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PWI PWI PWP PWR PWS PWT PWU PWV

17

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

299

PWW PWW QHI QHI

18

303

QHQ QHR QHR QHS QHS QHS QHT QHU QHU

19

F F F F F F F F F F F F F F F F QHW F F F F F F F F F F F F F F F F F QHW F F F F F F F F F F F F F F F F F F F F F F F F QIH F F F F F F F F F F F F F F F F F F F F F F F F QII F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

309

22 23

311 313

QIQ QIQ QIR QIS

. .

xi

SRFS SSFI SSFP STFI STFP STFQ STFR STFS SUFI SUFP SVFI SVFP SWFI SWFP SWFQ THFI TIFI TIFP TPFI TPFP TPFQ TQFI TQFP TQFQ TQFR TQFS TRFI TRFP

F F F F F F QIS F F F F F F F F F F F F F F QIU F F F F F F F F F F F F F F F F QIV F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

25

317

26

319

QIW QPH QPH QPI QPP

27

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QPS F F F F F F F F F F F F F F F F F QPT F F F F F F F F F F F F F F F F F F QPW X F F F F F F QPW F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QQI F F F F F F F F F F F F F F F F F F F QQP F F F F F F F F F F F F F F F F F QQP F F F F F F F F F F F F F F QQS F F F F F F F F F F F F F F F F F F QQU F F F F F F F F F F F F F QQV F F F F F F F F F F F F F F F F QRI F F F F F F F F F F F F F QRP F F F F F F F F F F F F F F F F F F F F F F QRQ F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

325

32

329

34

331

35 36

335 337

37

341

38

345

QRS QRS QRT QRU QRU

39

F F F F F F F F F F F F F F F F F F QSI F F F F F F F F F QSP

351

40TSFI TSFP TSFQ

F F F F F F F F F F F F QSQ F F F F F F F F F F F F F F QSQ F F F F F F F F F F F F F F F F F F F F F F F F F F F QSR

353

357

E E F F F F F " D D D D D " D F F F D D " E D F " E D F D F D D E F D D F D D E E F D F D " D E D 4 4F E F D D F D F ED F D E F D F ED E D D D F I

2

.

D F " E D @ A F ED " D F D D " D F @ A F D @ AD F D E @D nE E AF D D F D F D D E " D F D D F D E F D D D F D F D F D F D E " E D D D F D F E F D " E F D D E F D E D F F E E E D E F D E F D E F D E

. .

3

F " D D E F D D F D F D E D F D D F E @D IID IUD PHD PSAF D D E D E F D E E F E D " D F D F 3 F F F D F D D F F D F F F F D " E D F D D E D " D D F D F F F F D F F D F F D F F D F F D F F D F

4

.

11.1 F @E D AF " F F D y1 , . . . , ym x1 , . . . , xn X y1 = a11 x1 + . . . + a1n xn , ... () ym = am1 x1 + . . . + amn xn , F E AY E x1 , . . . , xn y1 , . . . , ym X a11 . . . a1n x1 y1 A = ... ... ... , x = ... , y = ... . am1 . . . amn xn ym F X m nD n 1 m 1F ()D y xD X y = Ax. () D A E xD E y F D D () () A xF m = nD F n n nF

1.2

y1 , . . . , ym x1 , . . . , xn x1 , . . . , xn z1 , . . . , zk Xy1 ym = a11 x1 + . . . + a1n xn , ... + . . . + amn xn , x1 xn = b11 z1 + . . . + b1k zk , ... + . . . + bnk zk .

= am1 x1

= bn1 z1

S

6

1

D y1 , . . . , ym z1 , . . . , zk F C F

y = Ax,

x = Bz y = Cz.

C D x1 , . . . , xn z1 , . . . , zk D y1 , . . . , ym x1 , . . . , xn D z1 , . . . , zk F Xn

C = [cij ], cij =l=1

ail blj .

()

. C () A B E C = AB F

. y = A(Bz) = (AB)z F

D F D D F C = AB D D A B F

1.3

. (AB)C = A(BC)F . A " m nD B " n k D C " k lF k k n

{(AB)C}ij =p=1 n

{AB}ip cpj =p=1 k q=1

aiq bqp

cpj

=q=1

aiqp=1

bqp cpj

= {A(BC)}ij .

1.4

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

AB = BA " F D

= =

, .

1.5

cij = aij + bij i, j.

C = [cij ] A = [aij ] B = [bij ]D

A, B C = A + B ! F X

A + (B + C) = (A + B) + C

@A,

. .

7

A + B = B + A @A. F " D C = A cij = aij F

1.6

A B . . . A1q B11 , ... ... ... B= . . . Apq Bq1 Aij Bij X . . . B1r ... ... , . . . Bqr

D A11 ... A= Ap1

Aij " mi nj D Bij " ni kj F C = AB D EX q C11 . . . C1r . . . . . . . . . , Cij = C= Ail Blj " mi kj . l=1 Cp1 . . . Cpr 3 D F D E F

1.7

nnE A B C = AB F @ E AXDO i = 1, n DO j = 1, n DO k = 1, n cij = cij + aik bkj END DO END DO END DO.

D cij F

1.8

?

F D E " F F D E D D D F

8

1

E D F D F E F F D 2n3 @n3 n3 AF c D c F D " F E D F D F E D F D F

1.9

E E " E F D @ THEAF X2m m m m

aik bkj =k=1 k=1

(ai 2k1 + b2k j )(b2k1 j + ai 2k ) k=1

ai 2k1 ai 2k k=1

b2k j b2k1 j .

n = 2mF D 1 i, j n D 2nm = n2 2nm = n2 F n2 m = 1 n3 3n2 m = 3 n3 F 2 2 " ED 2n3 @ n2 n3 AD E 1 n3 3 n3 3 " D 2 2 D F

1.10

IWTS 2 2E UE @ V AF D D F F E X 11 = (a11 + a22 )(b11 + b22 ), 2 = (a21 + a22 )b11 , 3 = a11 (b12 b22 ), 4 = a22 (b21 b11 ), 5 = (a11 + a12 )b22 , 6 = (a21 a11 )(b11 + b12 ), 7 = (a12 a22 )(b21 + b22 ),1 . 5.4.21 ...

c11 = 1 + 4 5 + 7 , c12 = 3 + 5 , c21 = 2 + 4 , c22 = 1 + 3 2 + 6 .

. .

9

D 2 E F

1.11

n n-

22E UE n nED 7 nlog2 7 F

7nlog2 7 0 n , n3 F D n = 2L A B 2 2EX

A=

A11 A12 A21 A22

,

B=

B11 B12 B21 B22 ,

,

Aij , Bij

"

n n . 2 2

D 2 2E F 2 2E3 D n UE n F E 2 UE UE IV n F 2 44 D

7log2 n = nlog2 7 F L

18k=1

7

k1

n 2k

2

18 2 = n 4

7 L 4 7 1 4

1

6 7L = 6 nlog2 7

@ D 4L = n2 7L = nlog2 7 AF F X 7 nlog2 7 > 2n3 n = 512F n = 1 024 F @A D F O (n )D < 2.42F D F D 2F

10

1

22.1 D E E F D D F a M D a M F X Y D X Y F X Y F D F F D M MF F D M D D E D D E F D D D F D 3 1 D D E F @D A D F F E D M = {1, 2, 3} " D ID PD QF D X Y X

A = X Y {a : a X a Y } B = X Y {b : b X b Y } C = X\Y {c : c X, c Y } / D = X Y {d = (a, b) : a X, b Y }

@ AY @ AY @ AY @ AF

2.2

, ,

" D x X 1 . . , , 1980.

II

12

2

y = f (x) Y F

= {(x, f (x)) : x X} X Y, @D A f F y = f (x) xD x " y f F D f X Y D X f : X Y F f (X) {y : y = f (x) x X} @ A f F M Y D f 1 (M ) {x : f (x) M } M F M = {y}D X f 1 (y) = f 1 (M )F f : X Y D g : Y X D f (g(y)) = y y Y g(f (x)) = x x X F g f g = f 1 F f -D y Y f 1 (y) F D F

2.3

f : X X X X F F c = a b D (a, b) X X c = f ((a, b))F f : M X M X X D f X F D D F g D ab = a bD E D ab @ A " a bF

2.4

X D a, b, c X ab bc a(bc)D (ab)c a(bc) = (ab)c. abc a(bc) = (ab)cF

x1 , . . . , xn X , x1 x2 , x2 x3 , . . ., xn1 xn . , x = x1 x2 . . . xn , x. D xF D (x1 ... xn2 )xn1 F E xn1 xn F D a = x1 . . . xn2 D b = xn1 D c = xn F

. X

. nF E

. .

13

a b F b = (x1 . . . xm )(xm+1 . . . xn ). a = (x1 . . . xk )(xk+1 . . . xn ), k < mF D D

a = (x1 . . . xk )((xk+1 . . . xm )(xm+1 . . . xn )) = ((x1 . . . xk )(xk+1 . . . xm ))(xm+1 . . . xn ) = (x1 . . . xm ))(xm+1 . . . xn ) = b. 2

2.5

1nD n1 1 nX c11 . . . [d11 . . . d1n ]. A = BCD = [b11 . . . b1n ] cn1 X

A = B(CD) = [b11

c11 d11 . . . c11 d1n ... ... , . . . b1n ] . . . cn1 d11 . . . cn1 d1n

(1)

A = (BC)D = [(b11 c11 + . . . + b1n cn1 )] [d11 . . . d1n ].

(2)

@IA @PA AF F 3 D 2n2 @IA 2n @PAF

2.6

G D X @IA e G D ae = ea = a a GY @PA a G b G D ab = ba = eF e @IA X e1 e2 " D e1 = e1 e2 = e2 F F b @PA aX b1 b2 " E D b1 = b1 (ab2 ) = (b1 a)b2 = b2 F b aF X b = a1 F a, b G ax = b @E xA ya = b @ y AF X x = a1 b y = ba1 F @AD ab = ba a, b GF

14

2

2.7

1. G = R " D " F E

0F 2. G = R\{0} " D " F 1F 3. G = Q " D " F E 0F 4. G = Q\{0} " D " F 1F 5. G " a+b 2D a, b " F " F D D G GX (a + b 2)(c + d 2) = (ac + 2bd) + (ad + bc) 2, a, b, c, d ac + 2bd ad + bcF D 1 = 1 + 0 2F a + b 2D D

a a2 2b2.G.

+

b a2 2b2

2.a2 = e

G

e.

,

a G,

2.8

A = [aij ] n n D aij = 0 i = j F A D aii = 0 1 i nF n nE E F E 1

I= F.

..

.

.1

B

A n n: AB = BA n. , A .2

2.9

A = [aij ] n n D aij = 0 i < j D D aij = 0 i > j F D aii = 0 1 i nF @A E @AF X2

.

. .

15

D @A E @A Y D I Y D @A A AX = I Y A = I D X Y @A F X = Y F

2.10

H G GD D GF D

ab H a, b H Y a1 H a H FD @A n nEF

2.11

a G E a H(a) GF D H(a) H H D aF D

H(a) = {ak : k " }. D a0 = eD ak = a . . . a @a k A k D ak = (a1 )k F D

ak+m = ak am

k D mF

2.12

H(a) D aF E k > 0 D ak = eD aF ak = e k > 0D a F

. . . H H(a) E aXH = {ai1 , ai2 , . . . }. m " i1 , i2 , . . . F D H amk F D H aF an H F n mX

n = qm + r,

q, r " ,

0 r m 1.

ar = an aqm H F r > 0 mF r = 0F 2.

Z

.

16

2

33.1 a11 x1 + . . . + a1k xk an1 x1 + . . . + ank xk = ... = b1 , (1) bn

x1 , . . . , xk F D a11 . . . a1k x1 b1 Ax = b, A = ... ... ... , x = ... , b = ... . an1 . . . ank xk bn n k aij RD R " E D Rnk F Rn1 Rk1 " ED D D n k F E Rn = Rn1 Rk = Rk1 E F A Rnk D b Rn " D x Rk " (1)F

3.2

(1)F D RD

b1 ... bn

b1 ... bn

.

a1 , . . . , ak " AX

A = [a1 , . . . , ak ],

a1 , . . . , ak Rn .

(1)

x1 a1 + . . . + xk ak = b.

(2)

x1 a1 + . . . + xk ak a1 , . . . , ak D x1 , . . . , xk " F E a1 , . . . , ak

L(a1 , . . . , ak ) = {1 a1 + . . . + k ak : 1 , . . . , k R}IU

18

3

a1 , . . . , ak F D (2) D

b L(a1 , . . . , ak ).

(3)

D (1) @A D E b @ A F

3.3

D D D F 0F D F @ D A D D F

1. a1 , . . . , ak k > 1 , am , m > 1, : am L(a1 , . . . , am1 ).

. 1 a1 + . . . + k ak = 0, m " D m = 0F m = 1D 1 a1 = 0 D 1 = 0D a1 = 0D F D m > 1F

1 a1 + . . . + m am = 0

am =

1 m

a1 + . . . +

m1 m

am1 . 2

3.4

D E F D a1 , . . . , ak D

1 a1 + . . . + k ak = 0

1 = . . . = k = 0.

2. .

. D F D E D F

. .

19

E D F F 2

3. . a1 , . . . ak b = 1 a1 + . . . , + k ak = 1 a1 + . . . , + k ak .

, .

(1 1 )a1 + . . . + (k k )ak = 0.

1 1 = . . . = k k = 0.

2

n a, -

A=

a 1

1 a.. .

1.. . .. .

1

a 1 1 a

n

.

.

(n + 1) n

aij > 0

,

n=3

.

i = j aij < 0 n = 4?

i = j.

3.5

L(c1 , . . . , cr ) L(b1 , . . . , bm ) L(b1 , . . . , bm ) L(a1 , . . . , ak ),

@ A X

L(c1 , . . . , cr ) L(a1 , . . . , ak ).

3.6

4. b1 , . . . , bm a1 , . . . , ak , ,

L(b1 , . . . , bm ) L(a1 , . . . , ak ). m k .

()

. ()D b1 , a1 , . . . , ak F I D E D

ak L(b1 , a1 , . . . , ak1 ).

20

3

D

L(a1 , . . . , ak ) L(b1 , a1 , . . . , ak1 ).

L(b1 , . . . , bm ) L(b1 , a1 , . . . , ak1 ),

b2 , b1 , a1 , . . . , ak1 F I D E D b1 @ b1 , b2 @ PAAF D D

ak1 L(b2 , b1 , a1 , . . . , ak2 ).D m > k F D D k E L(a1 , . . . , ak ) L(bk , bk1 , . . . , b1 ). D bk+1 L(bk , bk1 , . . . , b1 )D E b1 , . . . , bm F D m kF 2

3.7

b1 , . . . , bm V = L(a1 , . . . , ak ) V D L(b1 , . . . , bm ) = V F

. V .

. b1 , . . . , bm c1 , . . . , cr " EF D

L(b1 , . . . , bm ) = L(c1 , . . . , cr ). R D X m r r mF m = rF 2

. V dim V F

: dim L(a1 , . . . , ak ) k. . D

F 2 L(a1 , . . . , an ) E a1 , . . . , an F F

. a1 , . . . , an L = L(a1 , . . . , an ), , .

. D D k a1 , . . . , ak F D ak+1 , . . . , an E

. .

21

a1 , . . . , ak L L(a1 , . . . , ak ) L L = L(a1 , . . . , ak )F D a1 , . . . , ak LF LD D E D F 2 .

a1 , . . . , ak+1

.

,

-

L(a1 , . . . , ak+1 ) L(a1 , . . . , ak ).

,

3.8

.

b1 , . . . , bm L(a1 , . . . , ak ) .

. D a1 , . . . , ak F b1 , . . . , bm , a1 , . . . , ak F I D F F D E L(a1 , . . . , ak )F D D F D E D F b1 , . . . , bm a1 , . . . , ak F 2

3.9

c D D F D F D D E " F D F

3.10

A = [a1 , . . . , ak ],

1. Ax = b, ,

L(a1 , . . . , ak ) = L(a1 , . . . , ak , b).

. L(a1 , . . . , ak ) L(a1 , . . . , ak , b). D b L(a1 , . . . , ak )F D

()

L(a1 , . . . , ak , b) L(a1 , . . . , ak ).

()

22

3

() () F ()D D b L(a1 , . . . , ak )D Ax = bF 2

2. n = k , a1 , . . . , an . Da1 , . . . , an L(e1 , . . . , en ),

Ax = b .

e1 , . . . , en " n n @ iE ei 1D 0AF E r n D a1 , . . . , an F r nF a1 , . . . , an E L(a1 , . . . , ak , b)F b L(a1 , . . . , an )D F QF 2.

a0 a1 a2 ,

a1 a0 a1

a2 x1 1 a1 x2 = 0 0 a0 x3

x1 = 0. , .

44.1 a1 , . . . , an Rn " f (a1 , . . . , an )D E F f X f E F f X

(A) 1 i n iE @ AX

f (a1 , . . . , ai1 , a + b, ai+1 , . . . an ) = f (a1 , . . . , ai1 , a, ai+1 , . . . , an ) + f (a1 , . . . , ai1 , b, ai+1 , . . . , an ) a, b Rn , RY

(B) a1 , . . . , an D f (a1 , . . . , an ) = 0Y (C) E @ AX

f (e1 , . . . , en ) = 1, e1 , . . . , en " n nF f E F F

4.2

: x xD x = {1, 2, . . . , n}D @ A nF 1 2 ... n = , (1) (2) . . . (n) (1), (2), . . . , (n) 1, 2, . . . , n @ E AF PQ

24

4

a b D E @A b aX

(ab)(i) = a(b(i)),

i x.

nD F D @ E AF E 1 2 ... n e= , 1 2 ... n 1 F n n Sn F @ Y AF D Sn n! = 1 2 . . . nF X F (x1 , . . . , xn )D E X

F (x1 , . . . , xn ) = F (x(1) , . . . , x(n) ) Sn . @ k AXn

Fk (x1 , . . . , xn ) =i=1

xk . i

.

x1 x2 ... xnn

y1 y2 ... yn .

,

Sn

n

|xi yi | i=1 i=1

|xi y(i) |.

4.3

a Sn k D k i1 , . . . , ik x D @IA a(i1 ) = i2 , a(i2 ) = i3 , . . . , a(ik1 ) = ik , a(ik ) = i1 , @PA a(i) = i

i x \ {i1 , i2 , . . . , ik }.

a

a = (i1 , . . . , ik ). 2 F a = (i1 , . . . , ik ) b = (j1 , . . . , jm ) D

{i1 , . . . , ik } {j1 , . . . , jm } = .

. .

25

(1) a b : ab = ba. (2) Sn .

(3) k k 1 . (4) . (1) X a = (i1 , . . . , ik ) b = (j1 , . . . , jm )

(ab)(i) = (ba)(i) = a(i) i {i1 , . . . , ik }, (ab)(i) = (ba)(i) = b(i) i {j1 , . . . , jm }, (ab)(i) = (ba)(i) = i i {i1 , . . . , ik } {j1 , . . . , jm }. / (2)D j E j, (j), 2 (j), . . . F n " E k < l k (j) = l (j)D lk (j) = j F k " D k (j) = j F

a = (j, (j), 2 (j), . . . , k1 (j)),

(i) = a(i) i {j, (j), 2 (j), . . . , k1 (j)}.

D 1 = a1

i {j, (j), 2 (j), . . . , k1 (j)}.D j1 {j, (j), 2 (j), . . . , k1 (j)} / bD 2 j1 1 (j1 ) 1 (j1 ) . . . . l @D 1 (j1 ) = l (j1 ) lFA D E

a1 b1 . . . c1 = e,

= c . . . ba. a, b, . . . , c F (3) D D X

(i1 , . . . , ik ) = (i1 , i2 )(i2 . i3 ) . . . (ik1 , ik ). (4) (2) (3)F.,

n ,

.

26

4

4.4

F D 1 2 3 4 5 6 7 = 7 5 3 1 2 4 6

(1, 7) (7, 6) (6, 4) (2, 5) = (1, 7) (7, 6) (6, 4) (7, 2) (7, 5) (7, 2).D F

. .

. Sn= 1 2 ... n (1) (2) . . . (n)

(i, j)D i < j D (i) > (j)F () " F D ( ) () F = (i, j)D i < j F

=

1 ... i1 i i+1 ... j1 j j+1 ... n (1) . . . (i 1) (j) (i + 1) . . . (j 1) (i) (j + 1) . . . (n)

.

D k

(i, l), m (l, j),

l {i + 1, i + 2, . . . , j 1},

()

l {i + 1, i + 2, . . . , j 1},

()

s F j i 1 k () j i 1 m ()F D s + 1 D (i, j) D s 1 F D

() = k + m + s,

( ) = (i j 1 k) + (i j 1 m) + s 1. 2

( ) () = 2(i j 1 k m) 1. .

. . D D F

. (x1 , . . . , xn ) =i j iE j E AF

1. 1 = .

. .

51

2. Z = Zl Z : 1

Z 1

=

..

.1 (Zl )j+1 j ... (Zl )nj

..

.1

,

1 = I D ZZ 1 = I F

8.3

D S = [sij ] m n k D 1 j1 < . . . < jk mD X

1 i k D sij = 0 j = ji sij = 0 1 j ji 1Y k + 1 i mD sij = 0 1 j nF S k D S k F

@ A k F D S k F D F 1 , . . . , k X

. k k . . S D

[1 , . . . , k ]S = [0, . . . , 0].

1 s1i1 = 0D

1 = 0. 2 = 0.

0 s1 i2 + 2 s2 i2 = 0

D 1 = . . . = k = 0F E S D D S F 2

8.4

1. mn- A r P ,

, S = P A r.

. j1 AD

F @ D A = 0 E FA

52

8

1 (1, j1 )F E Z1 j1 E E F D @ 0pq p q A Z 1 1 A = 01(i1 1) 0(m1)(i1 1) u B , u R(ni1 +1)1 .

QD E m 1 D QB E F

P =

1 0 0 Q

Z 1 1 .

D P S = P A F k F D S k k = rank S = rank A = rF 2

2. mn- A r Q,

, AQ r. QA F

. I A D " D (QA ) = AQ

8.5

. m n- A r P

Q, , B = P AQ b11 , . . . , brr , . D SQ D PD F 2

. A S = P AD

8.6

A B D P Q D B = P AQF

. A B , .

A B " E DA DB F D DA DB D F D rank DA = rank DB F D rank A = rank DA rank B = rank DB F 2

. D

. .

53

8.7

LU -

F Ax = b AF D AD F F E @ AD n IE n 1E (Zn1 . . . Z2 Z1 A)x = Zn1 . . . Z2 Z1 b, () Z1 , . . . , Zn1 " D Zi I iE F Z1 , . . . , Zn1 D L = Zn1 . . . Z1 F ()

U = Zn1 . . . Z1 A = L A F L = L 1 F D A A = LU. L D U E @ AAF LU -F A E U F iE n i iE F iE lE l > i lE iE D D E F n i D @ A iE (n i)2 F

(n 1)2 + (n 2)2 + . . . + 12 =

1 3 n + O(n2 ) 3

Y O(n2 ) n 2F Ax = bD X

Zn1 . . . Z1 bY U F O(n2 ) " D F. :

A=

A11 A21

A12 , A22

A1 =

B11 B21

B12 . B22

54

8

,

A11

,

B22 .

8.8

LU -

D A LU EX A = LU F Ak D Lk D Uk k D AD LD U D D

A D

Ak P Q Ak

=

Lk 0 V Lk

Uk W 0 Uk

.

Ak = Lk Uk ,

k = 1, . . . , n.

D Lk UK @ E AF Ak F AD Ak D F D LU E A D F D F D LU E Ak = Lk Uk F

L1 0 k 1 QAk I

Ak P Q Ak

=

1 Uk Lk P 0 W

,

W = Ak QA1 P. k

(#)

W Ak AF (#) A D W F D W LU E W = Lk Uk F I 0 Uk L1 P Lk 0 k L= , U= . QA1 Lk I 0 Lk 0 Uk k L " F LU = A E F.

A

n

, .

A(I, J) I = (i1 , . . . , ik ) J = (j1 , . . . , jk ),

k < n,

I

J

. ,

det A1 (I , J ) = (1)i1 +...+ik +j1 +...+jk det A(I, J)/ det A.

99.1 @E A F E D F D E F D E F A B " D [AB] " D A B Y |AB| " [AB]F D E E x P (x)D P (0)D P (1) X

x = 0 P (x) P (1) P (0)D x > 0Y x < 0Y |P (0)P (x)| = |x| |P (0)P (1)|.D D D x " P (x)F D a x P (x + a) EF P (0) F l1 D l2 D l3 D O F

x P1 (x),

y P2 (y),

z P3 (z),

P1 (0) = P2 (0) = P3 (0) = O. (x, y, z) " D X = P1 (x)D Y = P2 (y)D Z = P3 (z) l1 D l2 D lz D F X

1 " D X l2 l3 Y 2 " D Y l1 l3 Y 3 " D Z l1 l2 FSS

56

9

D 1 D 2 D 3 M = M (x, y, z)F E

(x, y, z) M (x, y, z). X, Y, Z M l1 D l2 D l3 E D D 1 D 2 D 3 F x, y, z M = M (x, y, z)D l1 D l2 D l3 " F O @ A F D D [OP1 (1)]D [OP2 (1)]D [OP3 (1)] F D 1D F

9.2

A, B -

A B F X AB F OD OA - AF A E OAF A = B AB X [AB)D D B A D D @ A E AF D @ D AF AB CD D [AB) [CD) E D B D AC F D A = B D AB D C " F C D AB C AB F D1 D2 D |CD1 | = |CD2 | = |AB|F D {D1 , D2 }D

[AB) [CD) F CD AB F @ D CD AB FA

A = B F AA CC F X CD AB D AB CDc D D " D AB CD F D E D @ E A X AB CD D [AD] [BC] F

. .

57

F D E EF

9.3

M X X X X

xy.

M

(x, y) M.

X " D M " (A, B)D AB F D D E M M M F D A B D D B AF M X D E X

x x x X x y D y x x y y z D x zM M M M M

M

@AY @AY @AFM

X M x y D x y F X D a X D , aF

. X , , .

. X(a) D E

a X F x E X(a)F b, c X(a)D aD D E D b c @b a, a c b cAF D X(b) = X(c) = X(a) @ D AF D X(a) D aF E b X(a)D b aF D / X(a) X(b) X c X(a) X(b)D D b X(a) X(a) = X(b)F D a b X(a) X(b) D F D X = X(a)F E F 2 G " @ A F a, b G D h G @ a bA 1.

aX

58

9

a = hbh1 F " GD D D F Z " D p " @ A F x y pD p @ D x y pD x y = kp k AF X x = y (modp)F x y x y (mod)F Z F p 2.

Z(0), Z(1), . . . , Z(p 1), pF

9.4

. . F

. D D D F D a E F A B D AB aF D ED aF V " F a E V V X A V B V D AB aF D aF X AB a E a = AB @ AF

9.5

X AB a BC bD c = a + b D AC F D c AF D P Q a QR bF

ABC

P RD D F F 0 = AAF a =

P QR AC

. .

59

AB b = BAF b a b = aF

X AB aD a D AC D |AC| = || |AB| D = 0D AB > 0 < 0F D a AF D ( a) = () a , F D 1 a = aD (1) a D aF D X

( + ) a = ( a) + ( a),

(a + b) = ( a) + ( b).

9.6

F D E EF E F O " l1 D l2 D l3 X l1 D Y l2 D Z l3 D 1 F

e1 = OX,

e2 = OY ,

e3 = OZ

@ AF E

e1 , e2 , e3 . a = x e1 + y e2 + z e3 . x e1 D y e2 D z e3 a l1 D l2 D l3 @D D AF

1. x, y, z a = OA

2. xa , ya , za xb , yb , zb a = OA b = OB ,. c = a + b

xc = xa + xb ,

y c = ya + yb ,

zc = za + zb ,

d = a

x d = xa ,

y d = ya ,

zd = za .

60

9

D E D D D F. .

A1 , . . . , An

O.

,

OA1 + ... + OAn = 0 M

.

n-,

ABCD

,

MA + MB + MC + MD = 0 .

9.7

V " F ED E

OA " A F P E E V E R3 D E X a a R3 b b R3 D a + b a + b, a a.

E D D D D D F D V R3 F E D EF F D E E F D e1 D e2 D e3 E D X

V = L(e1 , e2 , e3 ),

dim V = 3.

9.8

1. D F

2. D F D D 1F X 1 F D D 2F 2 F EF D D

. .

61

D F D E D F l = AB D A B D @EA X

l = {M : OM = OA + tAB, t R}.

(1)

AB @ lA lF D A, B, C D @EA

= {M : OM = OA + uAB + v AC, u, v R}.

(2)

9.9

F E (x, y) @E A F A B " D F

Ax + By + C = 0 F

()

. F E . l " D (x0 , y0 ) (x1 , y1 )F D (1)D l (x, y) D

x, y D ()D D ()F

x = x0 + tpx , y = y0 + tpy ,

()

t R px = x1 x0 , py = y1 y0 . @ A x x0 px det = 0. y y0 py Ax + by + C = 0, A = py , B = px , C = py x0 + px y0 . (x, y)D ()F E D D

A B

x = C. y

A, B D IF ()D (x0 , y0 ) " D (px , py ) @ AF 1 21 , .

62

9

9.10

Ax + By + Cz + D = 0 (#)

AD B D C " D F

F

. F E x, y, z D (#)D D (#)F

. " D (x0 , y0 , z0 )D (x1 , y1 , z1 )D

(x2 , y2 , z2 )F D (2)D (x, y, z) D x = x0 + upx + vqx , y = y0 + upy + vqy , (##) z = z0 + upz + vqz , u, v " D

(px , py , pz ) = (x1 x0 , y1 y0 , z1 z0 ),

(qx , qy , qz ) = (x2 x0 , y2 y0 , z2 z0 ).

x x 0 p x qx det y y0 py qy = 0. z z0 pz qz

x, y, z D D (#)F (x, y, z)D (#)F D x y = 0. A B C z A, B, C D IF D (##)D (x0 , y0 , z0 ) " E D (px , py , pz ) (qx , qy , qz ) F 2

9.11

X O E e1 , e2 , e3 D O e1 , e2 , e3 F e1 e2 e3 = p11 e1 + p21 e2 + p31 e3 , = p12 e1 + p22 e2 + p32 e3 , = p13 e1 + p23 e2 + p33 e3 ,

@ A

P =

p11 p21 p31

p12 p22 p32

p13 p23 . p33

. .

63

E ei , ei E D [e1 , e2 , e3 ] = [e1 , e2 , e3 ]P. P F M (x, y, z) (x , y , z ) F D OM = OO + O M F O (x0 , y0 , z0 )F

xe1 + ye2 + ze3 = (x0 e1 + y0 e2 + z0 e3 )+ x (p11 e1 + p21 e2 + p31 e3 ) + y (p12 e1 + p22 e2 + p32 e3 ) + z (p13 e1 + p23 e2 + p33 e3 ) = (x0 + p11 x + p12 y + p13 z )e1 + (y0 + p21 x + p22 y + p23 z )e2 + (z0 + p31 x + p32 y + p33 z )e3 .D E Xx y z x0 z0 x

= y0 + P yz

.

D D E F Ax + By + Cz + D = 0D D D x

[A B C ] y = Dz

[A B C ]

x0 y0 z0

x

+P yz

= D.

D

[A [AB C ] = [A B

B

x C] y z

= D ,

C ] P,

D = D (Ax0 + By0 + Cz0 ).

9.12

l : Ax + By + C = 0F P = (x, y)

l = {(x, y) : Ax + By + C = 0}, + = {(x, y) : Ax + By + C > 0}, = {(x, y) : Ax + By + C < 0}.

D l + F P = (x1 , y1 ) Q = (x2 , y2 )D P Q

x = x1 + t(x2 x1 ) = (1 t)x2 + tx1 ,

y = y1 + t(y2 y1 ) = (1 t)y2 + ty1 ,

0 t 1.

D P Q + D P Q F

64

9

D E D F D l, + , F D P + D Q F

A(x1 + (t(x2 x1 )) + B(y1 + t(y2 y1 )) + C = 0

t =

Ax1 + By1 + C , (Ax1 + By1 + C) (Ax2 + By2 + C)

D 0 < t < 1F D P Q lF D l , lF D : Ax + By + Cz + D = 0 E

+ = {(x, y, z) : Ax + By + Cz + D > 0},

= {(x, y, z) : Ax + By + Cz + D < 0}.

, F.

ABC

O

-

. ,

M,

, -

OM = OA + OB + OC,

++ = 1

2

, , 0.

,

, ,

.

M . ABCD M , , , O , , , ,

2

OM = OA + OB + OC + OD,

+ + + = 1.

1010.1

a @E D D AF X |a|F (a, b) E a = OAD b = OB OA OB OAB F

a b

(a, b) =

|a| |b| cos (a, b), a = 0 b = 0, 0, a = 0 b = 0.

D

(a, a) > 0 a = 0; D

(a, a) = 0 a = 0. a, b.

(1)

(a, b) = (b, a)

(2)

a = OAD b = OB D D O, A, B O D OB B F |a| cos (a, b) A F X (a + b, c) = (c, a) + (c, b) a, b, (3)

( a, b) = (a, b)

a, b,

R.

(4)

(2) F D F

10.2

0, i = j, 1, i = j.TS

e1 , e2 , e3 F

(ei , ej ) =

()

66

10

a1 , a2 , a3 , b b1 , b2 , b3 .

. a (a, b) = a1 b1 + a2 b2 + a3 b3 . (#)

. a = a1 e 1 + a2 e 2 + a3 e 3 , b = b 1 e 1 + b 2 e 2 + b 3 e3 .

(2) (4) ()D

(a, b)

= =

(a1 e1 + a2 e2 + a3 e3 , b1 e1 + b2 e2 + b3 e3 )3 3

ai bj (ei , ej ) = a1 b1 + a2 b2 + a3 b3 . 2i=1 j=1

1. a b (#)D F (#)

2. (a, b) = a1 b1 + a2 b2 .

10.3

(#) (1) (4) " D F D a = [a1 , . . . , an ] , b = [b1 , . . . , bn ] " E Rn D (#)X

(a, b) = a1 b1 + . . . + an bn .

()

D " (1) (4) E a, bD (1) (4)F D F E X (a, b) cos (a, b) = . |a| = (a, a), |a| |b| D ()D a, b Rn F D

|(a, b)|

(a, a) (b, b).

@ !!A E ()D " F

. .

67

10.4

@ A X D D 1 Y D D F D E F D X D D F D F " F D D D D F 2 D D D F a E a1 , a2 , a3 D b " b1 , b2 , b3 D c " c1 , c2 , c3 F a, b, c D a1 b 1 c 1 det a2 b2 c2 > 0. a3 b 3 c 3 D F D F E n E Rn F

10.5

a b c D X

c a bY a, b, c Y |c| = |a| |b| sin (a, b)F a b D c = 0F X c = [a, b]F a = OA, b = OB D cD D E OA OB F D [a, b] cD a, b, cF X

(a, b, c) = ([a, b], c).1 , . 2 . . , .

68

10

. a, b, c V , a, b, c.

(a, b, c) =

V, a, b, c ; V, .

a, b, c ,

(a, b, c) = 0.

. D a = OA, b = OB, c = OC D OD = [a, b]F D

(a, b, c) = |OD| ,

= |OC| cos (OD, OC).

D || D C OAB @ AF > 0D D C OAB D < 0D F OA, OB, OC D " F 2

1.(a, b, c) = (b, c, a) = (c, a, b) = (b, a, c) = (a, c, b) = (c, b, a).

. D {a, b, c}, {b, c, a}, {c, a, b} D

{b, a, c},

{a, c, b},

{c, b, a}. 2

2. (a, b, c) . . F D {a, b, c}D {b, c, a}D {c, a, b} F

(a, b, c) = (b, c, a) = (c, a, b).D F

2

1. :[a, b] = [b, a]. F X d = [a, b] + [b, a]D (d, d) = (a, b, d) + (b, a, d) = 0 d = 0F

2. . . D [a + b, c] = [a, c] + [b, c]F d = [a + b, c] [a, c] [b, c].

. .

69

D

(d, d) = (a + b, c, d) (a, c, d) (b, c, d) = 0 d = 0.D d = [ a, b] [a, b]D (d, d) = ( a, b, d) (a, b, d) = 0 d = 0. E F 2 D F

[a, b] = 0.

. a, b ,

. a, b, c , (a, b, c) = 0.

10.6

e1 , e2 , e3 " F D [e1 , e2 ] = e3 , [e2 , e3 ] = e1 , [e3 , e1 ] = e2 . a = a1 e1 + a2 e2 + a3 e3 ,

b = b1 e1 + b2 e2 + b3 e3

[a, b] = + + =

a1 b1 [e1 , e1 ] + a1 b2 [e1 , e2 ] + a1 b3 [e1 , e3 ] a2 b1 [e2 , e1 ] + a2 b2 [e2 , e2 ] + a2 b3 [e2 , e3 ] a3 b1 [e3 , e1 ] + a3 b2 [e3 , e2 ] + a3 b3 [e3 , e3 ] (a2 b3 a3 b2 ) e1 (a1 b3 a3 b1 ) e2 + (a1 b2 a2 b1 ) e3 .

D X

e 1 e2 e3 [a, b] = det a1 a2 a3 . b1 b2 b3.,

a, b, c

[a, [b, c]] = (a, c) b (a, b) c.

.

,

x

[a, x] + [b, x] = [a, b] a b.

a

b.

10.7

a = a1 e1 + a2 e2 + a3 e3 , b = b1 e1 + b2 e2 + b3 e3 , c = c1 e1 + c2 e2 + c3 e3 .

70

10

D

(a, b, c) = ([a, b], c) = ((a2 b3 a3 b2 ) e1 (a1 b3 a3 b1 ) e2 + (a1 b2 a2 b1 ) e3 , (c1 e1 + c2 e2 + c3 e3 )) = c1 (a2 b3 a3 b2 ) c2 (a1 b3 a3 b1 ) + c3 (a1 b2 a2 b1 ) a1 b 1 c 1 = det a2 b2 c2 . a3 b 3 c 3 F . , , .

. D (a, b, c) D

X D Y (e1 , e2 , e3 ) = 1 D D D D F

10.8

D D F

Ax + By + C = 0, n = (A, B) F D l = (x2 x1 , y2 y1 )D (x1 , y1 ) (x2 , y2 ) " F D

A(x2 x1 ) + B(y2 y1 ) = 0 (n, l) = 0.D D D F Ax + By + Cz + D = 0, n = (A, B, C) " F D D E a bD X [a, b] a bF @D FA

10.9

l : Ax + By + C = 0 M0 = (x0 , y0 ) lF (M0 , l) M0 / lD X

. .

71

M0 l0 D lY M1 = (x1 , y1 ) l0 lY M0 M1 F D n = (A, B) lF l0

l0 = {(x, y) : x = x0 + At, y = y0 + Bt, tD (x, y) lX

t R}.

A(x0 + At) + B(y0 + Bt) + C = 0 t = D M0 M1 = (At, Bt)

Ax0 + By0 + C . A2 + B 2

(M0 , l) = |M0 M1 | =

|Ax0 + By0 + C| . A2 + B 2

10.10

: Ax + By + Cz + D = 0 M0 = (x0 , y0 , z) F (M0 , l) / M0 X |Ax0 + By0 + Cz0 + D| (M0 , ) = . A2 + B 2 + C 2

10.11

l : Ax + By + C = 0 v = (v1 , v2 )F D n = (A, B) lF v l D (v, n) = 0F D

v

l

Av1 + Bv2 = 0.

(1)

: Ax + By + Cz + D = 0 v = (v1 , v2 , v3 ) X

v

Av1 + Bv2 + Cv3 = 0.

(2)

D (1) (2) . D v : Ax+By+Cz+D = 0F A(x0 , y0 , z0 ) B(x1 , y1 , z1 ) D AB = v F v D

72

10

B Ax1 + By1 + Cz1 + D = 0F Ax0 + By0 + Cz0 + D = 0D A(x1 x0 ) + B(y1 y0 ) + C(z1 z0 ) = 0 Av1 + Bv2 + Cv3 = 0F - . ,

A

B

x C y = D, z

A

B

C

= A

B

C P,

P (. 9.11). (v1 , v2 , v3 ) v (v1 , v2 , v3 )

v1 v2 = P v3

v1 v2 v3

v1 v1 v2 = P 1 v2 . v3 v3

, ,

A v1 + B v2 + C3 v3 = 0

A

B

v1 C P P 1 v2 v3

Av1 + Bv2 + Cv3 = 0.

1111.1 D D D E D " E E F D F D E EF D E D E F @ A F D D E D X D Ax = 0 D D D F D " V D X @ a, b V D a + b V A @ a V RD a V A F D X

(a + b) + c = a + (b + c) a, b V

@ AY

0D D a + 0 = 0 + a = a a V ; a V b V D a + b = b + a = 0; a + b = b + a a, b V@ AY

( a) = () a , R, a V Y ( + ) a = (a) + (a) , R, a V (a + b) = (a) + (b) R, a, b VUQ @AY @AY

74

11

1a=a aVF

1

V F E " F D V E F F b D a + b = b + a = 0D a b aF D ED F

b = (0 a) @ 0 a)F 0 = b + (0 a) = (b + (0 a)) + (0 a) 0 = 0 aF 2

1. 0 a = 0 a V F . D 0 a = (0 + 0) a = (0 a) + (0 a)F D 0 = 0 RF 0 = (0 + 0) = 0 + 0 0 = 0.

2. . 3. .

2

(1) a = a a V F I D 0 = 0a = (1+(1))a = (1 a) + ((1) a) = a + ((1) a). 2

4. a = 0, = 0, a = 0. . = 0F 2

a=1a=

1

a=

1

( a) =

1

0 = 0.

2

D 1 , . . . , n w

w = 1 a1 + . . . + n an a1 , . . . an D E 1 , . . . , n E a1 , . . . , an L(a1 , . . . , an )F 0 0 0F

11.2

f + g f g (f + g)(x) = f (x) + g(x)F f D E (f )(x) = f (x)F D F

(1) [0, 1].

1 ,

a a = b b . , ,

b = a = 1. , , , a = b. , ( a) = () a, 1 (b) = (1 ) b = b 1 a = a. 2 1 a = a. , V 0 a = 0 a V . , .

. .

75

(2) {xk } . k=1 {xk } {yk } {zk } zk = xk + yk F {xk } {zk } zk = xk F D F

(3) {xk } . k=1 D E F D E D F (1)-(3) D E E F F D D F D E f1 , . . . , fn F f

f (x) = 1 f1 (x) + . . . + n fn (x).

()

n x1 , . . . , xn [0, 1] f

1 (f ) f1 (x1 ) + . . . + n (f ) fn (x1 ) = f (x1 ), ... ... ... ... 1 (f ) f1 (xn ) + . . . + n (f ) fn (xn ) = f (xn ). 1 (f ), . . . , n (f )F D E F () f F D F f 1 (f ), . . . , n (f ) E F x [0, 1] x1 , . . . , xn F g D g(xi ) = f (xi ) i = 1, . . . , nD g(x ) = f (x )F D i (f ) = i (g) i = 1, . . . , nD f = g D D f (x ) = g(x )F 2., ,

x1 , . . . ,

f1 (x), . . . , fn (x) n xn [fi (xj )]ij=1 .

11.3

D E D F

(1) nF

76

11

@A x n

f (x) = an1 xn1 + an2 xn2 + . . . + a1 x + a0 . ak = 0 ak+1 = . . . = an1 = 0D k f (x)F axi D F f (x) xF E D F D F D n xn1 , xn2 , . . . , x1 , x0 1.

(2) m n- m n. D E " F E kl = [(E kl )ij ] m n

(E kl )ij =

1, i = k, j = l, 0, .

mn D m nE F

(3) Ax = b. mnE A rD n r D F -D E AF X ker A @ null AAF

(4) y = Ax ( A). D E AF AF X imAF

11.4

V " F D E X

V = L(a1 , . . . , an ). EF E X

V a1 , . . . , an Y

. .

77

V D V Y V Y E V dim V Y V V Y D a1 , . . . , an F Q E " E F

11.5

W V V D D V F D W D D a, b W a+b W a W F a1 , . . . , an W D L(a1 , . . . , an ) W. V E OF ED E.

E OA " AF V D D F D E D V F D l " D D X A, B l OC = OA + OB Y D C lF /.,

Rn

-

,

Rn

.

11.6

P Q " V F P + Q E w = p + q D p P D q QF P Q F

. P + Q P Q V . .

@IA w1 , w2 P + QF P + QD w1 = p1 + q1 w2 = p1 + q2 D p1 , p2 P q1 , q2 QF 1 w1 + 2 w2 = (1 p1 + 2 p2 ) + (1 q1 + 2 q2 ) P + Q, P D Q @P Q " D AF @PA D w1 , w2 P QX

w1 + 2 w2 P

1 w1 + 2 w2 Q

78

11

w1 + 2 w2 P Q. 2

D X D D F

. W1 W2 V .

dim(W1 + W2 ) = dim W1 + dim W2 dim(W1 W2 ).

. g1 , . . . , gr W1 W2 W1

g1 , . . . , gr , p1 , . . . , pk , W2

r + k = dim W1 ,

g1 , . . . , gr , q1 , . . . , qm ,D

r + m = dim W2 .

W1 + W2 = L(g1 , . . . , gr , p1 , . . . , pk , q1 , . . . , qm ). D F

1 g1 + . . . + r gr + 1 p1 + . . . + k pk + 1 q1 + . . . + m qm = 0.

1 g1 + . . . + r gr + 1 p1 + . . . + k pk = (1 q1 + . . . + m qm ) W1 W2 . W1 W2 = L(g1 , . . . , gr )D E 1 , . . . , r

1 g1 + . . . + r gr = (1 q1 + . . . + m qm ).

1 g1 + . . . + r gr + 1 q1 + . . . + m qm = 0 1 = . . . = r = 1 = . . . = m = 0 1 = . . . = r = 1 = . . . = m = 0. 2n n-

.

n n-

.

1212.1 V " n f1 , . . . , fn " F v V

v = x1 f1 + . . . + xn fn . x1 , . . . , xn v F E D V Rn E x1 v x = ... . xn g1 , . . . , gn E E X y1 v = y1 g1 + . . . + yn gn y = . . . . yn

f1 = p11 g1 + . . . + pn1 gn , ... ... ... fn = p1n g1 + . . . + pnn gn

()

n nE P = [pij ]F () v f1 , . . . , fn D y = P x. () {fi } E {gi }F ()D P {fi } {gi }F {gi } {fi }X fi gi " E D D ()D [f1 , . . . , fn ] = [g1 , . . . , gn ]P @ E P D P 1 {fi } {gi }AF D " D P 3 UW

80

12

12.2

V W D E E : V W D E X

(a + b) = (a) + (b),

( a) = (a)

a, b V, R.

F D D D 3 V D E " W F D D D F

. (0) = 0,

(a) = (a) a V F

. (0) = (0 + 0) = (0) + (0). b = (a) @D (a)AX0 = b + (0) = (b + (0)) + (0) = 0 + (0) = (0) (0) = 0. 2

D D F D D E V D F D a1 , . . . , an V D E (a1 ), . . . , (an )F

. V n = dim V Rn . X

. E a, . . . , an V x1 (v) = . . . , xn

x1 , . . . , xn " v X

v = x 1 a1 + . . . + x n an . F 2

. .

12.3

Pn " n E F D Pn Rn F p(x) n

p(x) = pn1 xn1 + . . . + p1 x + p0 .

()

. .

81

D X p0 (p(x)) = . . . . pn1 D F E c () E xD F x ()D E F D () p(x) F

x0 , x1 , . . . , xn1 xF D F D D E X xk = 0 + 1 x + . . . + k1 xk1 . D D (, )X xk x Y 1D 0F D D D [0, 1]c D

p0 + p1 x + . . . + pn1 xn1 = 0 x [0, 1].

(#)

F x1 . . . , xn [0, 1]F (#) x [0, 1]D E x = x1 , . . . , xn X p0 1 + p1 x1 + . . . + pn1 xn1 = 0, 1 ... p0 1 + p1 xn + . . . + pn1 xn1 = 0. n E 1 x1 . . . xn1 1 A = ... ... ... ... . 1 xn . . . xn1 n D E A " n x1 , . . . , xn F X A = V (x1 , . . . , xn )F

. V (x1 , . . . , xn ) x1 , . . . xn

det V (x1 , . . . , xn ) =

(xj xi ).1i 0D k k

ti vi = t1 v1 + (1 t1 )i=1 i=2

ti vi , 1 t1

k

i=2

ti = 1. 1 t1

k F 2 @ A S @ D AF E D S F S F D S " D S F.

A Rnn

,

-

, 1. ,

n

(

, , ).

92

13

1414.1 D F D F D D F E D F 2 2E

z = z(a, b) =

a b b a

,

a, b R.

()

C F F E D F (1) u, v C, u + v C uv C. (2) z = z(a, b) C ,

z 1 =

c d d c

,

c=

a2

b a , d= 2 . 2 +b a + b2

. (4) C\{0} . (5) : z(u + v) = zu + zv u, v, z C.

(3) C

@QAE@SA C RD E F C E F F a b z = z(a, b) X e(z) = a, sm(z) = bF E ()X

e= D

1 0 0 1

,

i=

0 1 1 0

.

z = z(a, b) = ae + bi,WQ

a, b R.

()

94

14

e F ae aF e = 1 e 1D ()

z = a + bi, D D

i2 = 1

@1 eAF

D z 2 = 1 C z = i. D E @ A CF D E F D " D (a, b) D

(a, b) + (c, d) = (a + c, b + d),

(a, b)(c, d) = (ac db, ad + bc).

@QAE@SAF () D E F

14.2

F (a, b) " @E A a, bF D (a, b) z = a + bi E @EA F D @EA E D F z = a + biF ED a2 + b2 D z |z|F E z = 0 @ E AD z F X = rg z F D D 2 F z = 0 F D z = |z| (cos + i sin ), = arg z. F D EF @ E A |u + v| |u| + |v| u, v C,

. .

95

| |u| |v| | |u v| z

u, v C.

w = |w| (cos + i sin ),

= arg w,

zw = |z| |w| (cos + i sin ) (cos + i sin ) = |z| |w| ((cos cos sin sin ) + i (cos sin + sin cos )) = |z| |w| (cos( + ) + i sin( + )). D , F X ei = cos + i sin F 1 ei ei = ei(+) @ AF a bi z = a + biF X z = abiF E z E z F D z z = |z|2 F . -

n,

n

Sn =k=1

cos k,

,

Sn = Rek=1

zk

,

z = cos + i sin .z n+1 z z1

, :

Sn = Re

.

14.3

E E F D w C z z + wF @A D wF D z wz D |w| = 1F D |w| = 1D |wz| = |z|F E wz E z = rg wF D w 1 D wF > 0 " E @ AF w = 0 w = |w| wD w = w/|w| D E D |w| = 1D w = 0 @ A X E F z z EF E F D F D 1 , .

-

96

14

a, b z = a + bz D z CF 3

. T (z) = a + bz (z) = a + b, z a, b C,

|b| = 1,

.

. X

()(z) = ((z))F (z) = a + bz (z) = c + dz T F D |b| = |d| = 1F ((z)) = a + b(c + dz) = (c + bc) + (bd)z. |bd| = |b||d| = 1D T F E z z D D D T F D w = a+bz D z = a F | = 1D D bw b| D T F F T " D E F 2 E z (z) D X |(z1 ) (z2 )| = |z1 z2 | z1 , z2 CF D T D F F E T F " D @ D AF

X z 1/z F z = 0D E C\{0}F D C = C {}F z 1/z E CD D 0 D 0F z 1/z -

z (z) =

a + bz , c + dz

a, b, c, d " D D (z) X ad bc = 0F d = 0, - . , d = 0. (z) z = c/d. , (c/d) = () = c/d, - . (, !) .

.

, -

z a

z b , zb

|a| = 1,

Im(b)

< 0,

. .

97

( ) .

.

, -

z a

z b , 1 zb

|a| = 1, |b| < 1,

( ) .

14.4

z n D z n = 1F

. z = |z| (cos + i sin ), z n = |z|n (cos(n) + i sin(n)).

. D D F

2

. n n. zk = ei2k n

= cos

2 k n

+ i sin

2 k n

,

k = 0, 1, . . . , n 1.

. z = |z| (cos + i sin ) nFD D |z| = 1 cos(n) = 1 @ sin(n) = 0AF

=

2 k , n

k = 0, 1, 2, . . . .

D k

zk = cos

2 k n

+ i sin

2 k n

n F E D zk = zl D l = k + m n, m = 0, 1, 2, . . . . 0 k, l n 1D zk = zl k = l " D z0 , z1 , . . . , zn1 nED F 2

14.5

n

Kn n F D Kn n F Kn D D E F D Kn F D zk = k D = cos(2/n) + i sin(2/n) = z1 F zm = m n D

Kn = {(m )0 , (m )1 , ..., (m )n1 }.

98

14

D m " F mp = 1 0 < p n p = n @ p = 1D D p AF

, m n ( 1).

1. m Kn m 1

n d > 1X n = dp m = dq p, q d > 1F (m )p = mp = dqp = qn = 1 0 < p < n m D p < n m F m n F D m F D k = l 0 k, l n 1D k = lF D m(k l) nF m n D k l n k = lF 2 1 nD nD (n)D (n) F 2

. D m D m

2. n . . zk = k D Xn1 n1

zk =k=0 k=0

k =

n 1 = 0. 1

2

. .

n1,

(x + k y)n = n(xn + y n ),

= cos(2/n) + i sin(2/n).

k=0 , ,

n1

n1

(a) x2n 1 = (x2 1)k=1

(x2 2x cos(k/n) + 1);

(b)k=1

sin

k 2n

=

n

2n1

.

14.6

m n Cmn F A = [aij ] Cmn D A = [ij ]F a A F X A = A F A X

(AB) = B A Y det A = det AY A D A D (A )1 = (A1 ) @ A = (A )1 AF2 . ,

(ab) = (a)(b)

a

b

!

1515.1 D E F D D E X N Z Q R C,

N = {1, 2, . . . } " Y Z = {0, 1, 2, . . . } " D

1

Q = {p/q, p Z, q N} " D R " D C " F Z D E Q, R, C " F K X @ +A @ AD X

K Y X a(b + c) = ab + ac, (b + c)a = ba + ca a, b, c K;

F K @A F @ FA 0F D aD a aF

1. 0 a = a 0 = 0

a KF

1 , , .

WW

100

15

D 0 a = (0 + 0) a = (0 a) + (0 a)F b X 0 = b + (0 a) = (b + 0 a) + (0 a) = 0 + (0 a) = 0 aF 2

. b = (0 a) @D 0 aAF D K F D E F P " D P \{0} E F P F P \{0} P F 1F

2. K " D (1) a = a (1) = a . 0 = (1 + (1)) a = 1 a + (1) a = a + (1) aF 2.e ab

a KF

a

b

e. , e ba.

15.2

a, b D ab = 0F a, b F

3. X ab = 0 a = 0 b = 0F . ab = 0F a = 0D F D a = 0F a a1 @a1 a = aa1 = 1AF I D 0 = a1 0 = a1 (ab) = (a1 a) b = 1 b = bF 2:

(1) K " F "

F F F

(2) K = Rnn @ n nEAF "

F F F D n = 2

1 1 1 1

1 1 1 1

= 0.

(3) K " a + b 2D a, b QF "

F D E F D K " 1 = 1 + 0 2F K X K\{0} @F PAF

. .

101

15.3

D p " D p @F WFQAF p > 1F a Z Z(a) D p D a @E a pAF Z(a) pF p Zp F p E pX

Zp = {Z(0), Z(1), . . . , Z(p 1)}. X

Z(a) + Z(b) = Z(a + b),

Z(a)Z(b) = Z(ab).

X

Z(c + d) = Z(a + b),

Z(cd) = Z(ab)

c Z(a), d Z(b).

D Zp F

. p p . . p p = ab 1 < a, b < p Z(a) Z(b) = Z(ab) = Z(p) = Z(0) = 0F D Zp D QD Zp pF D p " F D Z(a) 1 a p 1 Z(b) D Z(a)Z(b) = Z(1) = 1F ka pX1 a = pq1 + r1 , 2 a = pq2 + r2 , ... , (p 1) a = pqp1 + rp1 , (1)

q1 , r1 , . . . , qp1 , rp1 Z,

0 r1 , . . . , rp1 p 1.

r1 , . . . , rp1 D a pF D F D rk = rm F (k m)a = p(q k qm )F a p D k m pF D k, m = 1, 2, . . . , p 1 D |k m| < p k m = 0F D

{r1 , r2 , . . . , rp1 } = {1, 2, . . . , p 1}.D k rk = 1

(2)

Z(a) Z(rk ) = Z(1) = 1F 2

.

X p a p, ap1 1 p. D (1) (2)D D

102

15

(p 1)! (ap1 1) pF (p 1)! p D p ap1 1F D Zp D p " @ D AF " E D D E F., .

15.4

M " K F K " D M D D K F K " D M D D K F D M K D K @A M F D F L M E : L M D X

(a + b) = (a) + )b),

(ab) = (a) (b)

a, b L.

D L M " F K @A L D L E @A M F 1 " P F D p p 1 = 1 + . . . + 1. D p 1 @p E D D AF p D p 1 = 0D P F

1. p 1, p .(m 1)(k 1)F D F 2 Zp .

. D p = mk F 0 = (mk) 1 =

2. p 1

. p D D p E k 1D k = 1, . . . , pF D F Zp (k 1) = Z(k)F 2 . -

. .

103

. 1. 2.

P

R P C. Q.

,

P =R

P = C.

,

15.5

p , m .

3. n = pm ,

. p " F D D E

PD F pX Zp F F E D a1 , . . . , am F Zp D E 1 a1 + . . . + m am = 0 1 , . . . , m Zp D 1 = . . . = m = 0F m " D Zp F v F

v = 1 a1 + . . . + m am ,

1 , . . . , m Zp . n = pm F 2

i p

F D E D n = pm F n c D F. .

15.6

K . a , a, b K b = 0. a b b a a ad = bc. , , ( = ) b b

. . .,

c d , . . ,

a c = b d

ad = bc,

c p = d q

cq = dp. aq bp = 0

(aq bp)(cd) = 0

, ,

aq = bp

a p = . b q

, . .

a a b , b . c , : d a c a K b , K d = K b ; .

K

104

15

:

K

a c +K =K b d

ad + bc bd

,

K

a c ac ) K( =K . b d bd

, -

a c b K d , . , 1 = 0 K a . 0 = K a . K a , a = 0. , a b b K a . a , K b . K ? ac - a K , c :

K

a+b K

ac +K c

bc c

,

ab K aK

ac c

K

bc c

. Kac c(,

c = 0). 2

, ,

,

K

K,

(

).

1616.1 P " D D V " D F D V X @ P AD @AD E " D E P F V P D P F P D F D D D E @ D F FAF D E F D V F F:

(1) V " @ AD P = R " E

F F R E " F RF D dim V = 2F

(2) V " D P = Q " F

F Q " F F D D 1 < p1 < ... < pn V log p1 , . . . , log pn F

1 log p1 + ... + n log pn = 0,

i = si /ti , si , ti Z.

t = t1 ...tn D i = i t

1 log p1 + ... + n log pn = 0

IHS

log(p1 ...pn ) = 0 1 n

p1 ...pn = 1. 1 n

106

16

D i F E D F i1 , . . . , ik > 0 j < 0D i pi1i1 ...pik k pj F D D

1 = ... = n = 0

1 = ... = n = 0.

D n n F

(3) V " m n P F E

X V = P mn F " X [aij ] + [bij ] = [aij + bij ]F P E X [aij ] = [ aij ]F V " P Y dim V = mnF

. . .

sin x, sin 2x, . . . , sin nx

-

[a, b].

,

- . 10 ?

16.2

p(x) = a0 + a1 x + . . . + an xn , a0 , a1 , . . . , an P. ()

x P "

x F an = 0D D p(x) " nF X deg p(x) = nF P F D E 0D F F D D D E p(x) x P F F D D P = Z2 = {0, 1}F x = x2 x Z2 . D E D D F D P E F x P [x]F

. D p(x) () ai

xi i 0 n 0 xi D i n + 1F x P D xF D x x2 Z2 @ x Z2 AF P [x]X

p(x) = a0 + a1 x + . . . + anp xnp ,

ai = 0 i np + 1,

. .

107

q(x) = b0 + b1 x + . . . + bnq xnq ,

bi = 0 i nq + 1.

p(x) + q(x) = s0 + s1 x + . . . D xi s i = ai + b i , i 0. p(x) q(x) = t0 + t1 x + . . . D xi i

ti =k+l=i

ak b l ,

i 0.

D E

(a0 + a1 x + . . . + anp xnp )(b0 + b1 x + . . . + bnq xnq ) = (a0 b0 ) + (a1 b0 + a0 b1 )x + (a2 b0 + a1 b1 + a0 b2 )x2 + . . . + (anp bnq )xnp +nq . @ A X

deg(p(x) q(x)) = deg p(x) + deg q(x).

(#)

16.3

. P [x] .

. P [x]

. D P [x] D F p(x) q(x)D r(x) = c0 + c1 x + . . . + cnr xnr , ci = 0 i nr + 1.

(p(x)q(x))r(x) = u0 + u1 x + . . . Y p(x)(q(x)r(x)) = v0 + v1 x + . . . F D D

ui =j+m=i k+l=j

ak b l

cm =k+l+m=i

ak bl cm =k+j=i

akl+m=j

bl c m

= vi .

D F E F F 1F (#)F 2 D P [x] P @ D P " D P AF P [x] @ AF Pn [x] n n + 1F

108

16

16.4

. f (x), g(x) P [x] g(x) = 0 q(x), r(x) P [x] , f (x) = g(x)q(x) + r(x), deg r(x) < deg g(x)

r(x) = 0.

()

g(x) = bm xm + . . . + b0 D bm = 0F deg f (x) < deg g(x)D X q(x) = 0 r(x) = f (x)F deg f (x) deg g(x)D f1 (x) = f (x) an nm x g(x) bm deg f1 (x) < deg f (x) f1 (x) = 0.

. f (x) = an xn + . . . + a0 D

f (x)F

f1 (x) = g(x)q1 (x) + r1 (x), ()

deg r1 (x) < deg g(x) r1 (x) = 0D

q(x) =

an nm x + q1 (x), bm

r(x) = r1 (x).

F q(x) r(x)D ()F

g(x)(q(x) q(x)) = r(x) r(x). q(x) q(x) = 0D g(x) deg(r(x) r(x)) deg g(x)F D q(x) = q(x) r(x) = r(x)F 2 r(x) () D q(x) " f (x) g(x) = 0F r(x) = 0D D f (x) g(x)D g(x) f (x)F

16.5

d(x) P [x] f (x) g(x) P [x]F D F X d(x) = (f (x), g(x))F E P D E F D E @ AD E P F f (x) g(x) D (f (x), g(x)) = 1F

. .

109

D deg f (x) deg g(x)F f (x) g(x) D E X

f (x) = g(x)q1 (x) g(x) = r1 (x)q2 (x) r1 (x) = r2 (x)q3 (x) ... ... ... rk2 (x) = rk1 (x)qk (x) rk1 (x) = rk (x)qk+1 (x).

+ r1 (x), + r2 (x), + r3 (x), + rk (x),

deg r1 (x) < deg g(x), deg r2 (x) < deg r1 (x), deg r3 (x) < deg r2 (x), deg rk (x) < deg rk1 (x),

F rk (x) " F . rk (x) = (f (x), g(x))F . D D rk (x) f (x) g(x)F d(x) " f (x) g(x)F D D d(x) rk (x)F D rk (x) = (f (x), g(x))F 2

. f (x), g(x) P [x] (x), (x) P [x] ,

f (x)(x) + g(x)(x) = d(x),

d(x) = (f (x), g(x)).

. E F

ri2 (x) = f (x) i2 (x) + g(x) i2 (x),

ri1 (x) = f (x) i1 (x) + g(x) i1 (x),

D ri (x) = f (x) i (x) + g(x) i (x),

i (x) = i2 (x) i1 (x) qi (x), i = k F

i (x) = i2 (x) i1 (x) qi (x). 2

. f (x), g(x) P [x] (x), (x) P [x] , f (x) (x) + g(x) (x) = 1.

. f (x)(x) + g(x)(x) d(x) = (f (x), g(x))@D D d(x)AF

16.6

f (x) = a0 +a1 x+. . .+an xn P [x] P F f () X f () = a0 +a1 +. . .+an n F D f () P F f (x) x = F f (x)D f () = 0F . f (x) P [x] f () = 0 P , f (x) x . 1 . D f (x) = (x )q(x) + r(x)D r(x) = 0 deg r(x) = 0F r(x) = 0D F deg r(x) = 01 ,

r() = f ()

r(x)

f (x)

x .

110

16

X 0 = f () = ( ) q() + r() = r() r(x) = 0F D D F 2 f (x) P [x] P D p(x), q(x) P [x] D f (x) = p(x)q(x)F f (x) D P F D P P D n P n F.. ,

.

,

Zp

p

(z 1) = z 1.

p

p

16.7

P D P F D D E F P F F P D E P F P P F P D F F F F - P D D P F X P ()F D 1 , . . . , k F D P (1 , . . . , k ) E D P 1 , . . . , k F D D P 1 , . . . , k F P D D P () P F / D P [x]F

. . n , - P P () = {s F : s = a0 + a1 + . . . + an1 n1 , a0 , a1 , . . . , an1 P }. ()

. D f (x) g(x) " @ nAF d(x) P [x] d(x) = f (x)(x) + g(x)(x), (x), (x) P [x].

d() = 0F deg d(x) = n f (x) g(x) d(x) F M D ()F D M P ()F D M " F p(x) P D p() M F

. .

111

p(x) f (x)X

p(x) = f (x)q(x) + r(x)

p() = r().

D r() 1, , . . . , n1 P F r() M F M D D E p(x) P [x] x = F M F D M p()D p(x) P [x] n 1F E f (x)D D D p(x) f (x) E F D (x), (x) P [x] D

f (x)(x) + p(x)(x) = 1.

p()() = 1.

2

P , Q 5 2. , P Q . . , p1 , . . . , pn p1 < . . . < pn .

112

16

1717.1 D " D @ A " X n > 1 D F F " F F D 1 F f (z) C[z] z CF E z F D F X D n D n Y f (z) = g(z) z D f (z) g(z) D F

17.2

zk D k = 1, 2, . . . F E z0 D > 0 N = N () D k N |zk z0 | F @ D E |zk z|FA X lim zk = z0 zk z0 Fk

-. zk -

= [A, B] [C, D] zki , z0 .

. zk = xk + i yk , xk , yk RF D xk [A, B] yk [C, D]F E E D xki D E x0 [A, B]F 1 , , .

IIQ

114

17

zki = xki + i yki F yki [C, D]D E ykij D y0 [C, D]F xkij x0 @ AF z0 = x0 + i y0 F |zkij z0 | |xkij x0 | + |ykij y0 | 0. 2

17.3

(z)D z C E F (z) z0 D zk D z0 D (zk ) (z0 )F

. (z) = [A, B] [C, D]. z , z ,

(z ) (z) (z )

z .

. z F D D

(z) F D E zk (zk ) > k F ED E zki z0 F D (zki ) (z0 )D (zki ) > ki D ki F M D (z) M z F M (z)F () = {x : x = (z), z }F E D M " D D F 2 D zk (zk ) M F ED zki D z F D

M = lim (zki ) = (z ).k

D (z) = (z) D (z) F D (z)F D D (z) z D (z) (z ) z F (z ) (z) z F 2

17.4

f (z) = a0 + a1 z + . . . + an1 z n1 + z n

(#)

2 .

. .

115

an = 1D n 1F

. (z) = |f (z)| z C. (z0 +h) h C h = 0F D f (z0 +h) hX

. (z) z = z0

f (z0 + h) = b0 + b1 h + . . . + bn1 hn1 + hn ,

b0 = f (z0 ).

|(z0 + h) (z0 )|

=

||f (z0 + h)| |f (z0 )|| |f (z0 + h) f (z0 )| |b1 h + . . . + bn1 hn1 + hn | |b1 ||h| + . . . + |bn1 ||h|n1 + |h|n . 2

, |z| R , |f (z)| M .

. M > 0 R > 0 . D |z i | = |z|i D |f (z)| |z n | |a0 + a1 z + . . . + an1 z n1 | |z|n |a0 | |a1 ||z| . . . |an1 ||z|n1 . A |a0 |, . . . , |an1 |F |z| 1

|f (z)| |z|n 1 M > 0

nA |z| n

.

R = max{1, 2nA, D |z| RD

2M }.

|f (z)| Rn 1

nA 2nA

= Rn /2

2M = M. 2

2

17.5

f (z) " (#)F

0, h C , |f (z + h)| < |f (z)|.

. z C |f (z)| > . n = 1F D n 2F z C f (z + h) hX f (z + h) = f (z) + b1 h + . . . + bn1 hn1 + hn . bm " @ b1 = . . . = bm1 = 0AF

f (z + h) = f (z) + bm hm + g(h)hm+1 ,

g(h) = bm+1 + . . . + bn1 hnm2 + hnm1 .

116

17

m = f (z)/bm h

h = t,D

t > 0.

|f (z) + bm hm | = |f (z)(1 tm )| = |f (z)|(1 tm ) < |f (z)| t > 0. 0 t 1 B > 0

|g(t) (t)m+1 | Btm+1 .D 0 < t 1D

|f (z + t)| < |f (z)|(1 tm ) + Btm+1 = |f (z)| + (Bt |f (z)|) tm . 0 < t min(1, |f (z)|/B) |f (z + t)| < |f (z)|F

2

. .

. M = |f (0)|F M = 0D F D

M > 0F g D |z| R |f (z)| M F = [R, R] [R, R]F |f (z)| z C D D z F D z D |f (z )| |f (z)| z F D |f (z )| M D D M |f (z)| z / |f (z )| |f (z)| z C. ()

|f (z )| > 0D D D h C |f (z + h)| < |f (z)|D ()F D |f (z )| = 0 z X f (z ) = 0F 2

17.6

F . f (z) n > 0 C[z] n :

f (z) = a (z z1 ) . . . (z zn ),

a, z1 , . . . , zn C.

()

.

. D f (z) E

" z1 F D f (z) z z1 X f (z) = (z z1 )f1 (z)F deg f1 (z) = 0D E F deg f1 (z) > 0D " z2 F D f (z) = (z z1 )(z z2 )f2 (z)F deg f2 (z) = 0D F D f2 (z) D F D a f (z)F D X

f (z) = a (z z1 ) . . . (z zn ) = a (z z1 ) . . . (z zm ).

. .

117

D D m m = nF D a = a @ f (z)AF D (z1 z1 ) . . . (z1 zn ) = 0 z1 E zi F D z1 = z1 F D

(z z1 ) ( (z z2 ) . . . (z zn ) (z z2 ) . . . (z zn )) = 0. C[z] D

(z z2 ) . . . (z zn ) = (z z2 ) . . . (z zn ). D @ A z2 = z2 D F 2

. f (x) n > 0 f (z) = a (z 1 )k1 . . . , (z m )km , i = j i = j,

k1 , . . . , km > 0,

k1 + . . . + km = n,

()

a, 1 , . . . , m C.

() E f (z)F ki i F i D ki > 1D D ki = 1F ()D f (z) m F () z1 , . . . , zm X zi = j D kj D j F X n > 0 n F

17.7

@ A f (x) = a0 + a1 x + . . . + an xn D z C F z @ ai = ai iAX

f (z) = a0 + a1 z + . . . + an z n = a0 + a1 z + . . . + an z n = f (z) = 0. z = z D @ 2A

(x) = (x z)(x z) = x2 (z + z) x + |z|2D D R[x]F

. f (x) n > 0 R[x] :

f (x) = a(x x1 ) . . . (x xM ) 1 (x) . . . N (x), a, x1 , . . . , xM R,

M + 2N = n,

118

17

i (x) = x2 + si x + ti , si , ti R,

i = 1, . . . , N.

.

. f (x) n z1 , . . . , zn

F M F n M @ n M X nM = 2N AF M D N F F D X

f (x) = a(x x1 ) . . . (x xM ) 1 (x) . . . N (x) = a(x x1 ) . . . (x xM ) 1 (x) . . . N (x).D a = a @ f (x)AF D

a(x x1 ) . . . (x xM ) = a(x x1 ) . . . (x xM ), R[x] D

M =M

N =N.

1 (x) . . . N (x) = 1 (x) . . . N (x). 1 (z) = 0 1 (x) = (x z)(x z)F D 1 (z) . . . N (z) = 0 F D D 1 (z) = 0 1 (x) = (x z)(x z)F D

1 (x) = 1 (x) F

2 (x) . . . N (x) = 2 (x) . . . N (x).

2

. . F D f (x) @ x RA x E xF f (x) F

. "

1818.1 f (x) n 1 X

f (x) = a0 + a1 x + . . . + an1 xn1 + xn = (x x1 ) . . . (x xn ). E xD X

an1 = an2 = an3 = ... ... ank = ... a0

(x1 + x2 + . . . + xn ), (x1 x2 + x1 x3 + . . . + xn1 xn ), (x1 x2 x3 + x1 x2 x4 + . . . + xn2 xn1 xn ), ... (1)k xi1 . . . xik ,1i1 < ... < ik n

... ... = (1)n x1 . . . xn .

k = k (x1 , . . . , xn ) =1i1 < ... < ik n

xi1 . . . xik ,

k = 1, . . . , n,

()

x1 , . . . , xn F E D f (x) E x1 , . . . , xn X

ank = (1)k k ,

k = 1, . . . , n.

18.2

n

x1 . . . xn D 1 , . . . , n " D 1 n 1 + . . . + n x1 , . . . , xn F i = 0 @ xi AF x1 , . . . , xn P x1 , . . . , xn P F E F D

f (x1 , x2 , x3 ) = x3 x2 x3 + x2 x3 x3 + x1 x2 x4 + x1 x2 + x3 3 1 2 1 2IIW

120

18

6F D f (x1 , x2 , x3 ) Q E F x1 . . . xn = x1 . . . xn D i = i iF E 1 1 n n P F D " E E D D F f g D F f +g D D f g F f = a1 ,...,n x1 . . . xn g = b1 ,...,n x1 . . . xn 1 1 n n f g D

(a1 ,...,n b1 ,...,n ) x1 +1 ... xn +n . 1 n D E F x1 , . . . , xn P P [x1 , . . . , xn ]F E F

18.3

x1 , . . . , xn @A X

x 1 . . . x n 1 n

@A

x 1 . . . xn , 1 n

1 k n

1 = 1 , . . . , k1 = k1 ,

k > k .

E D E E F D F D E F

18.4

f (x1 , . . . , xn ) D n f (x1 , . . . , xn ) = f (x(1) , . . . , x(n) ).

. .

121

E k D F

f (x1 , . . . , xn ) g n , f (x1 , . . . , xn ) = g(1 , . . . , n ), k = k (x1 , . . . , xn ) ().

. -

.

a x1 . . . xn " f (x1 , . . . , xn )F 1 n 1 . . . n F D X E x1 . . . xn 1 n x1 . . . xn F (1) (n) n1 (1 , . . . , n ) = a 1 1 2 . . . n1 n n n .

(1)

x1 , . . . , xn D D D

a x1 2 (x1 x2 )2 3 . . . (x1 . . . xn1 )n1 n (x1 . . . xn1 xn )n = a x1 . . . xn . 1 1 n

(2)

f1 (x1 , . . . , xn ) = f (x1 , . . . , xn ) (1 , . . . , n ) f (x1 , . . . , xn )F f1 f2 D F E E F g D g(1 , . . . , n ) = 0D f (x1 , . . . , xn ) = 0F D D k x1 , . . . , xn F g (1) 1 . . . n F x1 , . . . , xn D (2)F g

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