf suy ra f 2 = O.
Vi du 2.2a GiA si V la khong gian vac to tit' truong 1K. v?
f e EndK(V). Vol m61 da thfic P e IK[x], P = ±a ixi , dat P(f)
=0

'a; f' thu6c Endk(V). De thyIKPc] EndK(V) (13(P) = P(f
=o
la mat d6ng cau tuyen tinh.
Da thitc Pf &toe goi la da thew t6i tie'u cum to d6ng au f new
P1(0 = 0 vix n6u Q(f) = 0 thi Pf la vac cua da thew Q, voi Qe IK[x].
94

1) Gia sia V I la kh8ng gian con oda V bar bin do=i vdi f va
f, = f I V, la clYng au cam sinh. Chung to rang da thfic to) tiaai
P, ena f, la Pdc cua da thfic to) tiPu P r.
2) Gia sfi rang V phfin tfch throe thanh tong cac kheng gian
con baat bign del vdi f: V = la dying (Au cam sinh trim
=1
V, va P, la da thuc tot tiPu cPa f,. Chung LO rang P, la boi chung
nhO nhavt cua cic da thfic P.
Liii gidi:
1)Vi f = f1 nen vdi moi Q E K[X], La co Q(f) = WO fit VI.
kreh Pf lA da thfic ten tiUu rim f, thi Pr(f1) = 0. Do vhy Pf la bOi
da thfic toi ti6u P, oda f,.
Ta cling nhyn flay rang P, khang nhal thi6t la uoc thpc sp
Lim P,, khi V, la klieg-1g gian con thpc so cim V. Vi du ne).1
'= %H", thi La c6 Pf = P, = x - vdi moi V, c V.
2) V, la khong gian con f bi6n, nen do can 1), da thfic
:61 thi6u Pr chia hect cho vdi moi i = 1, 2, p. Do vyy Pf chia
Mich° bOi chung nhO nhiit P cua Pi.
Ta chUng minh P(f) = 0. That vyy, vdi moi i = 1. p to c6
I)
= Q . P, tit c16 vdi moi x c V, x = Ix ; , I'(f)(x) = LP(f)(x,) =
1 =1 1=1
P p
P(f; )(xi ) = EQ; (fi )13; (fi )(x; ) = 0 do Pr(f) = 0 vdi i.
r=-1 i=1
NMI vyy P(f) = 0, to do P chia 116t cho da thik toi tiPu Pf
ty dang can f.
95

X = (a,), A1 =
2.1. Cho ma tran X E in. ]K)
70,
6
)19 -H la cot tha j ca» ma tran X, =1,2, 1)
C - BAT TAP
§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINH
a )
B,, B.,. , B„ la cite cot cUa ma tran X. X'
Chang minh rang cite khong gian con rim R" sinh boa
A l , A„ va sinh bai 13 1 , , B„ trimg nhau, tit de suy ra
rangX = rang(X.
2.2. Chang minh rang mei ma ran hang r d ou via dude
duai clang tang cua r ma tran hang
2.3. Cho E 19 la hai K - khang gian vec to va u e
Hom w (E, F). Cac vec tel x 1 , , x, thuoc Kern; y l , y, la nhang
vec to brit k9 thuec E. Chang minh rang hai trong ha khang
dinh sau keo theo khang dinh MEI ha.
1) lx,, , la cd sa can Ecru.
2) B(y,), u(ydl la cd sa cUa Imu.
3) Kr , ys} la ed so ciia E.
Tit do suy ra nen E hau han chieu thi
dimE = dim(Keru) + dim(tmu).
96

2.4. a) GM sit E va Fla hai kheng gian vec to him han
chieu tren truring IK, dimE = dimF = n. u e HomK(E, F). Chung
minh rang u don cgu khi va chi khi u town
b) Neu vi du chUng to rang neeu E va F có so" chie"u vo han,
menh da tren kfrong dung MM.
2.5. Cho a la mat ph6p thgbac n va u E End(C) xac dinh bar
u(Z,, Za) = (Z.K0 , Zem •• • ;II) )-
a) Hay tam ma tran A cim u trong co sa to nhien te r,
&la C.
It) Tim tat ca the ma trait B e Mat(n, giao hoan duple vdi A.
2.6. X6t kh8ng gian vec td E him han chigu tren truang K.
a) GM s>i fe„ , e„) la mot co so cim E, a„ , a,, la nhiing
vo hudng doi mat klMc nhau, u e End(E) the dinh bai
u(ei) = ° ;e; = 1, , n). Chung minh rang nth v e End(E) ye
u.v = v. d thi ton tai nhUng vo hudng p„...,j3„ sao cho v(ei) = [3;e;.
b) ChUng minh rang nth to clang cgu tuygn tinh u giao
hoan vdi moi to clang eau tuyeIn tinh tha E, thi t6n tai ve hitting
E K u(x) = 7rx vdi moi x e E.
..2.7. Cho A =(aIii)e Mat(n, K); det(A) x 0; V la mat killing *4,
vec to tren truang K va uj e End(V), j = 1, ..., n. Chting minh
rang nth cac to deng cgu tuygn tinh vi = (i = 1, 2, n)
giao hoan vat nhau, thi cac u3 cling giao hoan vdi nhau.
97

2.8. Gia sil A e Mat(n, R), detA # 0 va trong moi (long efn
A co dung mot s6 khac khong, bang ± 1. Chung minh rang:
a) Al = A-1
b) Co seta nhien m de Arl = A'.
2.9. Cho V la khong gian vac to Hen truing K va u
End(V), x la vec to cria V them man IOW = 0 va u° 4(x) # 0 v6
mat se' nguyen throng q nao do. Chimg minh rang he {x = u°(x)

u(x), , 10-1(x)} lap thanh mat he [lac lap tuygn tinh.
2.10. Gin sir) V la lit - khong gian vec to n chialt; f, g
End(V), Id la anh xa clang nhdt cim V. Chung minh rang nei
Id - g o f la clang cdu dm V thi fog - Id cling lA (tang cdu ciaa V.
2.11. a) Hai tu citing cdu u, v E EndK(V) duac goi la Worn
during ngu c6 die dang cdu p, q cem V sao cho uep=q v
Chung CO rang u va v taking during khi va chi khi chung NS cum
hang.
b) Tit a) suy ra rang nalu X E Mat(n, K), hang X = r, thi tar
tai cac ma trnn khong suy bP6n P, Q e Mat(n, K) sao cho:

'Ir 0 I, 0
X = Q . . P a do la ma tran vuong cap n

0 0 0 0 ,
al gee tren b8n trai la ma trnn don vi h cap r.
2.12. a) Cho V a khong gian vac to n chigu tren trUang
Ira u, v e End(V) sao cho u o v = 0.
Chung minh rang hang(u) + hang(v) < n.
98

(^ri±ng 11111111 rang vo. mai hi ci6ng eau u e End(V) deu
Ong cdu V e End(V) sao cho u 0 v = 0 va
hang(u)+ hang(v) = n.
2.11 Cho E, F, G, H la cac khong gian vec td hitu Mtn cht4u
men trodng K, u c Hom(F. 0), u ce hang r. Hay tim hang dm
cac anh xa tuyeho t(nh sau:
a) cp: Hom(E. F) —> Horn(E, G)
u v
b) di; Hom(G, H) —> Bom(F, H)
F--> 0)01.1.
2.14. Cho E la khong gian vec to n chien, u, v e End(E) sao
cho rang (u - Id E) = r, rang(V - IdE) = s. Chiing minh rang
range o v - id s) r + s.
2.15. Cho E la khOng gian vec to tren trtiang K Nth u e
End(E), rang u = 1. ChUng minh rang t6n tai 1. e 1K a u2 =
Min ntm, neAl X # 1 thi Id E - u la clang mill.
2.16. Gia s>i V la khong gian vac td tren trtfong so thuc R;
dim V = n; f e End (V).
Chdng minh rang en sdN nguyen ding sao cho rang (fk) =
rang (F`') vol moi k 2 N.
2.17. Cho ma tran A e Mat (n, K), A = (a id ma aii = 0 vol
moi i = 1, 2, n. Chimg minh rang t6n tai ma trail B, C e Mat
(n, R) de A = BC — CB.
99

2.18. Cho hai ma tram A, B E Mat (n, K).
a) Chiang minh rang ma tren A d6ng clang vol X In thi A=X I„
X e ]K, In la ma tran ddn vi.
b) Neu A, B la hai ma train d6ng dang, thi A2 deing clang vet
B2, At &Ong clang vdi 13' va ndu A, B kha nghich, thi A -1 &ling
dang vdi B-'.
2.19. Chung minh rang m8i ma trail vuong cep hai tren
truring K &et' d6ng clang vdi ma tran chuydn vi eila no.
2.20. a) Chung minh rang to d6ng ceu tuyen trnh u e End(V)
thCla man didu kien
u2 (Id — u) = u (Id — u)2 = 0 thi u lily clang, nghia la u 2 = u.
b) Flay tim phep bign del tuygn tinh kluing lily clang u thda
man (Id — u) = 0.
c) Hay dm phep bign ddi tuygn tinh khong lily ding u th6a
man u (Id — u)2 = 0.
2.21. Cho V la kh8ng gian tree td tren trtiang K.
a) Neu f, g la nhfing dang tuyen tinh tren klreng gian vac td
V thOa man f(x) = 0 nett g(x)-e 0, thi co v8 Inking a e K M f=a. g.
b) Cho f, f„ f„ la nhung dang tuyen tinh trail khong gian
•vdc td V sao cho f(x) = 0 neM fl(x) = 0, i = 1, 2, ..., n. Chung minh
rang ton tai nhung vo hudng a l , c(2 , , an sao cho f = a1 f, +a2f2
++a„fn .
2.22. rot khong gian vec td n chigu V veil cd sa le l , e„) vk
V* la khong gian lien hop ens, V, nghia la khong gian cac dang
100

tuyen Utah tren V. Vdi moi = 1, 2, n xec dinh clang Wyk
tinh e V* bait
Qej) = Sij =
j
1 i = j
Chung minh rang (£1, ...,1„1 la cd sa cua V* va do do.dim V*
= dim V = n. Cd sa fb, f„} ciudc goi la cd sa lien hop hay ed sa
d6i ngau N:i ed sa ••.,
2.23. Chang minh rang nen fil, la nhfing dang tuy6n
tinh tren khong gian vec to n chieu V, vdi m < n thi ton tai
phiin ti x t 0 trong V sao cho fi(x) = 0 vdi moi i = 1, 2, ..., m. TU
do hay suy ra met k6t qua v6 nghiem cua he phudng trinh
thuan nht.
2.24. Cho V* Fa khong gian lien hdp cna khong gian vec td
V; u e End(V). Xet u*: V* —> V* the dinh bat u* (y) = you.
a) Chling minh rang u la phep Men dot tuyen tinh cua V*,
u* chive goi la phep Bien d6i tuyen tinh lien hop vdi u.
b) co sa{e..., en} (1) cila V va cd sa{fi , •••, in} (2) cua V*
lien hdp vdi ed sa (1). Chung minh rang ma tran (l u), cua u*
trong cd sa (2) la ma trait chuye'n vi cua (ctii)„ cua u trong Cd sa
(1), nghia la Po = a3; vdi moi 1 5 i, j n.
2.25. Cho V* la khong gian lien hop vdi khong gian vac td
V, S la be phan dm V. RI hie, u 8° la tap tal ca the phan to f E
V* sao cho f(x) = 0 vdi moi x e S.
a) ChUng minh S° la kitting gian con cua V*
b) ChUng minh rang nett dimV = n va S la kitting gian con
m chieu cua V thi dim S ° = n-m.
101

2.26. GiA s& V* IA khong gian lien hop vdi V, T la met be
phan ciia V. Ki hieu T° IA tap tat ca cat x e V sao cho f(x) =
vdi moi f E T.
a) Ch'ing minh I"' la khong gian con ciaa V.
h) Xet S c V va S') c V* duct dinh nghia a bai 2.25. Chung
minh rang (St la khong gian con oils. V shill bal. S. Dao biet.
S la khOng gian con thi (S")" = S.
c) Gia st V hfiu han chi6u. Chung minh (T") ) la Xining gian
con cfm V* sinh boi T.
D4c biet, neal T la khong gian con cUa V* thi (T") ) = T.
d) Chting t6 rang Meal V ve han °hien thi trong Wee khong
gian con T sao cho (11°)° # T.
2.27. Cho V* la khong gian lien hop ciia khong gian vet td
V. IV,I, €1 la ho nhfing khong gian con oda V. Chiang minh:
a)
n VID =
ier Biel
(
.0
1, fl y; = I Vi° neat V Mitt han chi6u.
IET / ki
c) Khang dinh b) van ching n6u I Mill han, con V ce the vo
han chigu.
d) Khang dinh b) khong con dting neal I ve han va V ve hat.
chi6u.
2.28. Cho (1',), e i IA ho the khong gian con ciaa khong gian
V* lien hop vdi khong gian vec to V.
102

Chung minh:
a) ETi I = nrE
iel iE=.1
n = n6u V heti hen chiau.
ieL
c) NC,u N.2 va han, thi trong V* cia hai khong gian
can T, (T,11T,r, Ti r12.
2.29. Cho E la te'ng true tipcim hai khong gian con V va
W; x e E viet dupe duy nhat dUdi clang x= y+ z; y c V va z e
W. Ta goi phep chigu cea E len V song song vdi W la clang eau
tuyen tinh p: E —> E xAc dinh bai p(x) = y.
Ching minh rang cl8ng cad tuyan tinh p: E E la phep
chi6u khi ve. chi khi p= = p.
2.30. Cho E la khong gian vec td va p e End (E). Chiing
minh rang p la phep chi6u khi va chi khi Id — p la phep chi6u.
Khi do Imp = Ker (1d-p); Keep = Im (Id-p).
2.31. Cho V Ea khong gian vec td tren tnidng K va f E End(V).
Chung minh rang P = 0 khi va chi khi Imf c Kerf.
ChUng minh rang trong trUCing hap nay g = hi + f la melt tkt
deng ceu dm V.
2.32. Cho V la khong gian vec td hilu han chiau va u e End(V).
b)
103

a) Cluing minh rang co hai co sa {e„ e„) va {ED ..., en} cna
V via soi nguyen s (0 s n) sao cho u(e,) = 6, vdi i i< s va u(e) = 0
vii s < i <
b) Tit do suy ra bin tai to clang caiu tuygn tinh v mart V sao
cho you la phep chigu.
2.33. a) Cho p la met phep chigu caa kheng gian huu han
chigu V. Chung minh rang trong V co cd sa trong do ma Wm A
= (a3) ena p co clang dac biet: a,4 = 0 ngu i # j;
ail = 1 neu 1<i< s va a;; = 0 ngu i> s.
b) Tu (Icing Pau tuygn tinh u duos got la del hop ngu u2 = id.
ChUng minh rang clang thuc u = 2p - id thigt lap mat song finh
tit tap tail ca cac phep chigu p trong V len tap Cal ca cac doi hop
u. 6 day V la khong gian vec to tren truing K voi dac s6 khac
hai.
c) Tit a) va b) co thd not gi lid ma tran cua phdp del hop teen
mat khang gian huu han chidu.
§2 HE PHUCNG TRINH TUYEN TiNH
2.34. Tim met he nghiem cd ban caa he phuong trinh sau:
x1 - 7x 2 + 4x3 + 2x4 -0
2x1 -5x.2 +3x3 +2x4 + X5 =0
5x1 -8x.2 + 5x3 + 4x4 +3x5 = 0
4x1 -x2 + X3 + 2x4 + 3X5 = 0.
•

2.35. Cho hee, phudng trinh thuan nhat:
2x1 5x, + x3 + 2x4 = 0
5x1 - 9x2 2x3 7x4 = 0
-3x1 7X9 X3 + 4x 4 = 0
4x1 6X2 + X3 - kX4 = 0
Hay giai va bien Man then A he phudng trinh Wen.
2.36. ret he phudng trinh:
1
a(b -c)x + b(c -4 + c(a. -13) z = 0
(b-c)x2 + (c -42 + (a. -b)z2 = 0
GM sit a, b, c cloi met phan Wet. Chung to rang he tren co
ghiem hoac x = y = z hoac
ab -be + ea bc -ca +ab
-
ca -ab +bc
2.37 Gigi he, plutAng trinh:
X-X1 + 2 4-1 X3 x 4 a
- x2
▪
X3 + X
xl + X9 + X3 + x4
X1 + X2 + X3 + x4
2.38. Giai va bien Man he phudng trinh tuyen tinh sau:

2x + y z = a

X + my z

3x + y mz = c
105

2.39. Tim digu kien din va du a 4 dim A,(x l, VI), Ai(xi,
312), A3(x3, 310, Ai(xi. y1) khong ce 3 digm nao thang hang ngm
tren met &tang trOn.
2.40. Tim da the bac 3 vdi he so thoc f(x) sao cho £(-1) = 0,
f(1) = 4. f(2) = 3 va f(3) = 16.
2.41, Ch9 A = (aig e Mat (n. K) sao cho dot A = 0. Goi la
phan phu dal ce cUa phgn to a,,, gia thigt A 11 # 0.
Hay tam mat he nghiem co ban cila he phtiong trinh thuan
nhal sau:
LAuxi
1=1
§3. CAU TRUC CUA MOT TV DONG CAU
2.42. Ching minh rang m6i gig tri rieng ciga phep chiegi
dgu bang 0 hoar 1 va m6i gia tri rieng oda phep dM hop dgu
bang 1 hogc bang —1. (xem bai 2.29 va 2.33).
2.43. Cho A, B la hai ma trail vuong gang
a) ChUng minh rang neu A, B deng Bang thi cluing ca clang
met da fink dac trting.
b) Chung minh rang n6u A, B ce cling da thug dac
thi det A = det B.
c) Cac khang dish dao dm va cua b) có dung khong?
2.44. Gia sit ip la met to d6ng ca/u cua khong gian vec to
thkic n chigu V, trong mot ca sa nao do co ma Han A. Chung
minh rang da tilde dac trong oua pee dang:
106

PTO,) = (-2,)" + c,(-2,)"1 + + + 'Prong do ck
Yang cua tat ca the dinh thew con chin)) cap k cua ma Han A,
)inh thew eon chinh la dinh thfic con lap nen tit the (long va
re c.0t vdi chi s6giang nhau).
2.45. Gia sa ( la to deng Ha.' cua khong gian yen to V n
nen trru trtfong K.
Gni la mot nF,Thi .em cua da ',hue dac Hang cua 9, 2,„ co
hieu r= rang (9-2,0M).
Hay cluang minh I < n — r 5 p.
Y.46. Havain) nghiem dac trung cua ma tran AeMat (n, C).
0 -1
1 0 -1
1
0
1
2.47. Tim da the dac bung cua ma tran vuong cap n.
0 1
0 1
0
ava a
1
an
Tit do suy ra m6; da thin; bac n yea he t>i cao nhat (-1)" den
a da their dac Hung cua met ma trdn valuing cap n nao do.
A=
107

2.48. GM V lk khong gian vec W n chieu troll trUang K,
cp E End (V), X° la mOt gia tri rieng ena gyp, 700 la khong gian
con rieng cna V, ling vdi gia tri rieng A 0. ChUng minh rang so
chigu cim g1 /.0 khOng vuyt qua sei bOi p cem nghiem 700 cua da
thdc (lac trung dm. T. Hay chi ra VI du trong do dim "91 A, < p.
2.49. Cho o la mot phop thg bac n va u e End (0 1) xac dinh
ban
u (Z„ Z„) = (74,„„), ...,
Hay tim da thfic dac trung va the gia tri rieng ctia tu clemg
tau u.
2.50. Cho f e End (R"), trong co se, to nhien {o„ ..., e n} co
ma tran A=(a d),tldoa1 = a vOi moi i va ad = b yen ix j; a # b.
Hay tim mOt ccI so cem R" a ma tran cua f trong cd sa do co
Bang
2.51. Cho A, B e Mat (n, C).
a) Chung minh rang the ma train AB va BA co cling tap hdp
the gia tri rieng
b) HOi AB va BA co clang da thne dac trung hay killing?
2.52. Gia sU V la khOng gian vec td phito him han chieu, f e
End (V) ma co se N, nguyen during dg fN = Id. Chtng minh rang
trong V co co sä gom nhfing vec td rieng cua f.
2.53. GM sif ma tran S e Mat (n, K) co anh chgt: Kai moi
ma tran A e Mat (n, 1K), to Mon viet duye mot each duy nhfit
108

dual. dang A = A, + A2, a do A, giao haw vdi S va A2 phan giao
hoan vdi S, nghia la A,S = SA„ A 2S = -SA2.
Chung minh rang 52 = X. I„ la ma tran dun vi.
2.54. Cho E la khOng gian vec td huu han chieu va f e End(E).
Vai m8i i = 1, 2, ... dat V; = kern va W, = Im(P).
a) Chung minh V, c Vm, voi moi i = 1, va t6n tai s6 m
V„, = v„,+, va khi do V. = V,„„ vdi moi j x 1.
b) Chung minh E = V„, W„, vdi m tim chide trong can a).
c) Vm va tim duo trong a) la nhUng khong gian con bOt
Bien doi vdi f. Hon nua, f Iny linh tren va kha nghich tren W„,.
2.55. Cho V la khong gian vec to tren truOng 1K (1K bang R
hoac C), 9 e End (V) va p lily linh. Chting minh rang:
a) Moi gia tri rieng cua cp deli bang 0.
b) Neu dim V = n thi da thlic dac trung cem 9 la oon.
2.56. Chiang minh rang neu W la tv d6ng ca'u cua khong
gian vec td phac han han chieu ma moi gia tri rieng deli bang
khong thi linh.
2.57. Gia s>t V la Haling gian vec td tren &Jiang ]K va dim V
= n. f e End (V) la tv clang cali lily linh dm V. ChUng minh
rang PI = O.
2.58. Gin sO V lh khong gian vec td tren trUang K va f e
End (V), rang f = 1.
109

Chung minh rang ton tai met co so dm V de trong do r
tran cern f c6 dang:
hoac
0 0 0 0 0 ... 0
Neu f co ma Den a dang thu hai, thi f lay linh va c2 = O.
2.59. Chung to rang neu hai tp deing cau L g gem kh6)
gian V Dacia man he thew f.g—g f = Id (0 thi vai moi k e
ta — &c.f. = k.g" (2).
Tit d6 suy ra rang khong the t6n tai the tp dgng cit'u tiv
man he theic (1).
2.60. Chung minh rang, d6i yeti ho bat kY CAC torn to tuy
tinh giao hoan vdi nhau Ding doi mgt trong khong gian vac
hem han chigu- tren trang so" Ode, tan tai yen to rieng chui
cho tat cA the Loan to tuyen tinh cria ho do.
D. HU6NG DAN HOAC DAP S6
§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINI
2.1. GM sU V, W Ian hurt 1a cac kheng gian sinh bai A„
ka B1 , ..., Dr Veli mei = 1, 2, n thi B, = a,,A, + a2A2 +
u„,A„.. Do do W c V. De chUng mini) W = V, ta chi can char
minh dim W dim V.
Ta c6 dim W = rang (X . X') = n — d, trong do d bang
chigu cua khong gian H the nghiem ena phudng trinh:
'a 0 . 0 '
0 0 .

to p 0

0 0 0
110

t„). (X.Xt) = (0, 0, .., 0).
Neat Y = (t,, t) E H Y. XX' = 0
X.)4' . = 0o Y . X. X' . = (Y.X) . (Y.X)' = 0
Y.X = 0. Nhtt vay Y e M la khOng gian nghiem cim
phuong trinh t„) . X = 0 to do dim W=n—cln— dim M
= rang X = dim V.
Do do W = V.
2.2. Gia su A„ ..., la the vec to cot. (-Ala ma tran X hang r.
C6 th6 gia thi6t A„ (lac lap tuy6n tinh. Khi do cac Ai; r <
5 n bie'u thi tuyett tinh qua A„ ..., A,.
Al = a,,A, + cii2A2 + + a„ A,
N611 dung ki hieu (A„ A2, ..., A,) (16 chi ma tran co cac vec
to cot la Al, A2, A„ thi
aciAl)
Mei ma tran 6 v6phili c6 hang 1. 'Pa co digu phai cheing minh.
2.4. b) Xet R [x] la khong gian vec td the da thew sr& he s6 thitc.
Xet u: R [x] —> R[x]
f(x) --) x .
116 rang u la chin caM, nhting khong than eau, vi Imu khong
chiia nhiing da theic bac khOng.
2.5. a) Theo each the dinh caa u thi trong ea so tu nhien
(el, ..., en} cern C", to ca u(e) = e,„„ Vay A = (ad la ma trait cua u
trong co so {e„ thi
111

{1 i (j)
auo =
0 voi # a (j)
b) Gia sii B = e Mat (n, C)
Xem B IA ma tram cila phep bien den tuygn tinh trong cd
to nhien cna en : v(ei )= Ebna . Khi do AB = BA a uV = vu <
uv(ej) = vu(a) j = 1, 2, in n Eb u(a)= v(ea0)), j = 1, n <
n n n
Ebijecro) = Ebi,wei = <=> b;,= bc,(;) „„,
i=1
1, ..., n, nhu vay, the ma tran giao hoan dime vo
tran B = (130 sao cho bu = bum on) voi myi i, j = 1, 2, .
2.6. a) De dang.
11) Gia stl u e End s (E) giao hoan voi
EndK (E). Theo a) u (e) = p ie, voi {e„ cd so
Gia sV i # j, 1 g 3 , j < n to hay chfing minh 13, =
EndK (E) sao cho v(ek) = Vk 1, 2, ..., n. Tit vou
vou(e) = uov(e,). Nhu vay j3, e3 = u (a) = rija a (Di
P: -pj=A. Tit dou(x)=Axvoimoixe E.
2.7. Ta co the bieu din n clang (laic Ea nu
jet
n) &MI dang clang (Mk ma tran sant
an1 ant ann
( ail 312 aIn ‘ Ill 1
[a21 a22 v2
/ n
, . . v
Vn
voi mci i, j
i A la cic m
n.
d6ng eau v
nao do dm I
pi. Clam v
= uov say
-13i) = 0 =
i=
112

VI ma Han A = (an) khong suy men, nen co ma tran nghich
dge 11 = A-' = (bd.
Do b(au) = (ba) u voi nun vo Imang a, b va voi moi phep bi6n
den tuyen tinh u nen La co:
ul (ui
B. =[B .A]. •
Aun j
Tie do to co:

1
..
0

0
0

l 0 1
u l
u2
Lon)
=

1 bin

621 b2n
••-

bn bnn
v2
vni
Hay u; = ybi,v;
Theo gig thigt, eac vi giao &ban voi nhau, nen suy ra the u,
thing giao honn voi nhau.
2.8. a) Vi A khOng suy bign nen mei cot ena no cling co
dung met phAn to klthe khong.
Gig set A = (an), A t = fat) at _ a ,,
ij
Hat A.AL = B = (b0). Ta c6
bii = L aik -a ki = L aikaik=sij
k=1
Vi vgy A.AL = In = A-1 .
113

b) Cacti 1: Goi G la tap cac ma tran clang troll
Ta thdy A e G, B E G thi A. B e G, A-' =A° e G. Do do C
lam thanh nhgm con cim nhian GL (n, R) cite ma Dan vuong
cdp n khong suy bign. Vi s6 cac phial to °Pa nhOm G la lulu
han, nen vdi A e G, t6n tai k e N k 21, dd = Ak-' = =
At, nhtt vay A"' = At vdi m = k — 1 e N.
Cad' 2: Gia sit (e), ..., en) la co sa W nhien cim f e End(R")
c6 ma trait A trong co sa {e l , enL Ta e6 f(e,) = ± no) 6 d6 a la
phep thg bac n. Vdi m61 i = 1, 2, n xet a(i), a2(i), tai
k,deg ak'(i)=i, khi do f(e,) = eap o = e, PLi(6) = e,
Dat k = 2 . k, kn thi gc(e) = e, V; = 1, n hay f" = Id Ak
= In, ldy m = k — 1 to c6 = AL-1 = At.
2.10. GM sit f.g — Id khong la clang cdu claa V, khi do t6n tai
x E V, x # 0 a (fog — id) (x) = 0. Hay fog (x) = x. Dant g(x) = y e V,
to co f(y) = xz 0= yx 0.
Xet (Id — gof) (y) = y — g (fly)) = y — g(x) = y — y = 0. Digu nay
chfing to rang c6 y x 0 dO (Id — gof) y = 0, trai vdi gia thief Id —
gof la (fang cdu.
2.11. a) D6 thdy u, v Wong ding thi co gang hang. Ngudc
lai, n6u hang u = hang v = r, theo bai 2.3, trong V c6 cac cd sd
{6, ..., en}, (e),, e'„1 t(ei) = e't = 1, r), u(u) = 0 > r), va
c6 cac cc' sa e„}, e'„} v(e) = e,' vdi i = 1, r, v(el)
= 0, j > r. Xgt p E End (V) p (e,) = e; voi moi i= 1, n; Xet q e
End (V) sao cho q(s) = e', (i = 1, n) thi uop = qov.
b) Suy ra tit a)
114

2.12. a) De dang do Tiny c Keru va dim Imu + dim keru =
dim V.
b) Xet co so ..., en/ va {c1, c„} rim V sao cho u(e,) = e, (i
= 1, 2, r) va u(e,) = 0 (j > r). Xet v E End (V): v (c,) = 0 (i 1,
r); v(c,) = ei vai r + 1 5 j < n. Khi (id vou = 0 va hang u + hang
2.13. a) rang rp= dim E x r
b) rang ty = dim FI x r.
2.14. WE( u, = u — IdE , v, = v —
Tit do u.v —IdE = + IdE) (v, + IdE) — Id E =
= u, . v1 + u, + v, = u, (vi-lIdE) + v, nhung rang u, (v, + IdE) ic rang
u, va rang fu l(v, + IdE) + rang u, (v, +14) + rang v1.
Do vdy: rang (uov — H E) < rang u, + rang v, = r + s. (xem vi
du 2.14).
2.15. Vi rang u = 1 dim (Imu) = 1. Gia sii {x,4 la ed sd GM
Imu. Do u(x0) e Imu nen co 2, e 1K: u(xo) = Ax„.
Ta hay thing minh u= = Au. Vol x e E u(x) = a . xo
u2(x) = a u (x0) = a . Xxo = X a. xo = Au(x) vdi moi x e E u2 = Au.
Gia # 1, tim 71 a (Id — u) (Id - nu) = Id.
Di5u do tudng &tong vdi (fik
1
x_1 .14.y
Id-u Icha nghich va nghich dao mid no la: Id +
1
u.
1-X
115

2.17. HD: Chon B la ma Dan cheo vdi cac phan to tren
during cheo chinh khac nhau doi mgt: b,, Chon C = (au)
a
dsaoChoC v itt va Cli tuy y, tin A = BC - CB.
bi - bi
a b'
2.19. GM X =
d 1
E Mat (2, K). Hay chUng to c6 P =
x y
e Mat (2, K), det P t 0 de XP = P.Xt.
z t
bz = by
(a-d)y= cx - bt
-d)z = cx - bt
cy= cz , xt zy
(1)
Xet cac trudng hop sau:
+) b = c = 0, chon y = z = 0, x t = 1
+) a = d, chon y = z = 1 x = t = 0
+) b2 +cz> 0 va a t d: chAng ban b s 0,
chon x = 0, t = 1,y=
b
-z .
a -d
Nhtt gay trong moi trUdng hop, hP (1) dgu c6 nghigm
2.20. a) Tit u2 (Id - u) = 0 suy ra 112 =
Tit u (Id - u)2 = 0 suy ra u - u 2 - u2 + = 0
W vgy uz = u.
=
116

b) Vi du V = K3, u (xi , x2, x3) = (x„ x3, 0)
Ta thSy u2 * u nhung u2 =u3 .
e) V = K3, v(x„ x2, x3) = (0, x2 - x3, xa)
Ta có v2 * v nhting v (Id - v)2 = 0.
Chu Y: Co the neu nhieu vi du khac nfia.
2.21. a) Ngu g(x) = 0 voi moi x c V thi f(x) = 0 raj moi x e V
va f = a . g voi a e K Neu có xo e V ma g(x0)* 0.
Data-
f(x)
.
g(x{,)
Ta chUng minh f(x) = a g (x) voi moi x c V.
Vi g(xo)* 0 nen g (x) = g(x), khi do:
g(x -;xo) = 0 f (x - )(0) = f(x) - 4f(x0) = 0
f(x) = 4 . 1(x0) = g(x) f(x ) = a.g(x).
g(x0 )
b) HD: chtlng minh quy nap theo n.
n = 1 dilng vi day la trUang help a).
Gia eu bai town dung voi k = 1, 2, .. n - 1. Ta chUng minh
dung voi n.
Trubng help 1: Neu co i de fl Kerf c Ker f, thi fl Kerfj =
joi j#i
fl Kerf, c Kerf. Nhu vay theo gia thief quy nap, to co
i=1
f= ifJ=Eaifi (ai = 0)
i=1
117

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