07线性代数讲义

更新时间：2023-06-07 18:52:01 阅读量：实用文档文档下载

说明：文章内容仅供预览，部分内容可能不全。下载后的文档，内容与下面显示的完全一致。下载之前请确认下面内容是否您想要的，是否完整无缺。

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

5 êùÂ

ÜÆD

~^ÎÒ

Z=:{x|x´ ê}.

N0=:{x|x∈Z,x≥0}.N=:{x|x∈Z,x>0}.Q=:{x|x´knê}.R=:{x|x´¢ê}.C=:{x|x´Eê}.Pv«knê !¢ê !Eê"

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1

§1.1

n01 ª ½Â

1 ª

(1).10I ª| (a)|=a

a 11a12

(2).20I ª =a11a22 a12a21

a21a22

~X =4 6= 2

a11a12a13

(3).30I ª a21a22a23

a31a32a33

=a11 a22a33+a12 a23a31+a21a32a13 a31a22a13 a21a12a33 a12a23a31

2 12

~X 110 =2

312 123 246 =0 312 (4)p0I ª

ds 5'Vg:

½Â1.1d1,2,···,n|¤' kSê|.i1,i2,···,in ¡ n?ü , 1,2,···,n ¡ n?IOü .

~X:1,2,3,4 2,1,4,3 2,3,1,4þ´4?ü .

½Â1.2 i1,i2,···,in´ n?ü ,Xts<s is>is ,@o¡is is ¨¤ _S. ü '_Soê, ¡ ù ü '_Sê.P τ(i1,i2,···,in)

ü Ñy = _ _ ê 12345 Ã0 ~: 13245(32)1

(42),(43)2 14235

14352(43),(42),(52),(32)4 ½Â1.3Xt_Sê´óê,@o¡ù ü óü ;Xt_Sê´Ûê,@o¡ù ü Ûü .

½n1.4n?ü ²v gé & '_SêUCÛó5.

y².#i1,i2,···,in´n?ü ,²véxis is+m& j1,j2,···,jn

yyτ(i1,i2,···,in) τ(j1,j2,···,jn)ØÓÛóSµ¨m=1 @Øw,¤á.

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

(i1,i2,···,is,is+1,···,is+k,is+k+1,···,in)²vmg éx&(i1,i2,···,is+m,is,is+1,···,is+m 1,is+m+P²vm 1g éx1uj1,j2,···,jn.Ïdτ(i1,i2,···,in)'ÛóSUg 2m 1g,τ(j1,j2,···,jn)Ug τ(i1,i2,···,in)'ÛóS.

½n1.5(1).Xtü j1,j2,···,jn´dg,ü ²vtgé & ',@oτ(j1,j2,···,jn) tÓÛó.

(2).Xtü i1,i2,···,in²vtgé & g,ü ,@oτ(i1,i2,···,in) tÓÛó.

²vtgé

y².(1).j1,j2,···,jn→1,,2,···,n,@oj1,j2,···,jn'ÛóSUg tg.Xtτ(j1,j2,···,jn)´Ûê,Ï1,,2,···,n´óü ,Ïdt ´Ûê.Ó ,Xtτ(j1,j2,···,jn)´óü ,@ot ´óê,Ïdτ(j1,j2,···,jn) tÓÛóS.

(2).aqy².2

Ún1.6Xtü j1,j2,···,jn´n?ü ,¿ ji1=1,ji2=2,···,jin=n,@oτ(j1,j2,···,jn) τ(i1,i2,···,in)ÓÛó.

j1j2···jn12···n²vtgé

y².→.d½n1.5 &

12···nj1j2···jn

@Ø"

²vtgé ²vtgé

Xtj1,j2,···,jn→1,2,···,n,@o1,2,···,n→i1,i2,···,in.d½n1.5 ,τ(j1,j2,···,jn) τ(i1,i2,···,in)ÓÛó.2

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.2

Ï ,123 '¤kü 123 (ó),132(Û),213(Û),231(ó),312(ó),321

a11a12a13

(Û),Ïd a21a22a23 =( 1)τ(i1i2i3)a1i1a2i2a3i3.

a31a32a33

a110···0 a 21a22···0

~.O :(1).D= =a11···ann.

············ a n1an2···ann

0···0a1n

0···an(n 1)0 2(n 1)

(2).D= =( 1)an1···a1n.

············ a00 n1···

:P56,1.2"1.^I ª'½ÂO e I ª' .

0b1000 00···0a1 00b002 a0···00 2 (1). =a1a2a3 an.(2). 000b30 =a1b1b2b3b4. 0a3···00 0000b 4 00···a 0n

a1a2a3a4a5

a 11a12a13a14a15 a21a22a23a24a25 (3). a31a32000 =0.

00 a41a420

a51a52000

2.!Ñ40I ª¥¤k KÒ ¹kÏfa11a23' .

3.y²:eQ n0I ª¥1u0'£ ' ê un2 n,uTI ª 0.

4.©yÀtiÚj,¦&(1)¥1274i56j9¤Ûü ,(2)¥1i25j4897¤óü .§1.3

1 ª 5

=( 1)τ(i1,i2,···,in)a1i1···anin(ØÓIØÓ a

11a12

£ 'È).kw ,n0I ª: =( 1)τ(i1i2)a1i1a2i2.

a21a22

n01 ª ½Â

a11a12···a1n a

21a22···a2n

············ a

n1an2···ann

{zI ª'O , A S .XtòI ªD'I

p ,&¡ I ª'= & '5I ª,P D .

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

2 12 213 ~X 110 = 111

312 202

5 1.7D=D.

a11a 12···a 1n a11a12···a1n

21a22···a2n a21a22···a2n

y².PD= . ,D=

············ ············

an1an2···ann n1an2···ann

τ(i1,i2,···,in)

D=( 1)a1i1···anin,D=( 1)τ(j1,j2,···,jn)a1j1···anjn.d=

'½Â aij=aji.é?Û ü j1,j2,···,jn,-i1,i2,···,in,¦&ji1=1,ji2=2,···,jin=n.w( 1)τ(j1,j2,···,jn)a1j1···anjn

=( 1)τ(j1,j2,···,jn)aj11···ajnn=( 1)τ(j1,j2,···,jn)a1i1···anin=( 1)τ(i1,i2,···,in)a1i1···anin

¤±D=D.

5 1.8¢ 1 ªü1 ,& '#1 ªUCÎÒ.

··············· ···············

at1at2······atn as1as2······asn

y².#I ªD= ··············· ,D1= ··············· .

as1as2······asn at1at2······atn

··············· ···············

yD= D1.dI ª½ÂD1=( 1)τ(i1,···,it,···,is···,in)a1i1···atit···asis···anin

= ( 1)τ(i1,···,is,···,it···,in)a1i1···asis···atit···anin= D.2íØ1.9e1 ªü1 Ó,ud1 ª ".

5 1.10^êk¦±1 ªD', 1& '1 ªD1´D'k .=D1=kD.

y².^êk¦±I ªD'IsI& 'I ªD1.wD1=( 1)τ(i1,i2,···,in)a1i1···(kasis)···anin

=k( 1)τ(i1,i2,···,in)a1i1···asis···anin=kD,

············

a+a a+a ···a+a j1jnj2j2jn j1

5 1.11D= =

············ ············

············ ············ a j1aj2···ajn aj1aj2···ajn + . ············ ············ ············ ············

(dÚn1.6).

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

y².D= ( 1)τ(i1,···,in)aa

1i1···(jij+ajij)···anin

= ( 1)τ(i1,···,in)a1i n)1···ajij···aniτ(i1,···,in+( 1)a1i1···a jij···anin.¤±þª¤á.2

íØ1.12e1 ªkü1éA1¤'~,u1 ª ".

5 1.131 ª', 1( )\þ, 1'k ,u1 ª' ØC.y².dS 1.11ÚíØ1.12

&.2~.O D= 1201 132

50 015 .

36 1 234 5

1 A.D=1

1201

23100

0 110 2 1201 03518 =

518 0 110 2 = 515 003512 1201 1015

00 1201 1

0015 1

=1 0 110 2 0 110 30 0 =1

2 1

0514 30 00514 30

(( 1)×5×( 43))=

=0015 1 000 42 1

43.5.þãIn,o,Ê,8 1Ò'O v§ g´:I I~ I I'

!Ó IoI~ I I'Ê!;InI\þI I'n!;InI~ IoI' !;IoI~ InI' n!. xaa···a

:O n0I ªD axa···a ~n= ··············· ··············· aa···ax x+(n 1)aaa···a

x+(n 1)a00···

x+(n 1)axa···a 0x ···A.D= ···············

a0 ···· ··············· =

··········· x+(n 1)aa···ax ········· 00···

=(x+(n 1) a)(x a)n 1.

a1+b1b1+c1c1+a1 a1b1c1 ~:y² a 2+b2b2+c2c2+a2

ac =2 a2b2c2 .

3+b3b3+c33+a3 a3b3c3

0 0

··· ···

x a

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

a1b1+c1c1+a1 b1b1+c1c1+a1 a1b1+c1c1

y². = a2b2+c2c2+a2 + b2b2+c2c2+a2 = a2b2+c2c2 +

a3b3+c3c3+a3 b3b3+c3c3+a3 a3b3+c3c3

b1c1c1+a1 a1b1c1 a1b1c1 a1b1c1 b2c2c2+a2 = a2b2c2 + a2b2c2 =2 a2b2c2 . b3c3c3+a3 a3b3c3 a3b3c3 a3b3c3

:P64,2(2),1(4).2.y ²e ª.b1+c1c1+a1a1+b1 (1). a1 bcb

2+2c2+a2a2+2 ba 3+c3c3+3a3+b3 =2 a2 a3 (2). abc aa+ba+b+c a2a+b3a+2b+c =a3.

3.® I ªA=2.òA¥'z £ªB,¦I ªB.

b1c1

b 2c2 b

3c3 aij©y¦±ki j(k=0)& I

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.4

1 ª Ðm

½Â1.14D= a11a12···a1n aa 2122···a2n

············ uD¥'Ii1Ij & 'f1

an1an2···ann

ª¡ aij'{fª,P

a11···a1(j 1)a1(j+1)···a1n ···

···············

M= a···a (i 1)1···a(i 1)(j 1)a(i 1)(j+1)(i 1)n ij

a .¿ ¡( 1)i+jMij aij'(i+1)1···a(i+1)(j 1)a(i+1)(j+1)···a(i+1)n

·················· an1···an(j 1)an(j+1)···ann ê{fª,P Aij.

½n1.15D=as1A1+as2As2+···+asnAsn.

y².· Äky²e¡@Øµ

a11a12···a1n ea¥Øa a22···a2n st¤Q'Ist©Ù¦'£ Ñ´",uD=

a21 ············

=an1an2···ann

astAst.

Xts=1=t.wD= ( 1)τ(i1,i2,···,in)

a1i1···anin=a11 ( 1)τ(i2,··· ,in) a2i2···anin=

aa st a11···a1t···a1n s1··· a11···a1t ············ ····a11A11.é?Û1≤s,t≤n,wD=

as1···ast···asn =( 1)s 1 ····· as 1,1···as ············ 1,t as+1,1···as+1,t a n1···ant···ann

···

······ an1···ant

astas1···as,t 1as,t+1···asn a1t

a·a 11··1,t 1as1t+1···a1n

············ ( 1)s 1+t 1 ········· a s 1,tas 1,1···as 1,t 1as 1,t+1···as 1,n

a,tas+1,1···as+1,t 1a =astAst.

s+1s+1,t+1···as+1,n as+1,1···as+1,t···as+1,n ····················· antan1···an,t 1an,t+1···ann

··asn

··a1n···

··as 1··as+1···

··ann

·····

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

yQ· 5y²½n,w

11a12···

·········

D= as10···

·········

an1an2···

a1n ···

0 + ···

ann

a11···0···an1a12···as2···an2

···············

a1n

···

asn

···

ann

a11a12···a1n ············

=as1As1+

as2···asn ············

an1

an2 ···

a11a12···a1n ············

=as1As1+as2As2+

00···ain ············

an1an2···

=as1As1+as2As2+···+asnAsn.2

a11a12···a1n

íØ1.16 D= a

21a22···a2n

············ ,u

n1an

2···ann

i)adk=i

k1Ai1+·+aknAin=

0k=i;

ii)ads=j

1sA1j+·+ansAnj=0s=j

a11 ···

(i 1)1y².i)¨i=k ,dUIÐmª &"y#i=k.du

ak1

a(i+1)1

···

a k1

···.

an10, ¡d½n1.15,þª =ak1Ai1+·+aknAin,Ïdi)¤ª= ØUgI ª' ,Ïdii) di)íÑ.2

a12·········a(i 1)2···ak2···a(i+1)2·········ak2·········an2···á;Ï Ia 1n ··· a(i 1)n a kn a(i+1)n

··· =a kn ··· a nn

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

x ¦I ª' .

10 2

~D= 013

231

A.D=1A11+0A12+( 2)A13,Ù¥A11

( 1)1+3 =2,ÏdD= 12.

123

~D= 456 ,¦7A11+8A12+9A13"

789 A.7A11+8A12+9A13=0.

11···1 a1a2···an

2~.y²Dn= a2···a2n2 a1

········· ···

n 1n 1

1 a1a2···ann

b'I ª&¡ n0 % I ª.y².én^VB{,¨n=2 ,w,¤á.#n≤k @Ø¤á,¨n=k+1

2 13=( 1)

= 8,A13=

1≤i<j≤n(aj ai),Ù¥n>1.

11···1

0a2 a1···an a1

22Dn= ···an ana1 a2 a2a1 0

········· ···

1n 2n 2n 1 0an a2a1···an ana1 2

a2 a1a3 a1···an a1 a(a a)a3(a3 a1)···an(an a1) 22 1 ············ an 2(a a)an 2(a a)···an 2(a a)

31n12123n 11···1 aa3···an 2

(a2 a1)···(an a1)

··· ········· an 2an 2···an 2 23n

(a2 a1)···(an a1)(aj ai)=(aj ai)

2≤i<j≤n

1≤i<j≤n

=m.

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

111

~D= 234 ,¦D"

4916

A.D=(3 2)(4 2)(4 3)=2.

~ .¦n0I ª 2 10···00

21···00 1

0 12···00 .Dn= ············ 000···21 000···12

210 21 11

A.n=2,D2=4 1=3,Dn= 121 =2 =6 2=4.

12 02 012

#(n>3),DnUI I Ðm&

110···00

21···00 0

12···00 3 0 =2Dn 1 Dn 2.Dn=2Dn 1+( 1)

············

000···21

000···12

e¡^VB{y²Dn=n+1¨n=2 ¤á;#n≤k ¤á,¨n=k+1 Dn=2Dn 1 Dn 2=2n (n 1)=2n n+1=n+1,ÏdDn=n+1.

e I ª. µP71,1.1.O

xab0c 56000

0y00d 15600

(1). 01560 (2). 0cz0f

ghkut 00156

0000v 00015

1+x 111 31 12 1 513 4 1 x11 (3). (4). .

1 11+y1 201 1 1 1 53 3 111 y

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.5Gramer{u

yQ· 5?Øn£ §|

a11x1+a12x2+···+a1nxn=b1 ax+ax+···+ax=b2112222nn2 .............................................

am1x1+am2x2+···+amnxn=bm

(I)

'A¯K.x1,x2,···,xn&¡ (I)'n þ.

ùpn ±Ø1um.aij∈P,´~ê,§&¡ §|(I)'Xê.b1,b2,···,bn∈P,§ &¡ §|(I)'~ê .

Xtò~êc1,···,cn \ §¥x1,···,xn¦ §|(I)¤á,@o¡ù|ê(c1,c2,···,cn) §|(I)' |A.

Xté?Û |ê(c1,···,cn)∈Pn,§ÑØU÷v §|(I),@o¡ §|QP¥ÃA.

2x+3y=4

~X±eù §|´ÃA'µ

2x+3y=0

Xtb1=0,b2=0,···,bn=0,@o §|£I¤g

a11x1+a12x2+···+a1nxn=0 ax+ax+···+ax=02112222nn ............................................

am1x1+am2x2+···+amnxn=0

(II)

¿ ¡§ àg S §|.Ï (0,0,···,0)´£II¤'A(&¡ 0A),Ïdàg S §|o´kA'.àg S §|Ø"A© Uk "A.~X

x 2y+z=0x+y+z=0

x=1,y=0,z= 1 ´ §' |A.

A S §|' {kxõ,e¡· Ñ «^I ª5¦A' {.=Gramer {.ù« { ¦ §ê þ ê Ó(m=n).

a11a12···a1n a 21a22···a2n

PD= ,

············ a n1an2···ann

ù I ª&¡ §|(I)'XêI ª.Dj D¥Ij d~ê b1,···,bn O& 'I ª.=

a11···a1(j 1)b1a1(j+1)···a1n a 21···a2(j 1)b2a2(j+1)···a2n Dj= .

····················· abnan(j+1)···ann n1···an(j 1)

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

½n1.17XtD=0,@o §|(I)k Aµx1=D1/D,x2=D2/D,···,xn=Dn/D.

y²(1).ky²x1=D1/D,x2=D2/D,···,xn=Dn/D´(I)'A.Dj=b1A1j+···+bnAnj.òx1=D1/D,x2=D2/D,···,xn=Dn/D \ §|'I §µ

nnn 11

a1j(biAij)a11(D1/D)+a12(D2/D)+···+a1n(Dn/D)=((a1jDj))=1

=n n j=1i=1

bi(a1jAij)=n n i=1j=1

bi(a1jAij)=j=1n

bi(

j=1i=1

i=1j=1

a1jAij)=b1D=b1

^Ó ' { ±&µx1=D1/D,x2=D2/D,···,xn=Dn/D÷vI2

§,···,In §.Ïd,§´(I)'A.

(2).Py S.=Xt k(c1,···,cn) ´ §|(I)'A,@U÷v(I)¥'z §.=ai1c1+ai2c2+···+aincn=bi,i=1,2,···,n.· w

a11c1a12···a1n

aca

21122···a2n

c1D=

············

aca

n11n2···ann

a11c1+a12c2a12···a1n

ac+acaa2n òc2¦I2 \I1 211 22222···

············

ac+aca

n11n22n2···ann

a11c1+a12c2+···+a1ncna12···a1n

ac+ac+···+acaa2n 2112222nn22···

=···=

············

ac+ac+···+aca

n11n22nnnn2···ann

b1a12···a1n

b 2a22···a2n

= =D1.

············

ba···a

u´c1=D1/D,Ó ,c2=D2/D,···,cn=Dn/D.2½n1.18Xtàg §

a11x1+a12x2+···+a1nxn=0 ax+ax+···+ax=02112222nn ...........................................

am1x1+am2x2+···+amnxn=0'Xê1 ªD=0,@o(II) k"A.

(II)

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

~µA S §| x1+2x2+3x3=07x1 x2+6x3=0

2x1+2x2+x3=0

123 123

A:D= 7 16 = 0 15 15 =0.ÏdT §| k0A.

221 0 2 5

x1+2x2+3x3 2x4=6 2x x 2x 3x=81234

~µA §|

3x1+2x2 x3+2x4=4

2x1 3x2+2x3+x4= 8

123 2 3 2 12

2 1 2 3 2 1 2 3

== AµXêI ªD=

13 32 12 15

2 320 244 1

123 2 123 2 123 2

01 27 00 1815 0 5 81

=18×18. = =

00 1815 01 27 01 27

0 2400018 000184

64 4 4 4 23 2 20 20

0 1 1813 8 1 2 3 8 1 2 3

D1= = =

00 4536 02 910 42 12

8 32 00 1818 0 40 21

163 2

28 2 3

= 2(18)( 45+36)= 2(18)( 9)=182.D2= =

34 12

2 821

12 6 2 36 12 2 18 3 2 1 28

36×18.D3= = 182.D4= =

32 42 32 14 2 3 81 2 32 8

36×18.Ïd,x1=D1/D=1,x2=D2/D=2,x3=D3/D= 1,x4=D4/D= 2.

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.6 £{(pd {)

· ®²Æv ¦A §|'Gramer{u,yQ· H , « {.Q¥Æ· ÆvA §|

2x1 x2+3x3=1 4x1+2x2+5x3=4(I)

2x1+2x3=6

^'´Xe{u:I ª~ I ª& 2x1 x2+3x3=1 4x2 x3=2(II)

3x1+2x3=6Inª~ I ª& 2x1 x2+3x3=1

4x2 x3=2(III)

x2 x3=5òI In § §¢ &

2x1 x2+3x3=1

x2 x3=5(IV)

4x2 x3=2òInI~ I I'4!&

2x1 x2+3x3=1

x2 x3=5(V)

3x3= 18

§|(V)'A´(9, 1, 6).

a11x1+a12x2+···+a1nxn=b1a21x1+a22x2+···+a2nxn=b2

.............................................am1x1+am2x2+···+amnxn=bm@Üc¡'~f, £{´^e n«Ä)'g ,ò §|g¤FG.

1).ò, §'k!\ , §;

Ä Ó.yQ.Äk·VI)

X(V) '´½'(

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

2).¢ , §;

3).^ "'ê¦ §üb.

ùn«Ä)'g I ª¦ ¤^'g Ä)þ Ó.Ïd· ¡ù g Ð1g .e¡· 5y² §|£VI¤Q²vÐ1g & '5 §| ¦ §|k Ó'A.w,Xt¦^2),3)«Ð1g ,§ 'A´ Ó'.´y¦^1)Ð1g & '5 §| ¦ §|(VI)'A´ Ó'( ¡¬y²). Ò´`^ £{A §|´ (' {.· Pw §|(VI):ò§'XêÚ~ê ü¤XeGª

a11a21a31···an1

a12a22a32···an2

···a1nb1···a2nb2···a3nb3············annbn

ù«ÝG®² û½ §|(VI),Ïd· § AÏ'¶¡µÝ ,P

a11a12···a1n a 21a22···a2n

A= .· ò

············

an1an2···ann

¯= A

a11a21···an1

a12a22···an2

···a1nb1

···a2nb2

······

···annbn

¡ (VI)'yPÝ .Ù¢é(VI)cIÐ1g Ò´éÙyPÝ cIÐ1g ,Ïd· ' £{ ±QÝ ¥cI.~Xþ¡'~f [ ±^Ý 'Gª Ñ@t .

2x1 x2+3x3=1

~.¦A4x1 2x2+5x3=4

2x1 x2+4x3= 1 2 1312 1312 131

A.ÙyPÝ 4 254 → 00 12 → 001 2

2 14 1001 20000

2x1 x2+3x3=1x1=(7+x2)

& §|, & .

x3= 2x3= 2

x2´gd þ.( ù«¹kgd þ'A,¿ §|'?¿AÑUd§v«,ù«A& ¡ ÏA,½ö A.)

2x1+x2 2x3+3x4=1

~.¦A3x1+2x2 x3+2x4=4

3x1+3x2+3x3 3x4=5

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

21 23

A. 32 12

333 3

111 1

→ 0 1 45

014 5 4,¤± §|ÃA.

121

4 → 32501

131

5 → 020

2312

124 → 14 510

11 13

1 45 5 ,000 4

3 231

11 13 14 51 Ñy0=

·K1.19

a11x1+a12x2+···+a1nxn=b1 ax+ax+···+ax=b2112222nn2

............................................

am1x1+am2x2+···+amnxn=bm¯@o §|'yPÝ A

0···0c1i1

0···00

¯→ 0···00A ······

0···000···00

²vÐ1C C ·······c2,i2···0·······0···0

····

·······c3,i3·······0···0

····

···············cs,is···0

···

···············

(I)

(II)

Ù¥c1,i1,···,cs,is ""Ý (II) ¡ F/Ý "

¯=0,@o@Øw,"¯=0.ém^VB{"y².XtyPÝ Ay#A¨m=

1 @Øw,"#m>1.#I "' i1 "òi1 ¥' "£ (P c1,i1)Ïvx I'Ð1g x I I", P^ I¦ ê\ I IòIi1 'I2 £ g 0"Ó ±òIi1 'Ù§£ g 0"y Ä uI I e'Ý B"dVBb# B dÐ1g g 0···

0··· B→ ····

0···0···

00·0000·00

···c2,i2···0·······0···0

····

···c3,i3·······0···0

····

···········cs,is···0

···

············

(III)

¯ ²vÐ1g g (II).2Ù¥c1,i2,···,cs,is ""l A

·K1.20Q·K1.19'PÒe§· kXe@Øµ

(i)¨is=n+1 § §|ÃA;(ii)¨s=n § §|k A;

(iii)¨s<n § §|kÃê A"d ¡xi1,···,xis gd þ§ Ù§Ñ¡ gd þ"

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

y².(i)· k §0=cs+1,gñ"Ïd§ §|ÃA"

(ii)dGrammer{u½n1.17 &"(iii)4Cv«·K1.19¥'Ý (II)§¿ ±b#§é?Ûj,kcj,ij=1,ct,ij=0,¨t=j "=Ii1 ¥Øc1,i1=1©§Ù§£ Ñ´"§Ó G§=Iij ¥Øcj,ij=1©§Ù§£ Ñ´"§1≤j≤s.ù §d(II)éA' §|&

xit=ct,n+1

j=i1,···,is

ct,jxj,1≤t≤s,

Ù¥xj,j=i1,···,is,´gd þ(½ö`§xj,j=i1,···,is,´?¿~ê)"§ ´ §|(I)'ÏA"2

~¦ §|'ÏA

2x1 3x2+6x3+2x4 5x5=3

x2 4x3+x4=1.

x4 3x5=2¯´ FG§A.w, §|'yPÝ AÏdgd þ x3,x5.ÏA x1= 2+3x3 5x5

x2= 1+4x3 3x5,

x4=2+3x5Ù¥x3,x5´?¿~ê"

µP72,2"P147(1).2.^ 40{uAe S §| .

2x1+x2 2x3=10 x1+2x2 3x3=6

(1).3x1+2x2+2x3=1(2).2x1 x2+4x3=2 4x1+3x2 2x3=14 5x1+4x2+3x3=4

x1+x2 3x4 x5=0 x x+2x x=0 2x1 3x2+6x3+2x1 5x5=3

1234

(4).(3).x2 4x3+x1=1

4x1 2x2+6x3+3x4 4x5=0 x1 3x5=2

2x1+4x2 2x3+4x4 7x5=0

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.7 þ Vg

Ôn¥küaþ, a ^ êi5v«,XNÈ, Ý,§Ý¶, a #§ 'ê ¢ Ø , Uv #§ ' ,Xå.~ <. ,¦'å 20Úî,å G¡¤30ÝY . 40Z ,¦\ Ý.d FØak ¢ k . ùaþ&¡ ¥þ½ö þ, Ò´`k k ¢'þ. þα' Ý´ þ' ¢, &¡ ½ ê.P |α|. Ý´0' þ&¡ 0 þ.

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.8 þ IL«{

· kw²¡þ'X£Q sXe¤ ±(½ þ.?Û þα,§Ñ ±Àt±Xo åX' þ,§'ªXMdα (½,M&¡ α' s.¯¢þ,#e x,e y©y´x¶,y¶þ'ü þ,@oae x+b ey=α, Pα=(a,b).Q m¥,z þαÑ ±±oX åX,#ªX M.#e x,e y,e z©y´x¶,y¶,z¶þ'ü þN´y²α=ae x+b ey+c ez,¡(a,b,c) α' s,Pα=(a,b,c).yQ· *¿ù Vg,Xta1,a2,···,an∈P,· ¡α=(a1,a2,···,an) Pþn£ê| þ,a1,a2,···,an¡ α'©þ,ü þ 1´ ¦ 'z ©þéA 1,Xtα'z ©þÑ´",@o¡α " þ,¡( a1, a2,···, an) (a1,a2,···,an)'K þ,4Pnv«¤kPþ'n£ê|'8Ü,QPn¥½Âê¦Ú\~{:

α±β=(a1±b1,a2±b2,···,an±bb),

kα=(ka1,ka2,···,kan),Ù¥,α=(a1,a2,···,an),β=(b1,b2,···,bn),k∈P.Qù«6 e· ¡Pn Pþ'n£ê| þ m,

大学线性代数讲义前面有对知识的讲解,后面是习题。便于理解。不想挂科的同学们的必备之物

§1.9 5 ' 5Ã'

~α1=(1,0,0),α2=(0,1,0),α3=(0,0,1),β=(2,3,4)· B©β=2α1+3α2+4α3, Ò´β dα1,α2,α3v«Ñ5.,©,Xtk1α1+k2α2+k3α3=0,@o íÑk1=k2=k3=0,ùü þ'9XkXe½Â.

½Â1.21α1,···,αs´Pn¥' | þ,k1,···,ks∈P,¡k1α1+···+

ksαs α1,···,αs' 5|Ü.Xtα=k1α1+···+ksαs,u¡α dα1,···,αs 5LÑ, ¡α ¤α1,···,αs' 5|Ü.

~.α=(2, 3,0),β=(0, 1,2),γ=(0, 7, 4),uγØU&α,β SvÑ.A.XtγU&α,β SvÑ,#γ=k1α+k2β,@o(2, 7, 4)=(2k1, 3k1 2k2,2k2),=

2k1=0

(I) 3k1 k2= 7

2k2= 4100200

S §|(I) dpd {&: 3 1 7 → 0 1 7 →