Ch 0.I.ii Tap Hop Anh Xa

8/8/2019 Ch 0.I.ii Tap Hop Anh Xa

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1

CHNG 0. TP HP - NH X

A. TP HP

1. Tp hp Phn t- Tp rng.1.1.

Tp hp, phn tl nhng t ban u khng nh ngha c. Tp hp cnc gi tt l tp.

Chng hn ta ni: Tp hp 3 mu tu ca bng ch ci la tinh. Khi ta ni a l phn

t ca tp hp ny; b, c cng vy. Nhng d khng phi phn t catp ny.

Tp hp cc s nguyn. Khi 11, 2 l trong s rt nhiu phn t

ca tp ny. Nhng 32

, 5 khng l phn t ca tp ny.

Cc tp s bit , , ,

Ta ghi tp hp bng cc ch k t in , , , ,...B X Y v ghi phn t ca tpbng cc k t thng , , , ,...a b x y

Nu a l phn t ca tpXth ta ghi a X (c l a thuc X).Nu a khng l phn t ca tpXth ta ghi a X (c l a khng thuc X).

Trong th d trn, ta ghi 2 , 5 .

1.2.

Cch ghi tp hp.

Lit k: K ra tt c phn t ca tp gia hai du { , }.Th d:

{ , , }, {0,1,2,3,4,....}A a b c Dng biu :

Th d:

Nu c trng:Tp hp c vit di dng{ / ( )}x P x (c l tp nhng phn tx sao cho x c inh cht P)hay

Aa

b

c X


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2

{ / ( )}x A P x (c l tp nhng phn tx thuc A sao cho x c tnh cht P)

Th d: 2{ / 3 2 0}M x x v x x

Trong cch ghi

{ / ( ) }phan tu dactrung

A x P x , Pgi l c trng ca tpA.

1.3.Tp hp rng.Tp hp khng c phn t gi l tp rng. K hiu: , { } .Th d:

Tp 2{ / 1 0}x x l tp rng.

2. Tp hp con Tp hp bng nhau.2.1.Tp hp con:

o Cho 2 tpA vB. TpA gi l tp hp con caB nu mi phn t caAul phn t caB.K hiu B (c lA nm trong B)NuA khng phi l tp con caB th ta ghi B (c lA khng nmtrong B)

o Qui c: Tp rng l tp con ca bt k tp noo Tnh cht:

A B v B C A C .

2.2.Tp hp bng nhau.Hai tpA vB gi l bng nhau nuA l tp con caB vB l tp con caA.K hiu B .Nu tpA khng bng tpB th ta ghi B .Tm li:

B A B B A Th d:

2{ / 2 3 1 0}A x x x , 12

{1}, { , 1}B C .

A

B


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Ta c: ,B A C .

2.3.Tp cc tp con.Cho tpA. Tp hp gm tt c cc tp con caAc gi l tp cc tp con caA (hay tp cc phn ca A). K hiu ( )P .

Vy: ( ) { / }X X A P .Th d:Cho {1, }a th ( ) ,{1}, { }, {1, }a a P .

3. Php ton gia cc tp hp.3.1.Php giao:

Giao ca 2 tpA vB l 1 tp hp, k hiu B (c lA giao B), gm vch gm tt c nhng phn t va thucA va thucB.

{ / }B x x A x B

Tnh cht:

( ) ( ) :

A A

A B B AA

A B B B A

B C A B C A B C

3.2.Php hp: Hp ca 2 tpA vB l 1 tp hp, k hiu B (c lA hp B), gm v

ch gm tt c nhng phn t thuc v t nht 1 trong 2 tp A hayB.

{ / }B x x A x B

A

B

B

A B

B


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4

Tnh cht:A A

B B A

A

( ) ( ) :B B A BB C A B C A B C

Ngoi ra, gia php giao v php hp c tnh phn phi:

( ) ( ) ( )

( ) ( ) ( )

B C A B A C

B C A B A C

3.3.Php hiu: Hiu ca 2 tpA vB l 1 tp hp, k hiu \B (c lA trB), gm v ch

gm tt c nhng phn t thuc v tp A m khng thuc tpB.

\ { / }B x x A x B

Tnh cht:\

\\

\ \ ( )

A A

A AB A B

B A A B

Phn b.o Cho tpE v ( )EP .

Khi , hiu \E A c gi lphn b caA trongE, k hiu EC A .

AA

\B

E

A

EC A


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5

o Tnh cht:, ( )

( )E E

E

E

A B E

C C A A

C A A E

C A E A

P

( )

( )E E E

E E E

C A B C A C B

C A B C A C B

4. Tch Descartes cc tp.4.1.Tch 2 tp.

Tp { ,{ , }}a a b gi l cp th t ca 2 phn ta v b.K hiu ( , )a b .

Hin nhin tnh ngha ta thy:( , ) ( , )

a ca b c d

b d

.

Ta gi tch Descartes ca 2 tpA vB, k hiu B , l tp tt c cc cpth t ( , )a b vi a A v b B .Vy {( , ) / }B a b a A b B .Th d:

Vi {1,2}, { ,2, }B a c , th{(1, ),(1,2),(1, ),(2, ),(2,2),(2, )}B a c a c .

4.2.

Tch n tp ( 2n

) Bn th t:

B 3 th t ca 3 phn t , ,a b c l cp (( , ), )a b c , k hiu ( , , )a b c Tng qut : Bn th t ca n phn t 1,..., na a l cp

1 1(( ,..., ), )n na a a , trong 1 1( ,..., )na a l b 1n th t ca 1n phn t 1 1,..., na a . K hiu 1( ,..., )na a .Lu :

1 1 1 1( ,..., ) ( ,..., ) ,...,n n n na a b b a b a b . TchDescartes ca n tp 1 2, ,..., nA A , k hiu 1 2 nA A , l tp tt

c cc bn tht 1( ,..., )na a , trong i ia A vi mi 1,...,i n .Vy:

1 2 1( ,..., ) / , 1,n n i iA A a a a A i n


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6

Qui c:1

2

n

n

A A

A A A

A A

BI TPTP HP

1. Tm cc s thcx, y 2 tpA,B bng nhau:a. { 2, { 2}}, { ,{2 }}x y B y x

b. 2 2 3 2{ , {2 }}, {3 4,{ +3}}A x x B x x x 2. Cho tp {1,2,3,4,5,6,7}E v cc tp

{1, 2,3, 4,5}, {1,2,3}, {2, 4, 6}A B C .a. Tnh \ , , ( \ ) ,EB A C B A C C A C .b. Vit ( \ ( )), ( \ ), ( )B C C B C .c. Gii trong ( )E phng trnh X B .d. Gii trong ( )E phng trnh B X A .

3. ChoA, B, Cl cc tp con ca mt tpE. Chng minha. B A B A A B B b. \

E

B A C B c. ( \ )B B A B d. ( \ ) \ \ ( )B C A B C e. \ ( \ ) ( \ ) ( )B C A B A C f. ( \ ) ( )B A B A g. \A B A A B

4. Cho tpA c m phn t, tpB c n phn t. Bit tp B cp phn t.Hi s phn t ca tp \ ,B A B .

5. So snh cc tp:

a. v ( ) ( )C B D b. ( ) ( )B C D v ( ) ( )C B D

Chng minhc. B A B


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B. NH X

1. nh x.1.1.nh ngha 1. Cho 2 tp hpA vB khc rng.

Mt nh xftAnB l mt b 3 th t ( , , )f A B F , vi F A B , tha:i) ( , )x A y B x y F ii) ( , ), ( , )x y x y F x x y y

Nu ( , )x y F th ta niy l nh cax qua nh xf,x l to nh cay quanh xfv ta ghi ( )y f x .Tc l ( , ) ( )x y F y f x .

TpFc gi l thca nh xf. TpA gi l tp i,B gi l tp n ca nh xf.

1.2.Th d.a) Cho { , , }, {1,2,3,4}A a b c B v

{( ,1), ( ,2), ( ,5)}F a b c {(1, ), (2, ), (4, )}G b c a {( ,1), ( ,2), ( ,2), ( ,1)}H a b c a {( ,4), ( ,3), ( ,4), ( ,1)}K a c b c

Ta thy: ( , , )f A B F khng phi nh x v F A B ( , , )g B A G khng phi nh x v 3 khng c nh. ( , , )h A B H l nh x tA dnB.

( , , )k A B K khng l nh x v c c 2 nh !

b) Cho ( , , ) , vi 2{( , ) / 1}x y y x . l nh x t n .

c) Cho ( , , ) , vi 2{( , ) / 1}x y x y . khng l nh x t n .

1.3.Nhn xt. Ta thng hay cho mt nh xkhng phi bng th ca n (nh trong

cc v d trn) m bngc trng ca th ca n. c trng thngl qui tc tm nh y ca phn tx thuc tp i. Chng hn trong th d b) trn y, nh x c thc cho nh sau:

:

1y x

Do , ta c thnh ngha nh x mt cch khc nh sau.


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1.4.nh ngha 2:

Mt nh xftAnB l mt qui tc t tng ng mi phn tx ca tp iA vi mt phn t duy nhty thuc tp nB m ta gi l nh cax quaf.

K hiu

:( )

f A Bx f x

hay f B .

Th d: y l nh xf:

nh xf cn c thc m t nh sau:

x a b cy 3 1 1

1.5.nh xng nht.nh x

:( )

f A Aa f a a

c gi l nh xng nht trnA.K hiu , 1A AId .

1.6.nh x bng nhau.Cho 2 nh x ( , , )f A B F v ( , , )g C D G .Tnh ngha nh x ta thy:

C

f g B D

F G

Ni cch khc, cho :f A B v :g C D l 2 nh x, ta thy:

( ) ( )

A C

f g B D

A f x g x

A B

a

b

c

1

2

3


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2. Tch cc nh x2.1.nh ngha.

Cho 2 nh xf gB C .

Khi , qui tc

:[ ( )]

h A Ca g f a

l 1 nh x tAn Cv c gi l nh x tch ( hay nh x hp) ca 2 nhxfvg.K hiu g f ( c l g trn f )Tm li:

g f C [ ]( ) [ ( )]a A g f a g f a

Tnh cht:

Vi nh x :f A B ta c:i) Af Id f ii) BId f f

Qui c.

Cho nh x :f A A , ta k hiu:2f f f , 3f f f f ,..., n

n

f f f

2.2.Mnh .Tch cc nh x c tnh kt hp (nu chng tn ti), tc l

( ) ( )h g f h g f .Chng minh:

f g hB C D [( ) ]( ) ( )[ ( )] ( [ ( )]) ([ ]( ))x A h g f x h g f x h g f x h g f x

[ ( )]( )h g f x Do ( ) ( )h g f h g f .

3. nh v nh ngc ca cc tp hp3.1.nh ca tp.

Cho nh x :f A B Xt X A , tp hp

( ) { ( ) / }f X f x x X B gi l nh ca tp Xqua nh xf.

Nhn xt:( ) { / ( )}f X y B x A y f x


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3.2.nh ngc ca tp.

Cho nh x :f A B Xt Y B , tp hp

1( ) { / ( ) }f Y x A f x Y gi l nh ngc ca tp conYqua nh xf.

Nhn xt:1( ) { / ( )}f Y x A y Y y f x

Th d:

Cho nh x :f A B v X A .

Chng minh 1( ( ( ))) ( )f f f X f X .Gii:

1 1( ( ( ))) ( ( )) ( )y f f f X x f f X y f x .

V1

( ( ))f f X

nn ( ) ( )f x f X . Suy ra ( )y f X . ( ) ( )y f X x X f x y

V ( ) ( )f x F X nn 1( ( ))x f f X v do 1( ) ( ( ( )))y f x f f f X .

4. n nh Ton nh Song nh4.1.n nh

Cho nh x :f A B .

nh xfc gi l n nh

,

( ) ( )

x x A

x x f x f x

,

( ) ( )

x x A

f x f x x x

Ni cch khc,nh xfl n nh Mi phn t ca tp n c nhiu nht1 to nh qua f.Th d:

nh x :f , ( ) 2f n n l mt n nh v, ( ) ( ) 2 2n n f n f n n n n n .

nh x

AB

a

b

c

d

1

2

3

4


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11

khng phi l n nh v phn t 1 ca tp n c n 2 to nh!

Xt 2 nh x :[0;1] [0;1]f v :[0;1] [0;1]g xc nh bi thsau:

nh x no l n nh ? (Tr li:f).

4.2.Ton nhCho nh x :f A B .

nh xfc gi l ton nh ( )y B x A y f x Ni cch khc,

nh xfl ton nh Mi phn t ca tp n c t nht1 to nh qua f.Th d:

nh x : {1,2,3,4} { , , }f a b c x 1 2 3 4y a c a b

l mt ton nh.

4.3.Song nhCho nh x :f A B .

nh xfc gi l song nh fva l n nh, va l ton nh.Ni cch khc,nh xfl song nh Mi phn t ca tp n c 1 v ch1 to nh qua f.

Th d:

y l mt song nh tAnB.

0 1

1f

0 1

1

g

A

B

ab

c

d

12

3

4


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nh xng nht AId lun l song nh.

4.4.Tnh cht Tch 2 n nh (nu c) l mt n nh Tch 2 ton nh (nu c) l mt ton nh

Tch 2 song nh (nu c) l mt song nh.Chng minh tnh cht u:

Cho 2 n nhf gB C .

Khi

,

[ ]( ) [ ]( ) [ ( )] [ ( )] ( ) ( )

x x A

g f x g f x g f x g f x f x f x x x

Vy g f n nh.

4.5.nh x ngc ca mt song nh.

Cho song nh :f A B .Khi , qui tc

:g B A y x , sao cho ( )y f x ,

l mt nh x tBnA, c gi l nh x ngc ca song nhf.

K hiu: 1f Tm li:

1fB A

1( ) ( )y B f y x f x y

Hin nhin nh x ngc lun l mt song nh. Th d:

1

A AId Id

Song nhf

c nh x ngc l 1f

AB

a

b

c

1

2

3


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13

Song nh

:

2

f

x

c nh x ngc l

1

2

:x

f

x

.

Tnh cht:

Cho song nh :f A B .Ta c:

i) 1 Bf f Id

ii) 1 Af f Id

iii) 1 1( )f f

Bi tp NH X

6. Vit tt c nh x tAnB:

a. {1,2}, { , }B a b b. {1,2,3}, { , }B a b c. {1,2}, { , , }B a b c

7. Xt tnh n nh, ton nh, song nh ca cc nh x sau, tm nh x ngcnu c:

a. 1 :f

3x

b. 2 :f 1n n

c. 3 : \{2}f

2 1xx

x

d. 4 : \ {1} \{1}f

31

xx

x

e. 5 :f

xx e

BA

1

2

3

a

b

c


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14

f. 2 3:f

( , ) ( , , )x y x x x y

g.3 3:

( , , ) ( , , )

g

x y z x y z x y z

8. Tnh 1 3 5 1 1 5, , ( 1), ( 1)n nf f f f f n f n (xem cu 7).

9. Tnh g f v f g vi

khi

2 khi

:

2( )

2 2

x

f

e xf x

x x x

v

2

:

ln 0( )

0

g

x xg x

x x

khi

khi

10.Cho cc nh x : , :f A B g B C . Chng minha. Nu g f n nh th f n nh.

b. Nu g f ton nh th gton nh.


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I STUYN TNH

15

Chng I. MA TRN NH THC

I. MA TRN1. Cc nh ngha:

ChoKl trng s ( , hay ) Cho {1,2,..., }I m v {1,2,..., }n , vi ,m n l 2 s nguyn dng.Mt nh x tI J n K gi l mt ma trn cp m n vi phn tthuc K.

Tp cc ma trn cp m n vi phn t thuc Kc k hiu l( , )Kat m n hay gn hn: ( , )at m n .

Cho ( , )Mat m n , ta thng ghi ma trnA di dng 1 bng c mdngv n ct:

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a aA

a a a

vi qui c ija l nh ca cp ( , )i j qua nh xA.

Th d:

Ma trn : {1,2} {1,2,3}A

(1,1) 2

(1,2) 0

(1,3) 2(2,1) 1

(2,2) 1

(2,3)

c ghi l:

2 0 2

1 1A

Vy ta c th ghi ma trn di dng tng qut nh sau:

1,1,i mij j n

A a

hoc ija , nu khng snhm ln v cp caA.

Cho ma trn 1,1,

i mijj n

A a

.


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I STUYN TNH

16

Phn t ija gi l phn tdng i, ct j ca ma trn

A. 1 2{ , ,..., }i i ina a a gi l dng i ca ma trnA. 1 2{ , ,..., }j j mja a a gi l ct j ca ma trnA.

Cho 2 ma trn ( , )ija Mat m n v

( , )klB b Mat p q . Tnh ngha nh x bngnhau, ta thy ngay:

, ,ij ij

m p

A B n q

a b i j

Ma trn dng-Ma trn ct:Ma trn ch c 1 dng gi l ma trn dng.

Ma trn ch c 1 ct gi l ma trn ct. Ma trn khng:

Ma trn ( , )ija Mat m n vi 0 , ,ija i j , c gi l

ma trn khng, k hiu 0mn hay vn tt l 0 ( nu khng snhm ln v cp).

Ma trn chuyn v.

Cho ma trn 1,1,

( , )i mijj n

a Mat m n

.

Khi ma trn 1,1,

( , )i nijj m

B b Mat n m

xc nh bi

, 1,..., , 1,...,ij jib a i n j m

c gi l ma trn chuyn vca ma trnA, v k hiu l t.

Th d:

1 2

0 1

1 0 2

x

A a

v

1 1

0 0

2 1 2

ta

A x

.

Ma trn vung. Nu ma trn c s dng m bng vi s ct n th ta

gi l ma trn vung cp n. Khi ta ghi( , ) ( )at n n Mat n .


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I STUYN TNH

17

Cho ma trn vung ( )ija Mat n .

11 22{ , ,..., }nna a a gi l ng cho chnh ca matrn.

Ma trn vung cp nc bit n ijI , trong

1

0ijkhi i j

khi i j

( k hiu Kronecker), gi l

ma trn n vcp n.Th d:

2 3 4

1 0 0 01 0 0

1 0 0 1 0 0, 0 1 0 ,

0 1 0 0 1 00 0 1

0 0 0 1

I I I

,

2. Php ton gia cc ma trn.

2.1. Php cng 2 ma trn cng cp:

Cho 2ma trn , ( , )ij ija B b Mat m n .

Khi , ma trn ( , )ijC c Mat m n xc nh bi

, ,ij ij ijc a b i j

c gi l tng ca 2 ma trnA vB, k hiu B .

Th d:1 2 0 1 1 3

2 1 1 1 3 2

2 0 2 1 0 1

Tnh cht:

, ( , )B Mat m n A B B A

, , ( , ) ( ) ( )B C Mat m n A B C A B C ( , ) 0Mat m n A A

( , ) ( , )ij ija Mat m n A a Mat m n

0A A A A

, ( , ) ( )t t tB Mat m n A B A B


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I STUYN TNH

18

2.2.Php nhn mt svi mt ma trn.

Cho s K v ma trn ( , )ij Ka Mat m n .

Ma trn ( , )ij Ka Mat m n c gi l tch ca vi ma trn v

c k hiu l .

Th d:

Cho

1 3 0

1 2

0 1

x

b

. Khi

2 6 0

2 2 4 2

0 2 2

x

b

Tnh cht: , ( , ) ( )KB Mat m n K A B A B , ( , ) ( ) ( )KK A Mat m n A A , ( , ) ( )KK A Mat m n A A A

( , ) ( )t t

KK A Mat m n A A

2.3.Php nhn 2 ma trn thch hp.Cho 2 ma trn:

( , )ij KA a Mat m p ,

( , )ij KB b Mat n p .

Ta nh ngha mt ma trn ( , )ij KC c Mat m n nh sau:

1 1 2 21

p

ij ik kj i j i j ip pjkc a b a b a b a b

.Ma trn C ny c gi l tch ca ma trnA v ma trnB.K hiu B .

1

21 2 1 1 2 2

j

ji i ip ij i j i j ip pj

pj

b

ba a a c a b a b a b

b

Th d:

0 1 11 1 1 2 0 2

1 1 12 0 2 2 6 2

1 2 0


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I STUYN TNH

19

1 0 0 0 0 0

02 0 1 3 0 0

Tnh cht:

( , ) ( , ) ( , )

( ) ( )

Mat m p B Mat p q C Mat q n

AB C A BC

( , ) , ( , ) ( )Mat m p B C Mat p n A B C AB AC , ( , ) ( , ) ( )B C Mat m p A Mat p n B C A BA CA ( , ) ( , ) ( ) ( ) ( )K A Mat m p B Mat p n A B AB A B

( , ) ( , ) ( )t t tMat m p B Mat p n AB B A ( , ) n mMat m n AI A I A

II. NH THC1. nh ngha:

Cho ma trn vung ( )ija Mat n

1.1. S ( )D A nh ngha theo qui np nh sau c gi l nh thcca ma trnA:

Nu 1n th nh thc 11( )D A a . Gi s ta nh ngha c nh thc ca ma trn cp 1n .

Ta nh ngha nh thc ca ma trn cp n ( )ija Mat n

l:

11 11 21 211

1 1( 1) )( n nnD A a D a D a D

trong ijD l nh thc ca ma trn cp 1n nhn c t ma trnA bng cch b

dng i v ctj caA.

Ta cn ghi ( )D A l det( ) hay | | .

Nu ( )ija Mat n


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I STUYN TNH

20

th ta ghi

11 12 1

21 22 2

1 2

( )

n

n

n n nn

a a a

a a aD A

a a a

nh thc c n dng (tt nhin cng c n ct) gi l nh thc cp n.

1.2. Th d:

1n :

11| |a 11a 2n :

11 1221 22

a aa a 11 11 21 21 11 22 21 12a D a D a a a a

11 22 21 12a a a a . 3n :

11 12 13

21 22 23 11 11 21 21 31 31

31 32 33

a a a

a a a a D a D a D

a a a

22 23 12 13 12 1311 21 3132 33 32 33 22 23

a a a a a aa a aa a a a a a

11 22 33 23 32 21 12 33 13 32 31 12 23 13 22( ) ( ) ( )a a a a a a a a a a a a a a a

11 22 33 12 23 31 13 32 21 11 23 32 13 22 31 12 21 33a a a a a a a a a a a a a a a a a a

Qui tc Sarrus tnh nh thc cp 3:


21/39

I STUYN TNH

21

nh thc11 12 1

22 20

0 0

n

n

nn

a a a

a aD

a

(nh thc tam gic)

22 2

11 11 11

0

n

nn

a a

a D a

a

11 22 11 11 22 nna a D a a a .

1.3.nh thc chuyn v

o nh thc ( )tD A gi l nh thc chuyn v ca nh thc ( )D A .o Th dnh thc

1 1 0

0 1 1

2 2 3

c chuyn v l nh thc

1 0 2

1 1 2

0 1 3

.

o

o Mnh : nh thc v chuyn v ca n l bng nhau.

Nhn xt: Do kt qu ny, nhng tnh cht no sau y ng cho dng ca nh thccng sng cho ct ca nh thc.

2. Tnh cht2.1 Tnh thay phin:i ch 2 dng th nh thc i du.

Th d:1 2 3 2 1 3

1 2 3

a b c

x y z x y z y x z

a b c b a c


22/39

I STUYN TNH

22

H qu: nh thc c 2 dng ging nhau th bng 0.

0

a b

D x y z x y z

a b c

a b c

D

c

a b c

D .

2.2. Tnh tuyn tnh:

Nhn 1 s vi 1 nh thc l nhn s vi 1 dng no ca nh thc.

12

2 2 2 2

2 2 2 2 4 2 2

2

a x m a x m a x m a x m

D b y n D b y n b y n b y n

c z p c z p c z p c z p

H qu:

nh thc c dng bng 0 th bng 0. nh thc c 2 dng t l th bng 0.

0 0 0 0

a b c

m n p

,

1 1 2

2 2 4 0

3 1 3

,

1 1 1

0 2 2 0

3 1 1

2.3. Tnh cng tnh:

Ta c th cng 2 nh thc cp n thnh mt nh thc cp n (m khng cntnh gi trnh thc) nu 2 nh thc c 1n dng, theo cng th t dng, i mtging nhau.

Khi , ta cng cc phn t tng ng trn dng cn li v gi nguyn 1n dng kia c nh thc tng.

1 2 3 1 2 3

a b c a b c a b c

x y z x y z

m n p m n p m n p

H qu:Nu ta ly kt qu nhn 1 dng ca nh thc vi 1 s cng vo 1 dng khc (cngcc phn t cng th t vi nhau) th nh thc khng thay i ga tr.

( hay ni cch khc: Nu cng vo 1 dng ca nh thc mt dng khc c nhn vi 1s th nh thc khng thay i gi tr.)


23/39

I STUYN TNH

23

1 2 3 1 2 3 1 2 3

2 4 6

1 2 3

a b c a b c a b c

x y z x y z x a y b z c

ng dng tam gic ha mtnh thc:

2 1 3 1 1 2 1 2

1 2 1 2 2 1 3 1

1 0 0 1 1 0 0 1

0 1 1 2 0 1 1 2

1 2 1 2

0 5 5 5

1 0 0 1

0 1 1 2

1 2 1 2

0 5 5 5

0 2 1 3

0 1 1 2

1 2 1 2

0 1 1 1

5 0 2 1 3

0 1 1 2

1 2 1 2

0 1 1 1

5 0 0 1 1

0 0 0 3

15

3. Khai trin nh thc3.1.nh thc con Phn b i s:

Cho nh thc, 1,ij i j n

D a

.

nh thc cp knhn c tD bng cch bi n k dng v n k ct no caDc gi l nh thc con caD.Ta cng gi nh thc l nh thc con ca ma trn ija .

Th d:

2

1 1 3

3 32 2

1

3 4

45 2 53

1 6

4

2

6

3

15 2

6 1

a x

b y

c z

c 1 trong cc nh thc cp 3 l

a x

b y

c z

.

nh thc con cp 1n nhn c bng cch b dng thi v ct thkcgi l nh thc con ghp theo phn tdng i ct j. K hiu ijD .


24/39

I STUYN TNH

24

Th d:Cho nh thc

2 1 0

0 2 3

1 1 2

D

.

nh thc con caD ghp theo phn t dng 2 ct 3 l 232 1

1 1D

.

S ( 1)i jij ijD c gi l phn b i s ng vi phn t dng i ct j.

3.2.nh l:

Cho ijD a v {1,2,..., }i n . Khi :

1 1

( 1)n n

i k

ik ik ik ik k k

D a A a D

.

Ta gi y l cng thc khai trin nh thcD theo dng thi.Tng t, ta c cng thc khai trin nh thcD theo ct thj:

1 1

{1,2,..., }

( 1)n n

k jkj kj kj kj

k k

j n

D a A a D

GHI CH: nh ngha nh thc nu chnh l khai trin nh thc theo ct 1.

Th d:

1 1 1 2 1 31 0 2 0 2 12 1 0 ( 1) 3 ( 1) ( 2) ( 1

3 2 1

) ( 1)1 2 0 2 0 1

0 1 2

6 8 2 0

3.1.nh l:( ) ( ) ( )D AB D A D B

III. HNG CA MA TRN1.nh thc con ca ma trn

Cho ma trn ( , )ija Mat m n .

nh thc cp rnhn c tA bng cch bi m r dng v n r ct no


25/39

I STUYN TNH

25

caAc gi l nh thc con caA.Th d:

Cho ma trn

2 4 0 2

1 3 2 5

0 1 1 0

A

.

nh thc4 2

1 0D

l 1 nh thc con cp 2 caA.

nh thc

2 0 2

1 2 5

0 1 0

E

l 1 nh thc con cp 3 caA.

nh thc chnh:Ta gi nh thc chnh ca ma trnA l mt nh thc conD cp rno caAm:

o 0D o Mi nh thc con caA c cp ln hn ru bng 0.

Th d:

Ma trn0 0 0

0 1 0

c 1 nh thc chnh duy nht l 1 .

Ma trn

1 0 2 3

1 1 2 3

1 1 2 3

1 2 2 3

c 1 trong cc nh thc chnh l

1 0

1 1.

2.Hng ca ma trnnh ngha:Hng ca mt ma trnA l cp ca nh thc chnh ca ma trn .K hiu: ( ), ( )rg A r A Th d:

o (0) 0rg

o

1 0 2 31 1 2 3

21 1 2 3

1 2 2 3

rg


26/39

I STUYN TNH

26

o Nhn xt: Nu ( )ija Mat n th :

( ) ( ) 0rg A n D A

3.Php bin i scp dng3.1.Cc php bin i sau y p dng cho mt ma trn (c mt ma trn khc)gi l php bin i scp dng:

Loi 1: i ch 2 dng ca ma trn cho nhau Loi 2: Nhn 1 dng ca ma trn vi 1 s khc 0. Loi 3: Ly kt qu nhn 1 dng ca ma trn vi 1 s cng vo 1 dng

khc (cng cc phn t cng th t vi nhau).

Th d:

2 0 0 1 1 2 2 2 1 2 2 2

0 1 1 1 0 1 1 1 0 1 1 11 2 2 2 2 0 0 1 0 4 4 5

1 2 2 2

0 1 1 1

0 0 0 9

3.2.nh l:Php bin i scp dng khng lm thay i hng ca ma trn.

Th d:

2 0 0 1 1 2 2 2 1 2 2 2

0 1 1 1 0 1 1 1 0 1 1 1

1 2 2 2 2 0 0 1 0 4 4 5

1 2 2 2

0 1 1 1 3

0 0 0 9

rg rg rg

rg

3.3.Ma trn rt gn theo dng tng bc:

Mt ma trn ( , )ija Mat m n c gi l c dng rt gn theo

dng tng bc nu tha 2 iu kin sau:o Tt c cc dng bng 0 (nu c) u nm di nhng dng khc 0

(nu c).o Nu rdng u c cha phn t khc 0 (dng 1r n dng m

u bng 0) v nu phn t khc 0 u tin (t tri sang) ca dng


27/39

I STUYN TNH

27

thi nm ct ik ( 1,...,i r ) th:

1 2 rk k k .Th d:

0 2 0 1 1

1 0 0 0 20 0 2 1 3

A

0 3 2 1 1 0

0 0 1 1 3 4

0 0 0 0 2 1

0 0 0 0 0 0

0 0 0 0 0 0

B

0 3 2 1 1 0

1 0 1 1 3 4

0 0 0 2 2 1

0 0 0 0 0 4

0 0 0 0 0 0

C

nh l:Nu ma trn A c dng rt gn theo dng tng bc vi s dng khc 0 lrth ( )rg A r .

nh l:Lun c th bin i mt ma trn v dng rt gn theo dng tng bcnhmt s hu hn php bin i scp dng.

Th d:

Tm hng ca ma trn:

1 1 2 3 1

3 1 2 2 4

2 3 1 1 6

1 2 3 1 4

A

.

Ta c

1 1 2 3 1 1 1 2 3 13 1 2 2 4 0 4 8 11 7

2 3 1 1 6 0 1 5 7 8

1 2 3 1 4 0 1 1 4 5

rgA rg rg


28/39

I STUYN TNH

28

1 1 2 3 1 1 1 2 3 1

0 1 1 4 5 0 1 1 4 54

0 0 6 3 3 0 0 2 1 1

0 0 4 27 27 0 0 0 25 25

rg rg

IV. MA TRN KHO MA TRN O1. nh ngha:

Mt ma trn vung cp n ( )ija Mat n c gi l kho

(hay kh nghch) nu ta tm c mt ma trn vung cng cp( )B Mat n sao cho:

nB BA I ( nI l ma trn n v cp n).

Ma trnB ni trn c gi l ma trn o caA v k hiu l 1 . Ma trn kho cn gi l ma trn khng suy bin. Th d:

o Cho2 1

3 2A

v

2 1

3 2B

.

V 2B BA I nnA kh nghch v1 2 1

3 2A B

.

o Tm , , ,y z t tha

2 1 1 1

3 2 0 2

x z

y t

2 1 2 1 2 1 1 1

3 2 3 2 3 2 0 2

2

2 0 3

3 1 0

1

x z

y t

x

x z y

y t z

t

2. nh l:Cho ( )ija Mat n .

A kho ( ) 0D A


29/39

I STUYN TNH

29

Lu :

Vi ( )ija Mat n :

A kho A khng suy bin ( ) 0D A ( )rg A n

3. Cch tm ma trn o.3.1. Dng phn b i s:

Cho ( )ija Mat n .

Khi :

11 21 1

12 22 21 1( )

1 2

n

nD A

n n nn

A A A

A A AA

A A A

3.2. Dng php bin i scp dng:

Cho ( )ija Mat n .

Ta vit mt ma trn nB A I cp 2n n (na bn tri l matrnA, na bn phi l ma trn n v) v p dng php bin i scp dng trn matrnB sao choA trthnh nI . Khi ma trn thu c na bn phi l ma trn o caA.

Th d:

Chng minh ma trn

0 1 1

1 0 1

1 1 0

C

kho v tm ma trn o ca n.

Gii:

Ta c

0 1 1 0 1 11 1

( ) 1 0 1 0 1 1 2 01 1

1 1 0 1 1 0

D C

nn Ckh nghch.

v Cch 1:

11 21 311 1 1

12 22 32( ) 2

13 23 33

1 1 1

1 1 1

1 1 1D C

C C C

C C C C

C C C


30/39

I STUYN TNH

30

Cch 2:

0 1 1 1 0 0 1 0 1 0 1 0

1 0 1 0 1 0 0 1 1 1 0 0

1 1 0 0 0 1 1 1 0 0 0 1

A

1 0 1 0 1 00 1 1 1 0 0

0 1 1 0 1 1

1 0 1 0 1 00 1 1 1 0 0

0 0 2 1 1 1

1 1 12 2 2

1 0 1 0 1 0

0 1 1 1 0 0

0 0 1

1 1 12 2 21 1 12 2 21 1 12 2 2

1 0 0

0 1 0

0 0 1

Vy

1 1 1

2 2 21 1 1 1

2 2 21 1 12 2 2

C

= 12

1 1 11 1 1

1 1 1

BI TP CHNG I.TNH TON TRN MA TRN

1. Cho cc ma trn2 0 10 1 2

1 1 0

A

v1 0 1 11 1 2 0

1 1 3 1

B

.

Tnh cc ma trn , , ( )t t tB B A B A A .

2. Chocos sin

sin cosA

.

a. Chng minh , ( )n

nA A A A

.

b. Tnh

20053 1 1 1

,1 11 3

n

3. Tm ma trn 2( )X M sao cho X XA vi:


31/39

I STUYN TNH

31

a.1 1

0 1A

b.

1 2

1 1A

NH THC

4. Tnh cc nh thc:

2 1

3 1,

13547 13647

28423 28523,

0 1 2

2 2 0

1 0 3

,

246 427 327

1014 543 443

342 721 621

,

3 0 1 1

1 3 0 1

1 1 3 0

0 1 1 3

,

1 2 3 4

2 1 4 3

3 4 1 2

4 3 2 1

5. Chng minh:

a.

1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3

2

a b b c c a a b c

a b b c c a a b c

a b b c c a a b c

b.

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

( 1) ( 2) ( 3)

( 1) ( 2) ( 3)0

( 1) ( 2) ( 3)

( 1) ( 2) ( 3)

a a a a

b b b b

c c c c

d d d d

6. Tnh nh thc cp n :

1 2 2 2 22 2 2 2 2

2 2 3 2 2

2 2 2 1 2

2 2 2 2

n

n


32/39

I STUYN TNH

32

7. Gii phng trnh:

2 31

1 1 1 10

1 2 4 8

1 3 9 27

x x x

HNG MA TRN MA TRN KH NGHCH

8. Tm hng ca ma trn:

1 3 4 5 2

0 1 3 4 6

3 5 2 3 4

2 3 5 6 4

,

3 1 3 2 5

5 3 2 3 4

1 3 5 0 7

7 5 1 4 1

,

2 0 3 1

1 2 2 3

3 2 5 4

5 2 8 5

,

1 1 2

2 1 5

1 10 6 1

a

a

,

3 1 1 4

4 10 1

1 7 17 3

2 2 4 3

a

,

1 1 1

1 1 1

1 1 1

1 1 1

m

m

m

m

9. Cho ma trn

2

3 2 3

1 0 2 3

2 1 3

5 3 3 7

m

mA

m m

m m

.

Tm m sao cho

a.

0 4 10 1

4 8 18 7

10 18 40 17

1 7 17 3

rankA rank


33/39

I STUYN TNH

33

b.

2 1 1 1

1 3 1 1

1 1 4 1

1 1 1 5

1 2 3 4

1 1 1 1

rankA rank

c.

1 0 0 1 4

0 1 0 2 5

0 0 1 3 6

1 2 3 14 32

4 5 6 32 77

rankA rank

10.Tm ma trn o ca:

a.1 2

2 2

b.

2 0 0

1 1 0

2 1 1

c.

1 0 3

2 1 1

3 2 2


34/39

I STUYN TNH

34

Chng II.H PHNG TRNH TUYN TNH

I.nh ngha1. H phng trnh11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

( )

n n

n n

m m mn n m

a x a x a x b

a x a x a x bL

a x a x a x b

,

vi , , ( 1, , 1, )j i ijx b a i m j n v jx l n s, c gi l hphng trnh tuyn tnh.

Ma trn

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a aA

a a a

gi l ma trn h sca h phng

trnh tuyn tnh (L).

Ma trn

11 12 1 1

21 22 2 2

1 2

n

n

m m mn m

a a a b

a a a bA

a a a b

gi l ma trn bsungca h

phng trnh tuyn tnh (L). Nu ta chn 1 nh thc chnh ca ma trn h sA th cc n ca h phng

trnh tuyn tnh (L) ng vi cc ct ca nh thc chnh ny sc gi l ccn chnh, cc n cn li l n khng chnh.

2. Dng ma trn ca h phng trnh tuyn tnh

Nu t

1

2

n

x

x

Xx

v

1

2

m

b

b

Bb

th ta thy h (L) c vit thnh

( )L AX B Ta gi y l dng ma trn ca h phng trnh tuyn tnh.


35/39

I STUYN TNH

35

Nghim ca h phng trnh tuyn tnh (L) l nhng bn th t

1( ,..., )n

nc c sao cho tt cng thc trong (L) l ng khi thay

j jx c .

Gii h phng trnh tuyn tnh l tm tt c nghim ca n. Hai h phng trnh tuyn tnh gi l tng ng nu chng c cng

tp nghim.NHN XT:

p dng php bin i scp dng ln cc dng ca 1 h phng trnh tuyntnh ta c 1 h phng trnh tuyn tnh khc tng ng.

II. H phng trnh Cramer.1. H phng trnh tuyn tnh (L) c gi l h Cramer nu n tha 2 iu kin

sau: m n ( ) 0D A

2. nh l (qui tc Cramer):H phng trnh Cramer lun c nghim v c duy nht 1 nghim xc nh bi:

jDj D

x , trong :

( )D D A jD l nh thc nhn c bng cch thay ct thj caD bi ct t do

1,...,

nb b .

3. Th d:Gii h phng trnh tuyn tnh:

1

1

2

x z

y z

x y

Gii:H cho c dng ma trn:

11 0 1 1 1 0 1 1

0 1 1 1 0 1 1 1

1 1 0 2 1 1 0 2

x x

y y

z z


36/39

I STUYN TNH

36

12

1 1 1 1 2

1 1 1 1 0

1 1 1 2 1

x

y

z

III.iu kin c nghim ca h phng trnh tuyn tnh.1. nh l (Kronecker-Capelly):

Cho h phng trnh tuyn tnh (L).

H (L) c nghim rgA rgA .

2. Gii h phng trnh tuyn tnhChng minh trn cho ta mt phng php gii h phng trnh tuyn tnh (L), gil phng php Gauss.

Vit ma trn v a n v dng rt gn theo dng tng bc.

Nu rgA rgA th h v nghim.

Nu rgA rgA r , th ta xc nh 1 nh thc chnh caA. Sau chuyn cc n khng chnh sang v phi lm tham s ly gi tr ty . Ta tnhc cc n chnh nhqui tc Cramer. Lu rng trong trng hp ny h sc v s nghim ph thuc vo n r tham s ly gi tr ty trong . Vnu 1 1,..., { ,..., }n r ny x x cc n khng chnh, th mt n chnh kx no s

c dng tng qut: 1

n r

k k kj jjx y

. Do , h phng trnh tuyn tnh (L) c nghim duy nht khi v ch khi

rgA rgA r v khng c n chnh, tc l khi v ch khi rgA n .

IV.H phng trnh tuyn tnh thun nht1. nh ngha:

H phng trnh tuyn tnh

11 1 12 2 121 1 22 2 2

1 1 2 2

00( )

0

n nn n

m m mn n

a x a x a xa x a x a xH

a x a x a x

c gi l h phng trnh tuyn tnh thun nht.


37/39

I STUYN TNH

37

H thun nht lun lun c nghim, chng hn (0,0,...,0) n l 1 nghimca h. Ta gi nghim (0,0,...,0) ny l nghim tm thng.

Theo kt qu trn, h (H) c nghim duy nht khi v ch khi rgA n .

2. H nghim cbn ca h thun nht:Xt h phng trnh tuyn tnh thun nht (H) vi rgA r n .Gi s 1 2, ,...,r r nx x x l cc n khng chnh ca h.Suy ra cc nghim ca h (H) c dng tng qut l:

1 2 11 1 1

, ,..., , ,...,n n n

j j j j rj j r nj r j r j r

u c x c x c x x x

.

Khi , n r nghim c th:

1 1 1 2 1 1( , ,..., ,1,0,...,0)r r rr u c c c ,

2 1 2 2 2 2( , ,..., ,0,1,...,0)

r r rr u c c c

,

1 2( , ,..., ,0,0,...,1)n r n n rnu c c c (nhn c bng cch gn cho cc tham s 1( ,..., )r nx cc gi tr thuc tp

{(1,0,..,0),(0,1,0,..,0),...,(0,0,..,0,1)} n r )c gi l 1 h nghim cbn ca h (H).

Th d:Tm mt h nghim cbn ca h

1 2 3 4

1 2 3 4

1 2 3 4

2 5 7 0( ) 4 2 7 5 0

2 5 0

x x x xH x x x x

x x x x

Gii:

2 1 5 7 2 1 5 7 2 1 5 7

4 2 7 5 0 0 3 9 0 0 1 3

2 1 1 5 0 0 4 12 0 0 1 3

2 1 5 7

0 0 1 3

0 0 0 0

Vy 1 2 3 4

3 4

2 5 7 0( )

3 0

x x x xH

x x

2 3 1 4

3 4

5 2 7

3

x x x x

x x

1 1

2 1 2

3 2

4 2

2 8

3

x t

t t

x t

x t

Nghim tng qut ca (H) l:


38/39

I STUYN TNH

38

1 1 2 2 2 1 2( , 2 8 , 3 , ) , ,u t t t t t t t

Cho 1 2( , ) {(1,0), (0,1)}t t , ta c mt h nghim cbn l:

1

2

(1,2,0,0)

(0, 8, 3,1)

u

u

BI TP CHNG II.

H PHNG TRNH TUYN TNH

11.Gii cc h phng trnh sau:

a.1 2 3

1 2 3

1 2 3

2 4

3 4 2 11

3 2 4 11

x x x

x x x

x x x

b.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

2 3 1

3 2 4

2 3 6

2 3 4

x x x x

x x x x

x x x x

x x x x

c.

1 2 3 4

2 3 4 5

1 2 32 3 4

3 4 5

0

0

2 3 22 3 2

2 3 2

x x x x

x x x x

x x xx x x

x x x

12.Gii cc h sau:

a.

1 2 3

1 2 31 2 3

1 2 3

3 1

2 2 1

3

2 3 1

x x x

x x x

x x x

x x x

b.

1 2 3

1 2 3

1 2 3

1 2 3

2 2

3 5

5 7

2 3 3 14

x x x

x x x

x x x

x x x

c.

1 2 3 4

1 2 3 41 2 3 4

1 2 3 4

2 1

3 2 2 3 2

5 2 1

2 3 4

x x x x

x x x x

x x x x

x x x x

d.

1 2 3 4

2 3 4

1 2 4

2 3 4

2 3 4 4

3

3 3 1

7 3 3

x x x x

x x x

x x x

x x x

e.

1 2 4 5

1 2 3 4

1 2 3 4 5

1 2 3 4 5

2 3 0

2 04 2 6 3 4 0

2 4 2 4 7 0

x x x x

x x x xx x x x x

x x x x x

f.

1 2 3 4 5

1 2 3 4 5

2 3 4 5

1 2 3 4 5

7

3 2 3 22 2 6 23

5 4 3 3 12

x x x x x

x x x x xx x x x

x x x x x


39/39

I STUYN TNH

13.Tm nghim tng qut v h nghim cbn (nu c) ca cc h thun nht sau:

a.1 2 3 4

1 2 3 4

1 2 3 4

2 4 5 3 0

3 6 4 2 0

4 8 17 11 0

x x x x

x x x x

x x x x

b.

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

6 2 2 5 7 09 3 4 8 9 0

6 2 6 7 0

3 4 4 0

x x x x xx x x x x

x x x x x

x x x x x

c.

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

2 0

2 2 3 0

3 2 2 0

2 5 2 2 0

x x x x x

x x x x x

x x x x x

x x x x x

d.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

0

2 3 4 0

3 6 10 0

4 10 20 0

x x x x

x x x x

x x x x

x x x x

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Ch 0.I.ii Tap Hop Anh Xa