Bài tập đại số tuyến tính có giải chi tiết

PHAN HUY PHIJ • NGUYEN DOAN TUAN

BAI TAP
DAI SO TUYEN TINH

NHA XUAT BAN HAI HOC QUOC GIA HA NOI


Chin trach nhiem xual bcin

doe:

Gicim

NGUYEN VAN THOA

Tong bien Op:
Bien tap:



NGUYEN THIEF N GIAP

HUY CHU
DOAN 'MAN
NGOC QUYEN

Trinh bay Ilia:

NGOC ANH

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


Lai NOI DAU

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

sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang
khi bier, soon, nhUng chat than con có khigm khuygt. Cluing
toi rat mong nhan dude nhUng y kin clang gap cna dee gia.
Ha N0i, thcing 3 !Lam 2001

NhOni bien soan
3


rvikic LUC
Chubhg .1: DINH THOC - MA TRA:N

7

A - Tom tat ly thuyeet

7

§1. Phep th6

7

§ 2. Dinh thitc
§ 3. Ma tram

10

B - Vi dn

12

C - Bei tap

35

D HtiOng dein hoac clap so

43

Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH

•

PHUGNG TRINH TUYEN TINH

57

A - TOrn tat ly thuyeet

57

§1. Kh8ng gian vec to

57

§2. Anh xa tuyeen tinh

61

§ 3. He phydng trinh tuy6n tinh

64

§4. Can true caa tai ding cku

67

B Vi dtt

71

C - Biti tap

96

§1. 'thong gian vec to va anh xa tuyeen tinh

96

§2. He pinking trinh tuy6n tinh

104

§3. Cau tit cna melt tu thing calu

106

D. Illidng sign ho(tc clap s6

110

5


§1. Khong gian vec td va anh xn tuyin tinh

11(

§ 2. He phudng trinh tuyeit tinh

12';

§3. Cau trite dm mot tg ang cau

12Z

Chtedng

DANG TOAN PHUONG - KHONG GIAN VEC TO

OCLIT VA KHONG GIAN VEC TO UNITA

134

A. Tom Vitt 1t thuyeet

134

§1. Dang song tuy6n tinh aol xUng va dang town phuong

139

§ 2. Killing gian vec to gent

135

§3. Khong gian vec to Unita

142

B. Vi du

14E

C - Bai DM

174

D. Hitting dan hotic ditp so

179

Tai lieu them khan

192

6


Chuang 1

DINH THUG - MA TRAN
A - TOM TAT Lt THUYET
§1. PHEP THE
Met song anh o tit tap 11, 2,
met phep the bac n, ki hieu la
'1 2 3
\

G

I

a2

G

n} len chinh no duet goi la

3

15 del a, = a(1), 02 = a(2),..., a„ = a(n).

Tap cac phep the bac n yeti phep nhan anh xa lap thanh
met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen
t3 cua nhom S„ bang n! = 1, 2... n.
Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich
the cem a n6u s6 - j) (a, a) am. Phep the a &foe goi la than
ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6
-

nghich the ciaa a le.
Ki hieu sgna =

1 neM s la phep the chan

-1 net} a la phep th6 le

va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the

cling bac, thi sgn(a

= sgn(a) . sgn( ).

Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„
• - • , i k doi mot khac nhau dr

coo = 12 , coo = i3,

a(ic) = i1

7


va a(i) = i vdi moi i x i„
i k . Vong )(felt do dttoc ki hieu IA
ik ). M9i phep th6 dau &tan tfch the thanh tfch nhung
yang xfch doe lap.
Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong
••• , ik) phan tfch chive thanh tfch 0 1 ,

xfch

§ 2. DINH THUG
I. Gia sit K IA met trueng (trong cuan sich nay to din yau

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

zsgn(a)a mo)
E

Sn

3. Tinh eh& ceta Binh that

a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram
A, thi dinh auk cim no ddi da:u.
b) N6u them veo met dong (hoac met cot) cim ma tran A
met to hdp tuygn tinh cim nhUng thing (hoac nhung
khac,
thi dinh auk khong thay ddi.
8

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

an

de

= det

al;

a 21

21

a2„

a,,, + ani

‘ a n„

a ll

an,

ail
a21 +alci

...a1,„
+ de t

all
a21

—a 1111/

d) Cho A = (Ito)

E

...a2 n

" S ' Ill " S IM /

Mat(n, K), thi

= b)

a do

= aij &toe

goi la ma tran chuy6n vi cim A.
Ta co detA = detA t.
4. Cdch tinh dinh that

a) Cho ma tran A

E

Mat(n, K). Kf hi'911 Mi; la dinh that cua

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

ngu i x k

det A ngu i = k
Nhu fly detA = EamAki (k = 1, 2, ... n)
1=1
heat detA = Z a ikAik
/=1
9


CUT thac tit throe goi la cang thdc khai trim dinh tilde
theo (long hay theo cot.
b) Dinh 1ST Laplace
Cho ma Iran A = (a, J) c Mat(n, K). Vo; rn6i bQ
;2.••, ix),

va Oh

ik),

1 s i, <1 2 < . <

1 =11 < j2 < •••
n, 1 s k< n, dat A.

ik

11-11,

la

dinh thac caa ma tran vuong cap k nam d cae &nig i„ ik va cac

cot j,... jk cim ma tran A; M.h

la dinh thdc cita ma tran vuong

cap (n - k) c6 dude bang each gach the clang thu
thu j 1 , , jk caa ma trail A. Ta c6 kgt qua:
detA = ZED'

Si k+31+

A.



ik va cac cot

ik

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

10





A=

811

a19

aln

a, 1

a99

aon

amt

amt

arnn,

2. Cac phep todn tren Mat(m, n,

Cho A = (a y ), B= (b,j ) thuOc Mat(m. n, K)
Ta có:
a) Ma tran C = (cg) a do cy = a ii +
&toe goi la tong cua hai ma tran A va B va ki hien la A +B.
Ma tran D= (d,,)

a do di; = a ij -

dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B.
b) Vdi k

E At,

ma trail kA c8 cac phAn to la (ka ii ) duoc goi

la tick cua ma tran A vdi ph&n td k cua trudng K.
A = (aii ) c Mat(m, n, K) va

c) Neu

B = (bp() e Mat(n, p, K) thi
ma tran A . B
AB = (c, k),

E

Mat(m, p, K) ma cac phAn tit &tele xac dinh INN

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 ; A t la ma tran chuya'n
vi cua A . Khi do:


At .
detA

B- VI DTI
Vi cla 1.1. Xac dinh clgu rim cac phep th6 saw
a)

a

11

2

2
3

r1 2 3
b) 5= I
ll 4 7
12

3
5

4
4

5
1

n n+1 n+2
ai-2 2
5

211

2n+1

31

Lai gidi

a) Phan tick) a thanh tich cac chuydn tri:
=

2 3 4 5)
= (1 2 3 5) = (1, 5) (1, 3) (1, 2)
(1
2 3 5 4 1

(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)
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 12
3

fn

en dgu la (-1) 1

Hay gag dinh da"u dm:
a) 1-1
b) g = (In

2


fn-) •

n)

Lift( gidi:

a) Vi sgn f. sgn

= sgn (f.
= sgn(Id) = 1 non

sgn (e 1)= sgn (f) = (-1) k

b) X4t phep the"a = 1 2
n -1 .

nj
1

thi g = f. a
Do gay sgn g = sgnf . sgn a.
Nhung sgn a = (-1)
14

n(n-1)
2 nen sgn g = (_])k+C;',

2


Vida 1.3
ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf

a clang

j) v6 ben trai Wring throng v6i viac dai cha car s6 i j
,

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 phop th6 cho

j) la phep chuy6n tri. Xet

truong hop nhan ben trai tile la f = (i, j). a.
(3

Gitisi2 a=

Theo d nh nghia (i, j) = (

_ (1

n

1 2
9 2

2a

ai

nj
n
)

Trunng hop nhan ben phai dude ant Wring fib
Vi dy 1.4. Cho f va g la hal phop th6cua n strtn nhien clAu tien.
a) Chung minh rang có the' cilia f va g bang khong qua (n - 1)
phop chuyan trf (nghia la ton tai k phop chuyan trf a l , cr2,
k 5.11 - 1

ak,

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
xich dOc lap T 1 ,

Tp .
go

e' =

...

e'

thanh tfch cac vang

T2 .T i

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
(a l , 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(m i -1)= n -p = k phep chuyen trf.
i=1
Nhungpa. 1

n-1:115n-1

Nhu vay g o f -1 = ak

(s, - chuyen tri)

TV do g = ak 0 ak.,, ... 0

0 f,

kn- 1

va cac a, la cac phep chuyen trf.
b) Cho g =

on f =

(

1

Oa

ri
0.

nj
2

...

la phep the ddng nhA

23
. Ta se chUng to rang killing dua
1 2... n-1)
n

f ve g duo Wang it hdn n - 1 phep chuyen trf.
16


'
WA met phep the" h = 1


1 2
2

ta not rang la
n

i. De" rang neat nhgn vao ben trai
phan tit chinh quy ngu
cua h met chuygn Lri thi A:C. 1)111in tit chinh quy tang cling lAm la
met don vi. That way, ngu ngudc kg, cheng hen i, j la hai phan
tit kheng chinh quy cim h ma nob d6i dig h i voi hj ta Jai &tog
hai phign tu chinh guy (cum phep th6 m6i) th6 thi: h, < i. h, < j
nhang hj 2 i, h; j vti 1Y.

Do f chi co met phAn tei chinh quy. ye g co n phan tV chinh
quy, vi vgy khong thg clua f vg g bring it hon n - 1 phep chuygn tri.
Vi du 1.5

Chung minh rang vdi mei so k (0 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,,
nghich thg, 0 < k <

la met hogn vi cda 1, 2, ... n va ace 1:
, 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&
vdi moi i = 1, 2, ..., n-1 thi
hgt ta nhan thy rang negi >
a co

nghich the'. Vi vay, do so' nghich th6 caa a la k <

,

nen ten tai i o dg oc ii) < a i0+1 ;

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

con p io = a ;0+1 , p i+, =a,„ thi HI rang 13 co nhigu hon a met
,

nghich the". Nghia la s6 nghich th6 ciga p la k + 1.
17




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

0

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

= (-1)" +1. (-1)"-'=1

Cdch 2.

Ta tha'y A= B. ca do
1
1

0
B=

t1

vi C=

1
11

ma detB =1, detC = 1 nen detA = detB. detC = 1.
Vi du 1.9. Hay tinh

cosa
1
1
2cosa
= 0
0

1
0

0

0

0

0

0

1
2cosa
0

2cosa
1
1
2cosa
21


Lai gidi:

Khai trim dinh thac Lheo cot cub" to co
D„ = 2cosa .

-

17,1 _2

De thay D, = cosa.

1
= 2coi2a - 1 = cos2a.
2cosa,

cosa

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

A„ =

1
0
+ a-9
1
e" +e 1
1
0

a

0

N x0

1
1 eP + e -(1)

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.
22


Khai trin theo cot tht nhEt, to c6:
A n = (e P +e -P)A n _i:
e 21' - e -2(P

Nlinn xet rang 4 1 = 6 9 +Cc =

A2 =

-

((i

6

e

e ro

36 - -39
e (P -

e (1+1)6 - e -(lrv1),p

Girt sit

AR -

e (-0 -e

,

(P

k - 1, 2,

, n - 1.

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

=(eP +e w)e

nip - e npe( n-1) n4)
ew

Nhu v 6. }.:

An =

e

- e

(:+1)o - e -(n+1)4'
—

e q) -

e (n+1)p

- e

-(n+1),p

e 1 -e

Vi du 1.11
Tinh:
ll

1

D = dot 1

a1

a,

an

a l: +h i

a,

an

a1

a, +b.,

an

a1

a.9

...

a n +b n

23


Lai gidi:

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

cac thing con

,

Vi du 1.12

Cho da thric P(x)=x(x+1)...(x+n)
Hay tinh Binh thdc:
P(x)
P(x)

P(x +1)
P(x +1)

P(x + n)
...

P(x +n)

d=

P(n-1) (x) P th-1) (x+1)
P thl (x)
Pthl (x +1)

P th-1) (x + n)
P (n) (x +n)

gidi:

Ta b6 sung de' dude ma Han dip (n+2):

P(x)
P(x +1)
P(x +n)
P(x)
P4x+1)
P4x+n)
D=
P (n ) (x)
Pthd(x +1)
P0P(x + n)
pg+0(
x +n )
P(„+l) (x) 1301+1) (x +1)

0
0
0
1

RO rang det D = d
(x +n
Nhan dOng 1111 k cua ma trail D vdi

dc-ix( 1) k-1 r6i
(k-1)!
Ong vao clang Hirt nhgt vgi tat ca k=2, .. n+2).
Khi do, phAn tii dung dau có clang:
poc + 0 +
k=1

24

P(k) (x +0.(x +11.) k
k!

ok

=

n).