-

.., ..

I

2010

512 1473-2 191

.., .. : 2 .: - / . . -. , 2010, . 1. 143 c.

ISBN 978-5-94356-912-8

- , - . - : , , , , . : -, () , , . - : . .

- .-. , . .

c , 2010

ISBN 978-5-94356-912-8 c .., .., 2010

4

1. 5 1.1. . . . . . . . . . . . . . . . . . . . 5 1.2. . . . . . . . . . . . . . . . . . . . . 9 1.3. . . . . . . . . . . . . . 14

2. , , 18 2.1. . . . . . 18 2.2. . . . . . . . . . . . . . . . . . . . . . . 26 2.3. . . . . . . . . . . . . . . . . . . 34 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5. . . . . . . . . . . . . . . . . . . . . 60

3. 67 3.1. . . . . . . . . . . . . 67 3.2. . . . . . . . 70 3.3. . . . . . . . . . . 80

4. 86 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2. . . . . . . . . 92 4.3. . . . . . . . . . 96

5. 100 5.1. . . . . . . . . . 100 5.2. . . . . . . . . . . . . . . . 102 5.3. . . . . . . . . . . . . . . . . 108 5.4. . . . . . . . . . . . . . . . . . 115 5.5. . . 118 5.6. . . . . . . . 127

133

137

138

142

- , - . -, , , , - . , 2.3.1 . - ., , . . - , - . - 201011 ( -). , , . - .

1

1.1.

- , - . , , . , . a - A : a A. a A, a 6 A. , - . , , , . - , P , , A -, P , :A = {x | x P} A = {x | P (x)}. - , . , a A : P (a), a A P , - a A : P (a) , a A, P .

: N = {1, 2, 3, . . .} , Z , Q - R .

B A, - B A. - : B A. , A, B, - B A, , : B A. - , , - ,

6 1.

. A - P (A) 2A.

. - A1, A2, . . . , An, A1 A2 . . .An ={x | x A1 x A2 . . . x An} , - A1A2. . .An = {x | x A1 x A2 . . . x An} A1, A2, . . . , An. A B A \ B = {x | x A x 6 B}. B A, A \ B B A. A , A B B.

, - . , {a, b} {b, a} , . , , , . n a1, a2, . . . , an (a1, a2, . . . , an) n-. - . n- (a1, a2, . . . , an) (b1, b2, . . . , bn) , a1 = b1, a2 = b2, . . .. . . , an = bn.

1.1.1. , {{a}, {a, b}} {{c}, {c, d}} , a = c b = d.

1.1.1. A1, A2, . . . , An,

A1 A2 . . .An = {(a1, a2, . . . , an) | a1 A1, a2 A2, . . . , an An}.

n- A

An = {(a1, a2, . . . , an) | a1 A, a2 A, . . . , an A}.

, n = 2, A.

X Y . f , - X Y , X Y ( f : X Y ). .

1.1. 7

, y Y x X f , :f(x) = y, xf = y, xf = y. -, , . . xf f(x). Z X, Zf {y Y | x Z : xf = y}, Z Y . Z = X, Xf Im f .

f : X Y g : Y Z. - ( ) f g f g : X Z, x X x(f g) = (xf)g. , f g - f , g. fg f g.

. - f g f g , g, f :(f g)x = f(g(x)). , f g , . , - . , - .

. , x X y = xf Y . : Y , X. , Y , .

1.1.2. f : X Y - X Y , - y Y x X , y = xf . : f : X Y .

, X Y . , xf =y1 xf = y2, y1 = y2. : Y X. , f : R R, xf = x2, 4 : 2 2. , .

8 1.

1.1.3. f : X Y X Y -, x1, x2 X x1f = x2f x1 = x2. : f : X

1-1 Y ., 1.1.2 1.1.3 -

, .

1.1.4. f : X Y - X Y , .

: f : X 1-1 Y . f : X Y , ,

, f1 : Y X, : x = yf1 , y = xf . . - X Y , , .

1.1.2. , 1) , ;2) , ;3) , ;4) .

1.1.3. , .

1.1.4. m,n N, X m, Y n . , .

1. : X Y .2. m = n. , , , ,

X ( ), - . , X , .

1.1.5. X . Y X ,

1.2. 9

: X Y . , - :

1. Z X : X Y , Y = X \ Z.

2. Z - X , : X Y , Y = X \ Z.

. 1.1.4 1.1.5, X , - , X. , , .

1.2.

, , . , N , m n , -. , , n+m m+n. m < n, x, x+m = n . . ( -) . ( ) -- ( - ), 800 . .., -, . , - , , . , , ( , ). , - , , . , , , ( ) . - . ,

10 1.

, . . , , -, . . , - , - . . , .

. 1. N - m n k = m + n, - m n. . -, - N2 N ( + : N2 N). , , , (m,n)+ = k m + n = k, . - : , ( m n), , ? , , - , m + n = n +m, , . , , . , - f : N2 N (m,n)f = mn ( m n), , , -, mn, , nm. .

2. S - A . - A. f g f : A A g : A A A, : S2 S, - , - S. , (f g 6= g f). , -, R2. f , g 90 .

1.2. 11

, , - , , .

x

y

A1

B1

O

C1

g

f

, A (1, 0) f g B (0, 1):

A(f g) = (Af)g = Ag = B. , A g f C (0,1):

A(g f) = (Ag)f = Bf = C. , f g g f .

- A ( , ) A2 A. - , .

3. Z , f . ,, x Z xf = x. x y =(x, y) = (x, yf)+ = x+(y) x, y Z. f , , . , , , .

4. . V A,B,C

12 1.

. , D, . - . , A,B,C , D ABC. f : V 3 V , , , V ., , n , n .

- .

1.2.1. A . f : An A n- ( n-) - A. , n- -, A, - n A A. n f .

.

1.2.2. A = A, f1, f2, . . . , fk, . . ., - A f1, f2, . . . , fk, . . ., . A A, - A.

, , - .

. 1. A = Z,+, , f, +, , f - .

2. A = S, , S A, .

3. A = R[x],+, , R[x] R , .

4. A = F,+, , F R R, - , x(f + g) = xf + xg.

1.2. 13

5. A = P (A),,, \,, P (A) A, , -, . ,, \ , .

6. : N,, Z, :, - . , Q , 0 -. Q Q \ {0} , Q, : . .

1.2.1. - :

1. Z ?2. Q+, , Q+ -

,

? , R+ - .

3. N, f, f : N2 N (m,n)f = ...(m,n) - m n?

1.2.3. A n- - f . B A, Bnf B (. . f , B, B), B f . - B n- f |B : Bn B, (b1, b2, . . . , bn)f |B = (b1, b2, . . . , bn)f , ( ) f B.

. B A - f , , , f B f , A.

. 1. Q, Z, N R - +, R. , - , , N .

14 1.

2. 1.1.3, - A - .

1.2.2. 2Z 2Z+1 . , Z , ? - : , , , -, ; : , , . - () , n 2n (3n ).

1.2.3. a b. M Z , a, b M M - . , M = Z.

1.2.4. B - A A = A, f1, f2, . . . , fk, . . . - A, - A. B = B, f1|B , f2|B , . . . , fk|B , . . . A.

. 1. B = N,+, , A = Z,+, , . A ( ), , N, .

2. A = F,+, , F R R, , - x(f + g) = xf +xg. F F . B = F ,+, A.

1.2.4. (. . - , ) - :

1. Z, f, f .2. Z, .3. P, f, P , f 3- -

, .

1.3. 15

1.3.

, . , -. - , . , .

1.3.1. A = A, f1, f2, . . . , fk, . . . B = B, g1, g2, . . . , gk, . . . , fi gi ni. A B - ( A ' B), : A B, i - (a1, a2, . . . , ani) Ani

((a1, a2, . . . , ani)fi) = (a1, a2, . . . , ani)gi.

, , - A B.

. A B - A B, 1 : B A, . - , , , B A. , A B, B A,. . A B.

. 1. A = R,+, B = R+, , R+ - . : R R+, x = 2x. , - A B. , R R+, . , x, y R

(x+ y) = 2x+y = 2x 2y = (x) (y), , , , - . , , x1 = log2(x), B A.

16 1.

2. A = Z,+, B = 2Z,+, , -, 2Z . : Z 2Z, x = 2x. , , , (x+ y) = 2(x+ y) = 2x+ 2y =x + y , . -, A ' B.

, - . , , , - , (, ).

, A B -. , . A B , , . A B -, , , , , . , - . , A B, . - . , . , () , () -, . .

. A = Z, ,B = 2Z, . Z 1, , x Z 1 x = x. 2Z . - , A B . , , - . 1 = 2k, 2 = 2n, k, n . , : 4kn = (2k) (2n) = 1 2 = (1 2) = 2 = 2n. n 6= 0, 2k = 1, . , 2 = 0, 1 3 = 2m, - 0, 3 6= 2 = 0.

1.3. 17

4km = (2k) (2m) = 1 3 = (1 3) = 3 = 2m. 2k = 1; . , , , A B .

1.3.1. -:

1) Z,+ Z, ;2) P (A), P (A),, A -

;3) Q,+ Q+, , Q+ -

( 1 );4) N, f N, g, f g -

?

1.3.2. a, b - . Aab R, f - f , xf = ax + b. {Aab | a, b R}?

2

, ,

2.1.

- -. . - : , . , - , . , . :(x+y)x. , , y (, x y ). -, , . , - , :

(x+ y) x = (x+ y) + (x), (1)(x+ y) + (x) = (y + x) + (x), (2)(y + x) + (x) = y + (x+ (x)), (3)

y + (x+ (x)) = y + 0, (4)y + 0 = y. (5)

(1) - . , - : x + (x) = 0 - (4). 0, , , , (5). (2) . - -. , (3) ( ).

2.1. 19

-, - ( ), . - ( ) ( ). . .

2.1.1. G = G, , ( ):

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

2. e G , a G a e = e a = a ( ).

3. a G a1 G , - a a1 = a1 a = e ( ).

., - . - , , . , - .

. 1. G = Z,+. . 0, - a, , . - : Q,+, R,+, Q, , R, .

2. S , , , G = S, - . - , .

, 1 -: a b = b a. , 2 (. - 1.2).

2.1.2. G = G, (

20 2. , ,

), a, b G a b = b a ( ).

. ab ab. , , , - , - . -, , - +, , 0, - a a. (- ) ( ) . , , - , , . . G = G, G.

2.1.1. G ( ), g . , g : G G, xg = xg x G, G .

2.1.2. G . , - , 2, . , g G , 3, .

g1, g2, . . . , gn G.

ni=1

gi = (. . . (g1 g2) . . .) gn.

2.1.3. ,

ni=1

gi =ki=1

gi n

i=k+1

gi.

. , - 2.1.3, : - ( ).

2.1. 21

. , - , .

2.1.3. G = G, , . , .

. N ( ), , (- ).

2.1.4. A . , P (A), . ?

2.1.3, - : , .

2.1.4. G = G, - . H = H, |H , |H - H.

., . , N,+ Z,+, , .

, - H = H, |H H. -, , H G, - : H 6 G.

. 1. G : G 1 = {e}, e G.

2. n nZ = {nk | k Z} , n, Z .

22 2. , ,

2.1.5. G . , - H G, a, b H - :

1) ab H; 2) a1 H. G G.

|G|. - . g - G. n , gn G, g n :

gn = g g . . . g n

.

g G n , gn = e, e G, n . , - g . g |g|.

2.1.6. , - .

, g0 = e gn = (g1)n, n , .

2.1.7. h G., H = {hn | n Z} - G.

. - + , , .

2.1.5. R = R,+, - , :

1. R .2. a, b, c R (a + b)c = ac + bc

( ) c(a + b) = ca + cb ( ).

2.1. 23

. , - R R.

, , - , . , , .

, , - , . -, , - 1. , , 1 6= 0, , . -, . R R.

2.1.6. F = F,+, - , (, -, ) , .

, .

1. a, b, c: (a+ b) + c = a+ (b+ c) .2. a, b: a+ b = b+ a .3. 0 a: a+ 0 = 0 + a = a .4. a (a): a + (a) = (a) + a = 0 -

.

5. a, b, c: (a+ b)c = ac+ bc .6. a, b, c: c(a+ b) = ca+ cb .7. a, b, c: (ab)c = a(bc) .8. a, b: ab = ba .9. 1 a: a 1 = 1 a = a .

24 2. , ,

10. a 6= 0 a1: aa1 = a1a = 1 -.

, , 17 ( 16, -, ), , -, 110, .

. 1. Z,+, , Q,+, ,R,+, . , . .

2. P (A) - A 4 : B,C A B 4 C = (B \ C) (C \ B). P (A),4, .

3. - ( ).

, () (), () - , - ( ).

. nZ - . - , , .

, () .

. , , 0 1 ( !), :

+ 0 10 0 11 1 0

0 10 0 01 0 1

, Z2, - Zn n (. ).

Z2, , a Z2 a 0 = 0 a = 0.

2.1. 25

, , (. ). , ( ).

2.1.1. R . a R a 0 = 0 a = 0.

. a0 = 0, . a0 = b.

b+ b = a0 + a0 = a(0 + 0) = a0 = b.

b + b = b b, (b+ b) + (b) = b+ (b). b+ (b+ (b)) = 0, , b+ 0 = 0, b = 0.

, ab = 0, , , a = 0 b = 0. -, ab = 0, . , , . .

2.1.2. .. F a, b F ,

ab = 0, a 6= 0 b 6= 0. ab = 0 - b1, b ( 10). (ab)b1 = 0b1. a(bb1), , 9 10 a. 2.1.1 0. .

2.1.1 - , , - 2.1.2 , ., , , - , . .

2.1.8. n. - k k , k n. Zn = {k | k Z}. , , : i + j = i+ j,i j = i j, Zn, ,

26 2. , ,

Zn . - n.

2.1.9. , Zn , n .

2.1.10. , .

2.2.

2.2.1. pi - M , . . M , - M .

. 1. pi N ,

npi ={

2k 1, n = 2k,2k, n = 2k 1,

, - n = 2n , , .

2. pi M = {1, 2, 3} 1pi = 2,2pi = 3, 3pi = 1, , - M 1 = 2, 2 = 3, 3 = 2, .

, M , - . , pi - 2 :

pi =(

1 2 32 3 1

).

, pi. - . -, . ,

pi =(

1 2 32 3 1

)=(

2 1 33 2 1

).

2.2. 27

2.2.1. S(M) M .

. pi, S(M)., -, pi -. , pi M . , pi . x, y M x(pi ) = y(pi ) (xpi) = (ypi). , xpi = ypi. pi -, x = y. , pi . -, pi , , pi1 1, , , . x M y = (x1)pi1 M , y(pi ) = ((x1)pi1)(pi ) = ((x1)(pi1 pi)) =(x1) = x. , pi . pi . , S(M), .

, . , - x M x = x. , pi - pi1, , , . 2 3 . , - . :

2.2.1. f : X Y , g : Y U , h : U W , (f g) h = f (g h).

. x - X. :

x((f g) h) = (x(f g))h = ((xf)g)h = (xf)(g h) = x(f (g h)).

, - 2.2.1.

S(M) , - pi , - pi pi pi .

28 2. , ,

. pi =(

1 2 32 3 1

), =

(1 2 32 1 3

).

pi =(

1 2 32 3 1

)(

1 2 32 1 3

)=(

1 2 31 3 2

).

2.2.1.1. pi

, pi 6= pi.2. pi .3. pi100 100.4. pi1 1, pi .

2.2.2. S(M) M M . - - M .

M M , - . , M n , - M Sn ( n) , M = {1, 2, . . . , n}. M - . , .

2.2.2. , |Sn| = n!. - S3. S3. - S3.

pi Sn

pi =(

1 2 . . . ni1 i2 . . . in

),

, {i1, i2, . . . , in} = {1, 2, . . . , n} 1pi = i1, 2pi =i2, . . . , npi = in. (

i1 i2 . . . inj1 j2 . . . jn

).

2.2. 29

, pi1 -:

pi1 =(

i1 i2 . . . in1 2 . . . n

).

M = {1, 2, . . . , n} - pi Sn. i M . ipi = i, . . i pi, i pi. ipi 6= i, -, i . pi supp(pi) pi. ,

pi =(

i1 i2 . . . is k1 . . . ktj1 j2 . . . js k1 . . . kt

),

il 6= jl l {1, 2, . . . , s}, supp(pi) ={i1, i2, . . . , is}. , supp(pi)pi = {j1, j2, . . . , js} supp(pi) pi , . .supp(pi)pi = supp(pi). , supp() = supp(pi1) = supp(pi), , pi1 pi .

i1 supp(pi) i1pi = i2, i2pi = i3 . ., s - , ispi {i1, i2, . . . , is}. - s - M , , supp(pi). , ispi = i1, i2, . . . , is i1, . . . , is1. {i1, i2, . . . , is} pi s ( - ). - : , , - pi. , , , M - ( ). , M pi - , supp(pi) .

.

pi =(

1 2 3 4 5 66 3 2 1 5 4

).

30 2. , ,

{1, 6, 4}, {2, 3}, {5} pi M , . M = {1, 6, 4} {2, 3} {5} supp(pi) = {1, 6, 4} {2, 3}.

supp(pi) - - , .

2.2.3. pi Sn (i1 i2 . . . is1 is k1 . . . kti2 i3 . . . is i1 k1 . . . kt

),

{i1, i2, . . . , is, k1, . . . , kt} = {1, 2, . . . , n} s > 2, s (i1, i2, . . . , is).

i1

i2

i3

is

, , .

. 1. (1 2 3 42 4 3 1

)=(

1 2 4 32 4 1 3

)= (1, 2, 4) = (2, 4, 1) = (4, 1, 2)

3.

2. (

1 2 3 44 3 2 1

) .

2.2.2. || s ( Sn) s.

. = (i1, i2, . . . , is). - k , i1pik = i1, s. ilpis = il l = 1, 2, . . . , s.

2.2.4. Sn , supp() supp() = .

2.2. 31

2.2.3. , = .. i M .

.1. i 6 supp() supp(). i = i = i.2. i supp(). i = i = i.3. i supp(). i = i = i.. 1, 2 . . . , k ,

1 2 . . . k - 1, 2 . . . , k.

2.2.3. 2.2.3.

2.2.2. pi Sn. pi = 12 . . . k 1, 2, . . . , k. c .

. , M = {1, 2, . . . , n} - pi. pi , {i1, i2, . . . , is} . 1 = (i1, i2, . . . , is) . - , pi = 1 , -. , supp(1) supp(pi), pi1 = 11 pi. i supp(1), . . i = il l {1, 2, . . . , s}, ipi1 = ilpi1 = il(11 pi) = (il

11 )pi = il1pi = il = i. -

supp(1) pi1. , i 6 supp(1), ipi1 = i(11 pi) = (i11 )pi = ipi. , pi1 pi. , pi1 supp(pi1) = supp(pi) \ supp(1), , supp(pi) . , pi1 = 2. . .k 2, . . . , k, . - pi = 1pi1 supp(1)supp(pi1) = , pi = 12 . . .k .

, pi = 12 . . .m pi - . i1 supp(pi), l , i1 supp(l). 1, 2, . . . , m, , , i1 supp(1).

32 2. , ,

i11 = i1pi = i11 = i2, i21 = i2pi = i1 = i3, . . . , is1 = ispi =is1 = i1. 1 = 1 pi = 1pi1 = 1pi1, - pi1 .

. , i - pi , - (i).

.

pi =(

1 2 3 4 5 66 3 2 1 5 4

) pi = (1, 6, 4)(2, 3) , - , pi = (1, 6, 4)(2, 3)(5).

, , M - pi, , pi . , - , .

- , .

2.2.5. 2 .

2.2.4. pi Sn .

. 2.2.2 - . = (i1, i2, . . . , is) s. , = (i1, i2)(i1, i3) . . . (i1, is) .

- . , .

. (1, 2)(1, 3)(1, 2) = (2, 3). -

. -.

2.2. 33

2.2.6. pi Sn, supp(pi) m , - k . d(pi) - pi m k. pi - sgn(pi) = (1)d(pi). , sgn(pi) = 1, , sgn(pi) = 1.

. pi d(pi) = 5 2 = 3, sgn(pi) = (1)3 = 1, . . pi .

. pi M M ( ).

2.2.3. - .

. pi , = (i, j) - Sn. , , . , , d(pi) = d(pi)1. pi = 12 . . . k - ( ). , supp() = {i} = (i). i j, - , pi, . .

i, j {k1, k2, . . . , ks}, . . = (k1, k2, . . . , ks) pi , i, j supp(). i = k1, j = km (-, ). - , (k1, . . . , km, . . . , ks)(i, j) = (k1, . . . , km1)(km, . . . , ks). , - (, - i, j, , ). , d(pi) = d(pi) 1.

i {i1, i2, . . . , is} i = i1, j {j1, j2, . . . , jt} j = j1 (, ). pi - (i1, i2, . . . , is) (j1, j2, . . . , jt). (i1, i2, . . . , is)(j1, j2, . . . , jt)(i, j) = (i1, i2, . . . , is, j1, . . . , jt) , - pi pi . ,d(pi) = d(pi) + 1.

34 2. , ,

. - . , - .

1. , - pi, - d(pi).

. , , , sgn() = 1. pi = 12 . . . k pi -. 2.2.3 sgn(1) = sgn(1) = (1)1. ,sgn(12) = (1)2 . . , sgn(pi) = (1)d(pi) = (1)k.

2.2.4. , , - pi, d(pi).

-.

2. pi, Sn. - sgn(pi) = sgn(pi) sgn().

2.2.5. 2 2.2.3.

2.2.6. , -M . -, An Sn - Sn. An - . , n > 3 An 3.

2.2.7. , Sn - 2.

2.3.

2.3.1. S. - S m n ( (m n)-) -

2.3. 35

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . . . . . . . . . . .am1 am2 . . . amn

, aij S, i {1, 2, . . . ,m} , j {1, 2, . . . , n} . , , . . m = n, , n . m n S Mmn(S), (n n)- Mn(S).

. , A = (aij) A = (aij)mn, . A B , , , : A = (aij)mn,B = (bij)mn, A = B aij = bij i = 1 . . .m, j = 1 . . . n.

.

1.(

1 1 11 1 1

) 2 3 {1}.

2.(1 2 3 4 5

) 1 5 N.

3.

1234

4 1 N. 1 n -

n, m1 m. - 2 5, 3 4.

. , , -, -. . , n S - n- S, . . n- Sn - S. Mmn(S) m n S

36 2. , ,

m-, - n- S, , , m- (Sn)m Sn. , . I = {(i, j) | 1 6 i 6 m, 1 6 j 6 n}. A = (aij) Mmn(S) I S, (i, j) aij S, Mmn(S) .

S +. :

a11 . . . a1na21 . . . a2n. . . . . . . . . . .am1 . . . amn

+

b11 . . . b1nb21 . . . b2n. . . . . . . . . .bm1 . . . bmn

=

a11 + b11 . . . a1n + b1na21 + b21 . . . a2n + b2n. . . . . . . . . . . . . . . . . . . . . . .am1 + bm1 . . . amn + bmn

.,

2.3.2. A = (aij)mn, B = (bij)mn S c . A B C = (cij)mn = A+ B S , cij = aij + bij .

S - , - .

2.3.3. A = (aik)ms B = (bkj)sn S , - . S - . A B C = (cij)mn = AB S m n,

cij =s

k=1

aikbkj = ai1b1j + ai2b2j + . . .+ aisbsj .

, , i- j- , i- j- .

2.3. 37

. S = Z .(1 2 34 5 0

) 0 12 3

4 5

= (1 0 + 2 2 + 3 4 1 1 + 2 3 + 3 54 0 + 5 2 + 0 4 4 1 + 5 3 + 0 5)=(16 2210 19

).

, A B , , .

2.3.1. R,+, , n - . Mn(R) - .

. - , - , .

1. R,+ , Mmn(R),+ - .

. A = (aij), B = (bij), C = (cij) Mmn(R). R , i, j aij + bij =bij+aij (aij+bij)+cij = aij+(bij+cij). , A+B = B+A (A+B)+C = A+(B+C). , -, , R. , A = (aij) - A = (aij). , Mmn(R) .

2. A = (aij), B = (bij) Mms(R) C = (cij) Msn(R). (A+B)C = AC +BC.

. D = (dij) = (A+B)C, F = (fij) = AC +BC. i, j

dij =n

k=1

(aik+bik)ckj =n

k=1

(aikckj+bikckj) =n

k=1

aikckj+n

k=1

bikckj = fkj .

, D = F . A(B + C) = AB + AC

, , -.

-. , .

38 2. , ,

2.3.1 ( ). S,+ - X = (xij)mn S.

mi=1

nj=1

xij =nj=1

mi=1

xij .

. S - , -. , X.

3. A = (aij) Mms(R), B = (bij) Mst(R) C =(cij) Mtn(R), (AB)C = A(BC).

. D = AB, F = BC G = (AB)C = DC,H = A(BC) = AF . , G = H. , -, G H m n. ,i {1 . . .m},j {1 . . . n}

gij =t

k=1

dikckj =t

k=1

(sl=1

ailblk

)ckj =

tk=1

sl=1

(ailblk)ckj =

=sl=1

tk=1

ail(blkckj) =sl=1

ail

(t

k=1

(blkckj)

)=

sl=1

ailflj = hij .

13 , , - , Mn(R) . -.

, R ( R ) Mn(R). -, n = 1, : R M1(R), a = (a)11, , . R M1(R). , R , M1(R) .

-, R , Mn(R) - n N. ,

E =

1 0 . . . 00 1 . . . 0. . . . . . . . . . . . .0 0 . . . 1

,

2.3. 39

eij = 1 i = j eij = 0 , , , Mn(R). E .

, n > 1, - R Mn(R)., , F , 0 1 - . M2(F )

A =(

0 10 0

) B =

(0 01 0

).

AB =

(1 00 0

)6=(

0 00 1

)= BA.

, M2(F ) . , -

A2 =(

0 00 0

)= 0,

A . , A1 , A1A = E, , A2 = 0 A1, A = 0.

, F , - , -. F , .

, - ( ) .

2.3.4. D(1, 2, . . . , n)

D = (dij)nn =

1 0 . . . 00 2 . . . 0. . . . . . . . . . . . . . . . .0 0 . . . n

, dii = i dij = 0 i 6= j. D(1, 2, . . . , n) , 1 = 2 = . . . = n = .

, - : E = D(1, 1, . . . , 1), 0 = D(0, 0, . . . , 0).

40 2. , ,

{aii | i = 1, . . . , n} A = (aij)nn, , A. - , - , 0.

, A = (aij) Mmn(F ) ( ) - D.

D(1, 2, . . . , m) A =

1 0 . . . 00 2 . . . 0. . . . . . . . . . . . . . .0 0 . . . m

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . . . . . . . . .am1 am2 . . . amn

=

=

1a11 1a12 . . . 1a1n2a21 2a22 . . . 2a2n. . . . . . . . . . . . . . . . . . . . . . . . . . . .nam1 nam2 . . . namn

= (iaij)mn. , A = (aij)mn

D(1, 2, . . . , m) i- A i i = 1, . . . ,m.

,

A D(1, 2, . . . , n) =

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . . . . . . . . .am1 am2 . . . amn

1 0 . . . 00 2 . . . 0. . . . . . . . . . . . . . .0 0 . . . n

=

=

1a11 2a12 . . . na1n1a21 2a22 . . . na2n. . . . . . . . . . . . . . . . . . . . . . . . . . .1am1 2am2 . . . namn

= (jaij)mn,. . A = (aij)mn D(1, 2, . . . , n) j- A j j = 1, . . . , n.

2.3.1. , D1, D2 Mn(F ) D1D2 D1D2 = D2D1.

2.3. 41

D = D(, , . . . , ), A = (aij) DA = AD = (aij). , ( ) A . , -, A. , D(, , . . . , ) E, E .

2.3.2. , - Mn(F ) , F Mn(F ), = E, .

2.3.3. , A Mn(F ) - () Mn(F ) , A .

2.3.5. A n n

A1 0A2

. . .0

As

,

i = 1, . . . , s Ai ni ni,

si=1 ni = n, Ai -

A A, Ai, 0.

2.3.2.

A =

A1 0

A2. . .

0 As

B =

B1 0B2

. . .0 Bs

, Ai Bi

42 2. , ,

i = 1, . . . , s.

AB =

A1B1 0

A2B2. . .

0 AsBs

.

2.3.4. 2.3.2.

2.3.6. F , r s - {1, 2, . . . , n}, r 6= s. Ers()

Ers() = (tij)nn =

s

r

1...

. . ....

. . . . . . 1 . . . . . . . . .. . .

...1...

. . .... 1

,

trs = , tii = 1 tij = 0 . - .

, , - E = Ers(0).

, A

2.3. 43

.

Ers()A =

1. . .

1 . . .

. . .1

a11 a12 . . . a1n

. . . . . .ar1 ar2 . . . arn

. . . . . .as1 as2 . . . asn

. . . . . .

am1 am2 . . . amn

=

=

a11 a12 . . . a1n

ar1 + as1 ar2 + as2 . . . arn + asn

as1 as2 . . . asn

am1 am2 . . . amn

.

, A Ers() r- A s- , .

,

AErs() =

a11 . . . a1r . . . a1s . . . a1na21 . . . a2r . . . a2s . . . a2n...

......

......

......

...am1 . . . amr . . . ams . . . amn

1. . .

1. . .

1

=

=

a11 . . . a1r . . . a1s + a1r . . . a1na21 . . . a2r . . . a2s + a2r . . . a2n...

......

......

......

...am1 . . . amr . . . ams + amr . . . amn

. , A -

Ers() s- A r-, .

44 2. , ,

2.3.3. Ers() - Ers(), . . Ers() Ers() = E.

2.3.5. 2.3.3.

, , ( ) ( ), , -. , - , - .

2.3.2. A Mn(F ) F . E1, . . . , Ek,Ek+1, . . . , Es Mn(F ) D Mn(F ) , A = E1 . . . EkDEk+1 . . . Es.

. - n A.

n = 1 A = (a11) = D(a11) ., -

A Mn1(F ), A Mn(F ). .

1.

A =

a11 . . . a1,n1 0

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

an1,1 . . . an1,n1 00 . . . 0 ann

=

0

A...0

0 . . . 0 ann

.

,

A =

a11 . . . a1,n1. . . . . . . . . . . . . . . . . . . . . .an1,1 . . . an1,n1

Mn1(F ). , - E1, . . . , Ek, Ek+1, . . . , Es Mn1(F ) - D Mn1(F ) , A = E1 . . . EkDEk+1 . . . Es.

2.3. 45

i = 1, . . . , s

Ei =

0

Ei...0

0 . . . 0 1

D =

0

D...0

0 . . . 0 ann

. Ei , D - Mn(F ). 2.3.2 E1 . . . EkDEk+1 . . . Es =

=

0

E1 . . . EkDEk+1 . . . Es...0

0 . . . 0 ann

=

0

A...0

0 . . . 0 ann

= A. . , -

ann A, , , ann = 0.

2. A Mn(F ) - ann 6= 0. A Mn1(F ) ,

A =

a1n

A...

an1,nan1 . . . an,n1 ann

. A,

, = a1na1nn ( - ann , ann 6= 0). - A1. A1 = E1n()A =

=

0a2n

A1...

an1,nan1 . . . an,n1 ann

,

A1 Mn1(F ). , - A1 (1, n) 0. 2.3.3

46 2. , ,

A = E1n(a1n/ann)A1, a1n/ann = = (a1na1nn). - A1 E2n(a2n/ann), A2 2- n- 0. , A = E1n(a1n/ann)E2n(a2n/ann)A2. , A, ann. - A, A( ann). , A = E1n(a1n/ann) . . .. . . En1,n(an1,n/ann)BEn,n1(an,n1/ann) . . . En1(an1/ann), B , . B -, A. 2 .

3. A Mn(F ) . ann 6= 0, , . , , ann = 0. - A 0, , . , , , , . , , a1n 6= 0. - A , A En1(1). B = En1(1)A , - (n, n), 0. B B = E1 . . . ElDEl+1 . . . Et . A = En1(1)E1 . . . ElDEl+1 . . . Et - A, .

2.3.6. , - , -

A =

0 1 21 0 12 1 0

.

2.4.

-, . , - .

2.4. 47

2.4.1. n , Sn - {1, . . . , n}, F ,A = (aij)nn Mn(F ). () A F , - det(A) |A| :

det(A) =Sn

sgn a11a22 . . . ann.

. - Sn, aii A, i- (i)- , i i .

. 1. n = 1. S1 = {}, A = (a11) det(A) = a11.2. n = 2. S2 = {, (1, 2)}, sgn = 1 sgn(1, 2) = 1.

A =(

a11 a12a21 a22

).

det(A) = a11 a12a21 a22

= a11a22 a12a21. (2 2)-

. {

a11x1 + a12x2 = b1,a21x1 + a22x2 = b2

.

x1 =b1a22 b2a12a11a22 a12a21 , x2 =

b2a11 b1a21a11a22 a12a21

, a11 a12a21 a22

6= 0. , -, 0, , .

. x y

48 2. , ,

(x1, x2) (y1, y2) . S ,

x y, x1 x2y1 y2

. , x y , 0.

2.4.1. -, .

3. n = 3. Sn = {, (1, 2, 3), (1, 3, 2), (1, 2), (1, 3), (2, 3)}, An = {, (1, 2, 3), (1, 3, 2)}. ,

A =

a11 a12 a13a21 a22 a23a31 a32 a33

, det(A) =a11 a12 a13a21 a22 a23a31 a32 a33

== a11a22a33 + a12a23a31 + a13a21a32 a12a21a33 a13a22a31 a11a23a32.

, - ( ), - . , n n - , n = 1, 2, 3, - , 0 ( - 4). n > 3 , n , - n . , .

2.4.1. A = (aij), B = (bij) Mn(F ), F ,r {1, 2, . . . , n}. j = 1, . . . , n brj = arj bij = aij i 6= r. det(B) = det(A).

, - .

. det(B) =Sn sgnb11 . . . brr . . . bnn =

Sn sgna11 . . . (arr) . . . ann =

Sn sgna11 . . . arr . . . ann = det(A).

2.4. 49

. A ( 0), det(A) = 0.

2.4.2. A Mn(F ), F . det(A) det(A).

2.4.2. A = (aij), B = (bij), C = (cij) Mn(F ), r {1, 2, . . . , n}. j = 1, . . . , n crj = arj+brj cij = aij = bij i 6= r. det(C) = det(A) + det(B).

, r- -, , , r- -, r- , , .

. det(C) =

=Sn

sgnc11 . . . crr . . . cnn =Sn

sgnc11 . . . (arr+brr) . . . cnn =

=Sn

sgnc11 . . . arr . . . cnn +Sn

sgnc11 . . . brr . . . cnn =

=Sn

sgna11 . . . arr . . . ann +Sn

sgnb11 . . . brr . . . bnn =

= det(A) + det(B).

2.4.3. A = (aij) Mn(F ), r, s - {1, 2, . . . , n}. j = 1, . . . , n arj = asj, det(A) = 0.

, , - 0.

. (r, s) Sn. 2.1.1 : 7 - Sn . Sn, pi = Sn. -, sgnpi = sgn() = sgn, An, pi Sn \An.

det(A) =Sn

sgna11 . . . arr . . . ass . . . ann =

50 2. , ,

=An

a11 . . . arr . . . ass . . . ann

piSn\Ana11pi . . . arrpi . . . asspi . . . annpi =

=An

a11 . . . arr . . . ass . . . annAn

a11 . . . arr . . . ass . . . ann =

=An

(a11 . . . arr . . . ass . . . ann a11 . . . arr . . . ass . . . ann) = 0.

. A = (aij), B = (bij) Mn(F ), r, s {1, 2, . . . , n}. j = 1, . . . , n arj = bsj, asj = brj aij = bij r 6= i 6= s. det(A) + det(B) = 0.

, .

. 2.4.2 2.4.3, :

... arj

... asj

...

+

... brj

... bsj

...

=

... arj

... asj

...

+

... asj

... arj

...

=

=

... arj

... asj

...

+

... arj

... arj

...

+

... asj

... arj

...

+

... asj

... asj

...

=

=

... arj

... arj + asj

...

+

... asj

... arj + asj

...

=

... arj + asj

... arj + asj

...

= 0.

2.4. 51

2.4.4. A = (aij), B = (bij) Mn(F ), F , r, s {1, 2, . . . , n}. j = 1, . . . , n brj = arj + asj, aij = bij i 6= r. det(B) = det(A).

, - .

. C = (cij) D = (dij) Mn(F ) , j = 1, . . . n crj = asj , drj = asj cij = dij = aij i 6= r. - 2.4.3 D ( r- s- ). 2.4.1 det(C) = det(D) = 0. , 2.4.2 , det(B) = det(A) + det(C) = det(A).

2.4.2. A = (aij) (n n)-, i, j {1, . . . , n} -, i > j, aij = 0. , , , 0. , -, , 0. , , - .

2.4.5. A = (aij) Mn(F ) . , - , . . det(A) = a11a22 . . . ann.

. , , A . .

, Sn : - i = 1, . . . , n i 6 i. n = n, . , (n1) = n1, n n. , , (n 2) = n 2, . . ., 2 = 2, 1 = 1. , - . , Sn i {1, . . . , n} , i > i.

A . - a11a22 . . . ann 0. det(A) =

Sn sgna11a22 . . . ann = a11a12 . . . a1n.

52 2. , ,

. D = D(1, . . . , n) -, T = Ers() . det(D) = 1 . . . n,det(T ) = 1. , -.

2.4.1. A,B Mn(F ). |AB| = |A| |B|.. , A = D(1, . . . , n)

. B A i {1, 2, . . . , n} i- B i., 2.4.1 |AB| == 1 . . . n|B| = |A||B|.

A = Ers() , - 1. , . - 2.4.4 |AB| = |B| = 1|B| = |A||B|.

A . - 2.3.2 E1, . . . , Ek, Ek+1, . . . , Es - D Mn(F ) , A = E1 . . . EkDEk+1 . . . Es. |A| = |E1(E2 . . . EkDEk+1 . . . Es)| = |E2 . . . EkDEk+1 . . . Es| =. . . = |D(Ek+1 . . . Es)| = |D||Ek+1 . . . Es| = |D|. , |AB| == |E1 . . . EkDEk+1 . . . EsB| = |D(Ek+1 . . . EsB)| = |D||Ek+1 . . . EsB| == |D||B| = |A||B|.

2.4.3.

A =

A1 0

A2. . .

0 As

. |A| = |A1||A2| . . . |As|. 2.4.3. A = (aij) (m n)- -

S. B = (bij) n m S - A, bij = aji i = 1, . . . , n j = 1, . . . ,m. , - A, A A>.

-.

2.4. 53

2.4.4. A B - :

1) A = (A) = A;2) (A+B) = A +B;3) (AB) = BA.. , D

, Ers() , D = D Ers() == Esr().

2.4.6. A Mn(F ). det(A) = det(A).. , -, 3)

2.4.4 : (A1A2 . . . As) == As . . . A2A

1. , A = E1 . . . EkDEk+1 . . . Es, E1, . . . , Ek,

Ek+1, . . . , Es , D , |A| = |D|. A = Es . . . Ek+1DEk . . . E1 |A| = |D| == |D| = |A|.

. - , .

2.4.5. - .

, - n .

2.4.4. A = (aij) Mn(F ). , - ars A, Mrs(A) Mn1(F ), A r- s-.

.

A =

1 2 53 4 67 8 9

M23(A) = ( 1 27 8).

2.4.5. A = (aij) Mn(F ). A ars Ars = (1)r+s|Mrs(A)|.

, (r, s) ,

54 2. , ,

, , r + s , , .

. a23 A -

A23 = (1)2+3 1 27 8

= (1) (1 8 2 7) = 6. 2.4.2 ( ).

A = (aij) Mn(F ). i, k {1, . . . , n}

nj=1

aijAkj ={ |A|, k = i

0, k 6= i.

. i- :

|A| =nj=1

aijAij ,

i = 1, . . . , n, , k = i.

. , i = k, . . , |A| = nj=1 aijAij i = 1, . . . , n. .

1. i = n, an1 = an2 = . . . = an,n1 = 0. A

Mnn

...

0 . . . 0 ann

,Mnn =Mnn(A) , ann A, .

=(

1 . . . n 1 ni1 . . . in1 n

) Sn =(

1 . . . n 1i1 . . . in1

) Sn1. |A| =

=Sn

sgna11 . . . an1,(n1)ann =

Snn = n

sgna11 . . . an1,(n1)ann =

2.4. 55

= ann

Sn1sgna11 . . . an1,(n1) = ann|Mnn| = ann(1)n+n|Mnn| =

= annAnn = 0 An1 + . . .+ 0 An,n1 + annAnn =nj=1

anjAnj .

2. i = n, anj = 0, , , ans. s = n, -

. s < n. A1 A - s- (s + 1)- . - 2.4.3 2.4.6 |A| = |A1|. s = n1, , - , Mnn(A1) = Mn,n1(A)., |A| = |A1| = an,n1|Mn,n1| = an,n1An,n1, . s < n 1, , (s+ 1)- A1 (. . s- A) (s+2)- , s- - A n- . n s . B

Mns(A)

...

0 . . . 0 ans

. |A| = (1)ns|B| = (1)ns+2sans|Mns(A)| = ansAns ==n

j=1 anjAnj , . , - - A. , , s- n- , Mns(A) .

3. i = n. Aj n, A, n1 A, (n, j) anj A, . 2.4.2 |A| = |A1|+|A2|+. . .+|An| =

nj=1 anjAnj .

4. . i = n, - . i < n. i- (i+ 1)- A , . (i + 1)- (i + 2)-

56 2. , ,

, , i- - A n- . B. 2.4.3 |A| = (1)ni|B|.

k- A B ak bk , : bk = ak 1 6 k < i, bk = ak+1 i 6 k < n bn = ai. ,Mnj(B) =Mij(A) j = 1, . . . , n. |A| =

= (1)ni|B| = (1)ninj=1

bnjBnj = (1)ninj=1

bnj(1)n+j |Mnj(B)| =

nj=1

aij(1)n+j+ni|Mij(A)| =nj=1

aij(1)i+j+2(ni)|Mij(A)| =nj=1

aijAij .

, -.

. , , i 6= k nj=1 aijAkj = 0. B n, A, : B, k- , - A, k- i- A. i- k- B ( i- A), 2.4.3 B . , B k- ,

|B| =nj=1

bkjBkj =nj=1

aijAkj ,

Mkj A B ( , A B, k, ). .

( ). A =(aij) Mn(F ). j, k {1, . . . , n} -

ni=1

aijAik ={ |A|, j = k

0, j 6= k.

2.4. 57

. - . , , - A.

2.4.6. A - : - , .

, - , .

2.4.6. A = (aij) Mn(F ). A =(aij) Mn(F ), aij = Aji i, j {1, . . . , n}, - A.

, - .

2.4.2 -.

2.4.2 ( -). A Mn(F ), A .

AA = AA = |A|E =

|A| 0 . . . 00 |A| . . . 0. . . . . . . . . . . . . . . . . .0 0 . . . |A|

.

. B AA,

bij =n

k=1

aikakj =n

k=1

aikAjk ={ |A|, i = j

0, i 6= j.

AA . ,

.

2.4.7. A -, det(A) = 0, .

58 2. , ,

2.4.8. A Mn(F ). A1 A, AA1 == A1A = E.

2.4.3 ( ). A Mn(F ) A . A ( - ) , . -

A1 =1|A| A =

1|A|

A11 A21 . . . An1A12 A22 . . . An2. . . . . . . . . . . . . . . . . . . . .A1n A2n . . . Ann

.. |A| = 0. ,

A A1. 1 = |E| = |AA1| = |A||A1| = 0;.

A , 1|A| A -. , A = (aij) A (aij). , E - . , 2.3.3 A = EA = A(E).

2.4.2

A(1|A| A) = A(

1|A|EA) =

1|A|E(AA) =

1|A|E|A|E = E

(1|A| A)A =

1|A| |A|E = E.

2.4.7. A - . -, - .

2.4.8. GLn(F ) = {A Mn(F ) | det(A) 6= 0}., GLn(F ) .

2.4. 59

2.4.9. GLn(F ), - 2.4.8, F , F .

, , : AX = B Y A = B , A (-, A, B, X, Y -). X = A1B Y = BA1.

-, , - . .

2.4.9. , A , , - (). , , - , ( ).

2.4.10. A,B Mn(F ) |A| 6= 0. () , () , , () ( ), , - (). , (n 2n)- (A | B), A B, - (E | X), E , X A1B, . . AX = B. -, B = E, X = A1. , (2n n)- (AB ) (EY ), Y Y A = B.

. -: . - det Mn(F ) F , , 2.4.12.4.3, 1 . , A = (a) M1(F ) a, A Mn(F ) det(A) -

60 2. , ,

n1, (. 2.4.2). 2.4.11. , -

( ) .

2.5.

2.5.1. C, .

1. C R, R .

2. C i , i2 = 1, 1 , C, , R.

3. z C a+bi, a, b R, R . 1, i . 2.

2.5.1. C - .

. C - (

a bb a

), a, b R.

:(a bb a

)+(

c dd c

)=(

a+ c (b+ d)b+ d a+ c

), (1)

(a bb a

)(

c dd c

)=(

ac bd (ad+ bc)ad+ bc ac bd

)(2)

C .

, C . - M2(R) C, M2(R) , - C, -

z =(

a bb a

) C.

2.5. 61

(2) , (

c dd c

)(

a bb a

)=(

ac bd (ad+ bc)ad+ bc ac bd

).

, C . ( a (b)

b a)

C, .

, z C C. 2.4.3 . -, a bb a

= a2 + b2 6= 0, a = b = 0, z .-,

z1 =1

a2 + b2

(a bb a

) C.

, C . , C - 13 2.5.1 , , - .

R ={(

a 00 a

) a R} ., R C. -

, 2.3.2 R M2(R) , R - . - , (aE) = a R.

i C, (0 11 0

).

,

i2 =(

0 11 0

)(

0 11 0

)=( 1 0

0 1).

62 2. , ,

, , - 1 R R.

z C, (a bb a

),

a =

(a 00 a

) b =

(b 00 b

).

z =(

a bb a

)=(

a 00 a

)+(

b 00 b

)(0 11 0

)= a+ bi.

, z z = a + bi == c+ di. a c = (d b)i. , (a c)2 = (d b)2. a, b, c, d - , a c = d b = 0. a = c b = d, , z = a+ bi . , C .

C , - . i (i)2 = 1. 3 C - a + bi, a b - 1 . - : C C, (a + bi) = a + bi, . , (a+ bi) + (c+ di) = (a+ c) + (b+ d)i (a+ bi)(c+ di) = (ac bd) + (ad+ bc)i (1) (2), .

. C , , a+ bi. , , a+0i = a - , , R C. Q R, - C. .

2.5.2. a b - z = a + bi - a = Re z b = Im z . i

2.5. 63

i2 = 1 .

C , 1. i, i. -, (a + bi)2 = 1 + 0i, C . - i = i, , z = a+ bi z = a bi -. , z z . , z u - : z + u = z + u zu = z u. , , z = z.

2.5.1. , - , : - .

. 2.1.10, , - . , - : a a > 0 b R : b2 = a.

C z u - , Re z = Reu Im z = Imu. C R2, - ( ). , z = a+bi (a, b). -, , .

z = a + bi - z (a, b). - - . , , , .

64 2. , ,

Re z

Im z

O

a

b

z = a+ bi

r

'

2.5.3. z = a + bi C. - z = a + bi r =

a2 + b2, z (a, b).

z |z|.. -

z = a+0i, - ,

a2 + 02

.

2.5.4. - z = a+ bi , z (a, b) . - 2pi. 0 . z arg z.

. 0 , , 0 .

r z = a + bi. , a = r cos b = r sin.

z = r(cos+ i sin).

- .

, z = r(cos+ i sin) u = s(cos+ i sin), - , , r = s = + 2kpi,k Z.

2.5.1. z = r(cos + i sin) u = s(cos + i sin). zu = rs(cos(+ ) + i sin(+ )).

2.5. 65

, , .

. .

( ). z = r(cos + i sin). zn = rn(cosn+ i sinn).

n- z - u , un = z. , n- .

2.5.2. z = r(cos+ i sin) - . xn = z n x0, x1 . . . , xn1 . k = 0, 1, . . . , n1

xk = nr(cos

+ 2kpin

+ i sin+ 2kpi

n).

. u = s(cos+ i sin) xn = z. sn = r n = + 2kpi(k Z). , s = nr ( ) = +2kpin ., k,m {0, 1, . . . , n 1} +2kpin

+2mpin , k = m.

, m = k + nj (j Z) +2kpin +2mpin =+2kpi

n + 2jpi 2pi. , {

nr

(cos

+ 2kpin

+ i sin+ 2kpi

n

) k = 0, 1, . . . , n 1} . -

n- .

2.5.2.1. Cn = {k | k = 0, 1, . . . , n1}, k = cos 2kpin +i sin 2kpin ,

xn = 1.2. x0 xn = z,

{xk = x0k | k = 0, 1, . . . , n1} xn = z.

66 2. , ,

3. Cn . < Cn, >'< Zn,+ > (. 2.1.8).

3

3.1.

- . - , . , : - . , , . R f, v v.

3.1.1. F . ( ) F V ( - ), + (- ) v 7 v ( ) F , .

1. < V,+ > .2. F u, v V (u+ v) = u+ v.3. , F v V (+ )v = v + v.4. , F v V ()v = (v).5. v V 1 F 1v = v.

, .

3.1.1. V F ; , F ; u, v V ; 0 F , 1 F , 0 V .

1) 0 = 0;2) (v) = v;3) (u v) = u v;4) 0v = 0;

68 3.

5) (1)v = v;6) ( )v = v v.. ,

F , 0.. 1. F .

Fn = {(1, 2, . . . , n) | i F} n- F F :

1) (1, 2, . . . , n) + (1, 2, . . . , n) = (1 + 1, 2 + 2, . . . , n + n);2) (1, 2, . . . , n) = (1, 2, . . . , n). Fn -

F . .

, ( ) - (-) , - R2 (R3).

2. F K. K F - K K F , K (, F K). , R Q , C R.

3. F (X,K) X K - : - (f+g)(x) = f(x)+g(x) (f)(x) = f(x).

4. F [x] - x F , .

5. M m n F - F A 7 A, - A, , , A .

, F -Mn(F ) . , Mn(F ) . -

3.1. 69

, , , .

3.1.2. F - A : , - F , :

1) A ;2) A

;3) F a, b A (ab) = (a)b = a(b). 3.1.2.1. , Mn(F ) F .2. , -

.

3.1.3. V - F . U V - V , -, V , . . u, v U F u+ v U u U .

. , U V F F . -, , , , U , U V . , 5 3.1.1 u U u = (1)u U , , - .

3.1.4. A F . B A A, , A, . . a, b B F a+ b B, ab B a B.

. 1. V -: 0 = {0} V . , , , , -, .

70 3.

A.2. R

() () C R.

3.1.3. :1. {(1, . . . , n) | 1+ . . .+n = 0} -

Fn F .2. Fn[x] x, -

n, F [x] x F . - F [x] F Fn[x] .

3. Mn(F ), . . A Mn(F ), A = A, - Mn(F ) F . Mn(F )?

3.2.

V F . - a1, a2, . . . , as ( !) .

3.2.1. ( -) a1, a2, . . . , as V F - 1, 2, . . . , s F 1a1 + 2a2 + . . . + sas, , . - , 1 = 2 = . . . = s = 0, .

. , V , , - 0.

3.2.2. a1, a2, . . . , as - , - , 0. - .

, a1, a2, . . . , as , 1a1 + 2a2 + . . . + sas = 0 , 1 = 2 = . . . = s = 0.

3.2.3. a

3.2. 71

a1, a2, . . . , as, , a.

. 1.

a1 = (1, 0, . . . , 0)a2 = (0, 1, . . . , 0)

. . .an = (0, 0, . . . , 1)

Rn, , - , 1a1 + 2a2 +. . . + nan = (1, 2, . . . , n) = (0, 0, . . . , 0) = 0 , 1 = 2 = . . . = n = 0.

2. , a1 = (1, 1, 1), a2 = (0, 1, 2), a3 = (1, 2, 3) R3 , 1a1 +1a2 + (1)a3 = 0. -, a3 a1 a2: a3 = 1a1 + 1a2.

.

3.2.1 ( ). - a1, a2, . . . , as , , . . - i {1, . . . , s} , ai = 1a1 + . . .+ i1ai1.

. , .

. , , . , .

. 1a1 + 2a2 + . . . + sas = 0. i {1, . . . , s} , i 6= 0 i+1 = . . . = s = 0 (, i = s). 1a1 + . . .+ i1ai1 + iai = 0.

ai = 1i (1a1 + . . .+ i1ai1) = (1i

)a1 + . . .+ (i1i

)ai1

. i, ai = 1a1+ . . .+i1ai1.

1a1 + . . .+ i1ai1 + (1)ai + 0ai+1 + . . .+ 0as 0 , ai = 1 - 0.

72 3.

3.2.4. A B . A B, A B. A B -, A B, B A.

3.2.5. A a1, a2, . . . , as , A. A - L(A) A.

,

L(A) = a1, a2, . . . , as = {1a1+2a2+ . . .+sas | i F, i = 1, 2, . . . , s}.

, - , . . , - -.

3.2.1.1. L(A) A -

V .2. A B ,

L(A) L(B). , A B , L(A) = L(B).

3. B A , L(A) = L(B).

, .

3.2.1 ( ). A B a1, a2 . . . , ar b1, b2, . . . , bs , A B. r 6 s - B , B a1, . . . , ar, br+1, . . . , bs.

, , r B A , B.

. r = 0. -, , A r 1, r . A A, a1, a2, . . . , ar1. , A -

3.2. 73

B. , - , A A. , A B - , , ,r 1 6 s B , B, a1, . . . , ar1, br, . . . , bs, B, . . L(B) = L(B).

ar L(B) = L(B). , ar B:

ar = 1a1 + . . .+ r1ar1 + rbr + . . .+ sbs. (1)

r1 = s r = . . . = s = 0, ar = 1a1+. . .+r1ar1 ar A, 3.2.1 A. - , r 6 s r, . . . , s . br, . . . , bs , r 6= 0. - B a1, . . . , ar, br+1, . . . , bs. , B B, L(B) = L(B) = L(B) .

a1, . . . , ar1, br+1, . . . , bs - B, B, , ar L(B) br L(B). (1). , r 6= 0, (1) , br (1r

)a1+. . .+

(r1

r

)ar1+

1r

ar+(r+1

r

)br+1+. . .+

(sr

)bs.

, br L(B). B B, , B B .

. , .

3.2.6. V F - , V v1, v2 . . . , vs, -, . . v1, v2 . . . , vs = V .

3.2.7. ( ) - V F V , V .

74 3.

3.2.2 ( ). V F . .

1. V .2. V

.3. V , -

.

4. A a1, a2, . . . , ar - V , V , A .

. 1. v1, v2, . . . , vs - V , V . ( ) , - . , , . - B. 3.2.1 B . , - B, V B., L(B) = V , .

2. B B V . L(B) =L(B) = V , B B . , .

3. B V , b1, b2, . . . , bn. , v V : v = 1b1 + 2b2 + . . . + nbn =1b1 + 2b2 + . . .+ nbn. (1 1)b1 + (22)b2 + . . . + (n n)bn 0. B , . -, 1 = 1, . . . , n = n v .

4. B V , b1, b2, . . . , bn. A B. , - B B, n A . B B, L(B) = L(B) = V . , B , , , . 1, , -

3.2. 75

, . 2. , B V , A .

3.2.8. - V dimV .

. . 2 , .

. V n F . :

1. m > n m V . n V .

2. m < n A m V . B n , L(B) = V , V .

3.2.2. .. 1. V = Fn -

F . V , dimV = n, , , :

a1 = (1, 0, . . . , 0),a2 = (0, 1, . . . , 0),

. . .an = (0, 0, . . . , 1)

Fn, Mn(F ).

2. E2 (E3) (), () . -, dimE2 = 2 dimE3 = 3.

3. C , - R , 2. , , , 1 i, a+ bi, a, b R.

4. V = R[x] - R .

76 3.

. , V , f1, . . . , fn. m - ( - ). , - f1, f2, . . . , fn, m. , xm+1 f1, f2, . . . , fn. -, V .

3.2.3. , b1, b2, . . . , bn Fn - , , n- b1, b2, . . . , bn, .

3.2.4. , Mmn(F ) . .

3.2.5. , F (X,K) X K , X . , |X| = n.

. a ( a X), :

a(x) ={

1, x = a,0, x 6= a.

, , , . , , .

3.2.2. B V . .

1. B V .2. B -

V .3. B , -

V .. ()

, ( -

3.2. 77

) , .

. (1 2). v - V B, v B , .

(2 1). B , , - B, B. ,L(B) = V B .

(1 3). B - L(B) = V . B v, - B. B.

(3 1). B , - B . , B .

3.2.9. V - F dimV = n. B - V , b1, b2, . . . , bn. v V v = 1b1 + 2b2 + . . . + nbn . n-[v]B = (1, 2, . . . , n) v - B.

3.2.3. V n - F Fn. : V Fn, v = [v]B B - V .

. B -, [v]B v [v]. v =[v]. V - , , [u] = [v] u = v. , . n- (1, 2, . . . , n) Fn v = 1b1 + 2b2 + . . .+ nbn V , . , .

[u] = (1, 2, . . . , n), [v] = (1, 2, . . . , n). u + v =[u]+[v] = (1+1, 2+2, . . . , n+n) = [u+v] = (u+v). , (v) = [v] = [v] = (v). -

78 3.

, , V , , , .

. .

. V U - F dimV = dimU = n. V U Fn, , .

, .

V n F . A B V , a1, . . . , an b1, . . . , bn . A , - i = 1, . . . , n bi B A:

bi = ti1a1 + . . .+ tinan. (2)

a b n, - A B. ,

a =

a1...an

b = b1...

bn

. (2) :

b = Ta, T =

t11 . . . t1n. . . . . . . . . . . . .. . . . . . . . . . . . .tn1 . . . tnn

. T A B. .

3.2.3. X = (xij), Y = (yij) (m n)-. a n, - a1, . . . , an. Xa = Y a X = Y .

. Xa = Y a , i = 1, . . . ,m xi1a1+. . .+xinan = yi1a1+. . .+yinan.

3.2. 79

(xi1yi1)a1+ . . .+(xinyin)an = 0. a1, . . . , an , i = 1, . . . ,m j = 1, . . . , n xij = yij , . . X = Y .

v V [v]A, [v]B v A B . - , v : v = [v]Aa = [v]Bb. , [v]Bb = [v]B(Ta) = ([v]BT )a ( -). 3.2.3 [v]Aa = ([v]BT )a [v]A = [v]BT . - :

T A B, v A v B T .

A, B, C V , a, b, c , - . T A B, S B - C, P A C. - c = Sb = S(Ta) = (ST )a 3.2.3 , P A C ST . ,[v]A = [v]CP = [v]C(ST ).

3.2.6. T A - B. T , T1 B A v [v]B = [v]AT1.

. A = C -, A A -.

3.2.7. V n F GLn(F ) (nn)- F . , - F q, GLn(F ).

. -. v B ( -). - , - : v = b[v]B . A

80 3.

v = a[a]A, T A B - -. , b = Ta b = aT = aT . , - v A - : [v]A = T [v]

B .

3.3.

-.

3.3.1. V - F . U V , U U .

. , , , , 3.1, -, U , -, , U .

3.3.1. U - V n F . - .

1. U , dimU 6 n.2. U

V .3. dimU = n, U = V .. 1. U = 0 = {0}, U (

, 0). U 6= 0, u1 U . L(u1) = U ,

dimU = 1 (, u1 V , ,dimV > dimU = 1).

U 6= L(u1). u2 U , - u1, u2 . L(u1, u2) = U , - . L(u1, u2) 6= U , u3 U \ L(u1, u2). u3 u1, u2, u1, u2, u3 . , U . V n, n (. 1 -

3.3. 81

). , -. , k, n, L(u1, . . . , uk) = U . dimU = k.

2. U V , . 4 .

3. dimU = n, B U n V . . 1 - B V .

3.3.2. V , U , - U .

. 2 3.3.1 - V , . , , .

3.3.3. U1, U2, . . . , Us - V F . U1, U2, . . . , Us

U1 +U2 + . . .+Us =si=1

Ui = {u1 + u2 + . . .+ us | ui Ui, i = 1, 2, . . . , s}.

3.3.1. U ,W V . .

1. U W , U +W V .2. U +W , U ,

W .3. U W , U W

W U .. -

- . - s, , U1, U2, . . . , Us V .

3.3.2. U , W V F . V , - .

82 3.

. v1, . . . , vr U W (-, ). v1, . . . , vr, ur+1, . . . , us U , v1, . . . , vr, wr+1, . . . , wt W . , U W .

v1

u2

w2 U

W U W

, v1, . . . , vr, ur+1, . . . , us,wr+1, . . . , wt , , -, . , - :

1v1 + . . .+ rvr + r+1ur+1 + . . .+ sus + r+1wr+1 + . . .+ twt = 0.

1v1 + . . .+ rvr + r+1ur+1 + . . .+ sus = (r+1wr+1 + . . .+ twt).

, , U , , , W . - , z, ( ) , U W . v1, . . . , vr U W ,

z = 1v1 + . . .+ rvr = (r+1wr+1 + . . .+ twt).

,

1v1 + . . .+ rvr + r+1wr+1 + . . .+ twt = 0.

, v1, . . . , vr, wr+1, . . . , wt W , - 1 = . . . = r = r+1 = . . . = t = 0. z = 0. -, 1v1 + . . . + rvr + r+1ur+1 + . . . + sus = 0. v1, . . . , vr, ur+1, . . . , us U , 1 = . . . == s = 0, .

3.3. 83

. U , W - V , dim(U +W ) = dimU + dimW dim(U W ).

. , U W U +W . ,, U +W = v1, . . . , vr, ur+1, . . . , us, wr+1, . . . , wt. - v U +W v = u + w = (1v1 + . . . + rvr + r+1ur+1 + . . . +sus) + (1v1 + . . .+ rvr + r+1wr+1 + . . .+ twt) = (1 + 1)v1 + . . .+(r + r)vr + r+1ur+1 + . . . + sus + r+1wr+1 + . . . + twt. -, v1, . . . , vr, ur+1, . . . , us, wr+1, . . . , wt U +W . dim(U +W ) = s+ (t r) = dimU + dimW dim(U W ).

3.3.2. U1, U2, . . . , Us - V S . dimS 6

si=1 dimUi.

. 3.3.2, - s.

3.3.4. U1, U2, . . . , Us - V F . U1 + U2 + . . . + Us , u1+u2+. . . + us = 0, ui Ui, i = 1, 2, . . . , s, u1 = u2 = . . . = us = 0. U1, U2, . . . , Us

U1 U2 . . . Us si=1

Ui.

3.3.3. U1, U2, . . . , Us - V F S = U1+U2+ . . .++Us . -.

1. S .2. v S

v = u1 + u2 + . . .+ us, ui Ui, i = 1, 2, . . . , s.3. , U1, U2, . . .

. . . , Us, S.4. dimS =

si=1 dimUi.

5. Sj = U1 + . . . + Uj1 + Uj+1 + . . . + Us. j {1, 2, . . . , s} Uj Sj = 0.

. (1 2). v S - v = u1 + u2 + . . . + us = w1 + w2 + . . . + ws, ui, wi Ui.

84 3.

(u1 w1) + (u2 w2) + . . . + (us ws) = 0, , u1 = w1, u2 = w2,. . . , us = ws.

(2 3). i = 1, 2, . . . , s Bi bi1, . . . , biti Ui. , B, B1, B2, . . . , Bs, S. S =

si=1 Ui, L(B) = S. ,

B . ,

si=1

tij=1

ijbij = 0. (1)

i = 1, 2, . . . , s ui =ti

j=1 ijbij . s

i=1 ui = 0. - - ui, i = 1, 2, . . . , s ui = 0. Bi , ij (1) 0.

(3 4). .(4 5). , j {1, 2, . . . , s}

S = Uj+Sj . dimS =s

i=1 dimUi. , 3.3.2 dimS = dimUj+dimSjdim(UjSj).

si=1,i6=j dimUi = dimSj dim(Uj Sj). -

3.3.2 , dimSj 6s

i=1,i6=j dimUi, dim(Uj Sj) = 0. , Uj Sj = 0.

(5 1). si=1 ui = 0 j {1, 2, . . . , s} , uj 6= 0. uj = u1+. . .+(uj1)+(uj+1)+. . .+(us), uj Uj Sj ; .

. . 5 , U+W - U W , UW = 0.

3.3.3. , - , -.

3.3.5. V = UW . v V v = u + w, u U , w W . - u v U W .

. u v - U , W ( U).

3.3. 85

. .

. 1. B b1, b2, . . . , bn - V F . V = b1b2. . .bn n . v V bi - ibi, i i- v B.

2. V = F [R,R] - . V+, V - V . - , V+ V V . f F [R,R]. f+ f : f+(x) =12 (f(x)+f(x)) f(x) = 12 (f(x)f(x)). f+ ,f f = f+ + f. , V = V+ + V. , V+ V = 0. V = V+ V. , , , , .

, A , A = A. A , A = A. (3) 3.1.3 - Mn(F ) -.

3.3.4. , - Mn(F ) . , Mn(F ) .

4

4.1.

. - . - , .

4.1.1. A a1, a2, . . . , as - V F . A - L(A). A - r(A).

A r. , , , - . 1 , r - A, L(A). , , .

4.1.2. A (m n)- F . a1, . . . , am , a1, . . . , an - A. A - Fn, Fm. A a1, . . . , am. A a1, . . . , an.

, - . , , , , . 2.3 , . .

4.1.3. A = (aij) (m n)- - F r , r 6 min{m,n}. r r A. - M , A,

4.1. 87

r r , , r A. , i1, . . . , ir j1, . . . , jr, mkl M aik,jl .

. , , . , , .

.

A =

1 2 1 02 4 0 00 0 0 0

. M = ( 1 12 0)

2,

A, , , .

4.1.1. Crk =k!

r!(kr)! k - r, r (m n)- Crm Crn.

4.1.4. A . , r, - r - r + 1.

. A 2. - 2 M .

4.1.1 ( ). A = (aij) (m n)- F . , - A .

. , -. , -, , , .

A r. , r r A , M r, 0. , r+ 1 A, - , . , ,

88 4.

M r r . , - i1, i2, . . . , ir 1, 2, . . . , r . . , M = (aij), i, j {1, 2, . . . , r}.

ai i- A, ai i- M . i {1, 2, . . . , r} ai - ai, r . , - a1, a2, . . . , ar - Fn, . . , . a1, a2, . . . , ar F r -, , . , . 3.2.3 detM = 0; -. , a1, a2, . . . , ar A , . . r.

m = r, . , m > r. , A , r, - r . , - - . , (r + 1)-. ar+1 = (ar+1,1, ar+1,2, . . . , ar+1,r). r a1, a2, . . . , ar - F r , -. , ar+1 :ar+1 = 1a1 + 2a2 + . . . + rar. b - Fn, ar+1 (1a1 + 2a2 + . . . + rar). , b = (1, 2, . . . , n) = 0, . , 1, 2, . . . , r r b . i (i > r) , i 6= 0. , i = r+1. M r + 1 A, r + 1 r + 1 .

a11 . . . a1r a1,r+1... M

......

ar1 . . . arr ar,r+1ar+1,1 . . . ar+1,r ar+1,r+1

a11 . . . a1r a1,r+1... M

......

ar1 . . . arr ar,r+10 . . . 0 r+1

detM = 0. M

4.1. 89

, 1, , 2, . . ., r- , r. M , - M. , r+1- -, 1, 2, . . . , r+1, , , , . M , 0 = detM = detM = r+1 detM . detM 6= 0, - r+1 = 0, r+1. ,b = 0 ar+1 = 1a1 + 2a2 + . . .+ rar.

4.1.5. (, ) - A r(A).

. (n n)- A n , detA 6= 0.

.M r+1, M r , , M . , .

4.1.2. - r A , A r.

, , ( ) . , - . C

a1j1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .a2j2 . . . . . . . . . . . . . . . . . .

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

arjr . . .

0

(1)

,

4.1.6. - . , :

90 4.

1) ;

2) , , .

, . , - , r - j1, j2, . . . , jr, (r + 1), , .

, - () :

1) () ( ), ;

2) () .

:

3) () .

, - - , . .

- .

4.1.1. () A .

. , ai, aj ai+aj aj , ai ai, 6= 0.

4.1.2. - .

. -m A = (aij)mn. j1 - A, . - , , , (1, j1), . - ,

4.1. 91

j1- . 0 . . . 0 a1j1 . . .

0

0 ... A10

.

A1 - . - A (1).

- .

4.1.3.1. A = (aij)mn, B = (bij)mn F .

r(A+B) 6 r(A) + r(B).2. A = (aij)ms, B = (bij)sn F .

r(AB) 6 r(A) r(AB) 6 r(B).. 1. C = (cij)mn = A+B. -

A,B,C Fn, . i = 1, 2, . . . ,m ci = ai + bi, L(C) L(A) + L(B). -, r(C) = dimL(C) 6 dim(L(A) + L(B)) 6 dimL(A) + dimL(B) =r(A) + r(B).

2. C = (cij)mn = AB. , r(C) 6 r(B). B,C - Fn, . i = 1, 2, . . . ,m ci = (ci1, . . . , cin) =

=

(s

k=1

aikbk1, . . . ,

sk=1

aikbkn

)=

sk=1

aik(bk1, . . . , bkn) =s

k=1

aikbk,

C B. L(C) L(B) r(C) 6 r(B).

r(C) 6 r(A) . A C.

92 4.

4.2.

4.2.1. F ( ) x1, x2, . . . , xn -

a11x1 + a12x2 + . . .+ a1nxn = b1,a21x1 + a22x2 + . . .+ a2nxn = b2,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .am1x1 + am2x2 + . . .+ amnxn = bm

(1)

ai1x1 + ai2x2 + . . . + ainxn = bi (i = 1, 2, . . . ,m), aij , bi F . - (1). aij , bi .

4.2.2. x0 = (x01, x02, . . . , x0n) Fn (1), i = 1, 2, . . . ,m - ai1x01+ai2x02+ . . .+ainx0n = bi. , . - ( ), .

aj (a1j , a2j , . . . , amj) - (1) xj j = 1, 2, . . . , n, b (b1, b2, . . . , bm) (1). (1)

x1a1 + x2a2 + . . .+ xnan = b. (2)

(1). (1) A = (aij)mn.

- (1). (1)

Ax = b. (3)

(1). (m (n + 1))- A = (A | b), -

A b , (1).

4.2. 93

4.2.1 ( -). (1) , - A (1) ,. . r(A) = r(A).

. (2) - (1). , x0 = (x01, x02, . . . , x0n) - (1) , b = x01a1 + x02a2 + . . .+ x0nan, . . b a1, a2, . . . , an - x01, x02, . . . , x0n. , (1) - a1, a2, . . . , an = a1, a2, . . . , an, b, , r(A) = r(A).

. .

, , . - A . A - A. - , , . r(A) = r(A) .

(1) - , , .

4.2.1. - (1), , .

. , - 4.1.1.

.

4.2.3. A -, :

1) 1;2) , , -

, 0.

4.2.1. .

94 4.

(1) . - A = (A | b) - . , , - C = (C | d).

Cx = d (4)

Ax = b. , r(A) == r(A) = r. C (r (n + 1))-, - . (

1 . . . r . . . nj1 . . . jr . . . jn

) , c1j1 = c2j2 = . . . = crjr = 1 C. xj1 , . . . , xjr , - xjr+1 , . . . , xjn . - (4) .

xj1 = c1jr+1xjr+1 . . . c1jnxjn + d1,xj2 = c2jr+1xjr+1 . . . c2jnxjn + d2,

. . .xjr = crjr+1xjr+1 . . . crjnxjn + dr,

(5)

(4), , (1). t1, t2, . . . , tnr n r

F . x0 Fn : x0i = fi(t1, t2 . . . , tnr) i = 1, . . . , n, fji(t1, t2, . . . , tnr) = cijr+1t1 . . . cijntnr + di i = 1, . . . , r fji(t1, t2, . . . , tnr) = tir i = r + 1, . . . , n. x0 - (5). , x0 = (x01, x02, . . . , x0n) - (5), , t1 = x0jr+1 , . . . , tnr = x

0jn, -

, (5) x0 = (f1(t1, . . . , tnr), . . . , fn(t1, . . . , tnr)). (1) (5), (1). - .

4.2.4. F . n- (f1(t1, . . . , ts), . . . , fn(t1, . . . , ts)), fi : F s F , -

4.2. 95

(1), (1) -

{(f1(t1, . . . , ts), . . . , fn(t1, . . . , ts)) | (t1, . . . , ts) F s}. .

4.2.2. (1) r. n fi : Fnr F fi(t1, . . . , tnr) = i1t1 + . . . + i,nrtnr + i, i = 1, 2, . . . , n, n r , n- (1).

, - - n r F . , , , , , , , - (1). - .

(1) (5).

4.2.2. x1 + 2x2 + x3 = 2,x1 + 3x2 + 2x3 x4 = 4,2x1 + x2 x3 + 3x4 = 2,2x1 2x3 + 4x4 = 4

(6)

. , -. - .

, - ( ) - , , . - , , , .

4.2.3. (1) (m = n) A detA 6= 0.

96 4.

x0 = (x01, . . . , x0n). - :

x0i =did, i = 1, 2, . . . , n, (7)

d = |A|, di , A i- .

. Ax = b. , (A1b) . -, A((A1b)) = A(A1b) = b. . , x0 , A(x0) = b. ,(x0) = A1b , , x0 = (A1b) = b(A1). , .

, b(A1) = (d1d , . . . ,dnd ).

(A1) = (Aijd ). b i- -

(A1), ( ) .

. (7) .

4.3.

4.3.1. n - F

a11x1 + a12x2 + . . .+ a1nxn = 0,a21x1 + a22x2 + . . .+ a2nxn = 0,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .am1x1 + am2x2 + . . .+ amnxn = 0,

(1)

, -.

. , - x0 = (0, . . . , 0).

4.3.1. X (1) Fn. - X n r, r -.

. (1) Ax = 0. x1, x2 ( - ). A((x1) + (x2)) = A(x1) + A(x2) = 0 + 0 = 0.

4.3. 97

, x0 = x1 + x2 . - , x0 , x0 , .

n- fi(t1, . . . , tnr), i = 1, . . . , n, (1), 4.2.2. k = 1, . . . , n r i = 1, . . . , n xki = fi(0, . . . , 0, 1, 0, . . . , 0), 1 k- . n r xk =(xk1 , . . . , x

kn) Fn. -, k = 1, . . . , n r

xk (1). -, -, - n r, . -, - -. , x0 = (f1(1, . . . , nr), . . . , fn(1, . . . , nr)) = 1x1 + . . . + nrxnr. , x1, . . . , xnr X.

4.3.2. - - .

. V = Fn - n F , U - k V . n F , U .

. ui = (i1, . . . , in), i = 1, . . . , k, U . - By = 0, B = (ij)kn . - B k, 4.3.1 - By = 0 n k. - : ai = (i1, . . . , in), i = 1, . . . , n k. A = (ij)(nk)n , - . i1j1 + . . . + injn = 0 i = 1, . . . , n k j = 1, . . . , k, uj Ax = 0 - j = 1, . . . , k. U X Ax = 0. , A n k, , - X n (n k) = k. , U = X .

98 4.

4.3.2. Ax = b A Ax = 0 . x0 Ax = b, X Ax = b, Z Ax = 0. X = x0 + Z = {x0 + z | z Z}.

. z0 . A(x0 + z0) = A(x0) +A(z0) = b+0 = b. , x0 +Z X. , x , A(x x0) = Ax A(x0) = b b = 0. ,z = x x0 . X x0 + Z.

. Ax = b , - , Ax = 0 .

.

4.3.3 (). - Ax = b Ay = 0, - . Ax = b , - y Ay = 0 yb = 0.

., Ax = b x0 . - A(x0) = b. , Ay = 0 - yA = 0 ( -). , y , Ay = 0, yb = y(A(x0)) = (yA)(x0) = 0.

y , Ay = 0, yb = 0, , by = 0. (

A

b

)y = 0

Ay = 0. ,

r(A) = r(A

b

),

4.3. 99

r(A) = r(A | b). - Ax = b.

4.3.1. , - 4.2.2. . - 4.3.2 4.3.3 .

, ( ).

4.3.2. F = Z2 2

x2 + x3 + x5 = 1,x1 + x2 = 1,x2 + x3 + x4 + x5 = 0,x1 + x2 + x4 = 0

. - .

5

5.1.

. ( -) - f : R R f(x) = a0+a1x+ . . .+anxn, ai R. ( -), () . , x x2 F = Z2 2, , , 02 = 0 12 = 1. , , , - . N0 N {0} .

5.1.1. k N0. ( -) f x R

f(x) =k=0

akxk = a0x0 + a1x1 + . . .+ akxk + . . . ,

ak R 0. f - , f deg f . -. x - R R[x].

. , , . . akxk, - , n f(x) = a0x0 + a1x1 + . . . + anxn. , - R(a0x0 = a0). ,

5.1. 101

f(x) = a0+ a1x+ . . .+ anxn. , 0 =

k=0 0x

k - 0. . ( ) , deg 0 = .

5.1.2. f(x) =

k=0 akxk g(x) =

k=0 bkxk , k N0 ak = bk.

. x x2 Z2 .

5.1.3. R

f(x) =k=0

akxk, g(x) =

k=0

bkxk R[x].

h(x) =k=0

ckxk p(x) =

k=0

dkxk R[x]

f g, k N0 ck = ak + bk dk =

i+j=k aibj .

: h = f + g p = fg.. , h = f + g

p = fg , , .

5.1.1. R . .

1. R[x] .

2. R , R[x] .

3. R , R[x] .. -

, - . , f(x) =

k=0 akx

k,g(x) =

k=0 bkx

k, h(x) =

k=0 ckxk, -

fh+ gh = (f + g)h i+j=k

aicj +i+j=k

bicj =i+j=k

(ai + bi)cj k N0,

102 5.

, , R.

. R R[x] ( ) -, R R[x].

5.1.1. 5.1.1 .

5.1.1. R f, g R[x], f, g 6= 0. .

1. deg(f + g) 6 max{deg f,deg g}.2. deg fg 6 deg f + deg g, R

, deg fg = deg f + deg g R[x] .., -

, . 2 , , deg 0 = .

. R , R[x] R[x] .

5.1.2. 5.1.1 . R, - deg fg = deg f + deg g .

F . , F [x] - . , F [x] F . , .

5.2.

5.1.1 - F [x]. . , , .

5.2.1 ( ). F , f, g - F [x] g 6= 0. q, r F [x], f = qg + r r = 0, deg r < deg g. q r, , .

5.2. 103

. . deg f < deg g, , q = 0, r = f , . - , , deg f = n > deg g = m. f(x) = anxn + . . .+ a1x+ a0 b(x) = bmxm + . . .+ b1x+ b0. n. n = 0 ( F ), . , , n > 0 , n, . f1 = f anbmxnmg. n, , q1 r, f1 = q1g + r r = 0, deg r < deg g. f = anbmx

nmg+ f1 = ( anbmxnm+ q1)g+ r, q = anbmx

nm+ q1 r .

f = qg+r = qg+r. rr = (qq)g. qq - , . 2 5.1.1 , , deg g. , , , . 1 r, r deg g. , q = q, , r = r.

5.2.1. q r, , () - f g.

5.2.2. g 6= 0 f , q , f = qg. g f , f g. g | f, g f , f

... g , f g.. , g f ,

: g - f .

5.2.1 ( ). F [x] .

1. g | f g | h, g | (f + h).2. g | f , h F [x] g | (fh).3. deg g = 0, h F [x] g | h.4. deg h = 0 g | f , (hg) | f . 5.2.1. 5.2.1, -

.

5.2.3. f, g F [x]. -

104 5.

f g d F [x], - :

1) d | f d | g;2) d F [x] , d | f d | g, d | d.: d = (f, g).

. 2 4 , d f g, - w f g , w = ud, u . , F . (f, g) = (u, v) - , .

5.2.2 ( ). f, g F [x] g 6= 0. d = (f, g) d = fu + gv, u, v F [x]. , f g 0, u v , deg u < deg g deg v < deg f .

. - .

. r f g. f g g r. , (f, g) = (g, r).

. h | g h | r, 1 2 - h f = qg + r. , h | f h | g, h | r, r = f qg. , , , .

. f g, d = g = f 0 + g 1 . f g, g , . . , . :

f = q1g + r1,

5.2. 105

g = q2r1 + r2,. . . . . . . . . . . . . . .rn2 = qnrn1 + rn,rn1 = qn+1rn,

(1)

ri 6= 0 i = 1, . . . , n. rn = (rn1, rn) = (rn2, rn1) = . . . = (r1, r2) = (g, r1) =

= (f, g). , f g rn .

, ,

r1 = fu1 + gv1,r2 = fu2 + gv2,. . . . . . . . . . . . . . .rn1 = fun1 + gvn1rn = fun + gvn,

(2)

ui, vi (i = 1, . . . , n) F [x] (,u1 = 1, v1 = q1). , d = rn fu+ gv.

d = fu + gv u g. u g: u = qg+r. , d = f(qg + r) + gv = fr + gv. deg r < deg g. deg f 6 deg v, deg fr < deg gv. , , g f , , , deg d < deg g 6 deg gv. ,gv = d fr, , deg gv = deg(d fr) 6 max{deg d,deg fr};. , deg v < deg f .

. - (1). - . , (f, g) f g . (f, g) f g.

5.2.4. f, g F [x] , (f, g) = 1.

5.2.3 ( ). - f, g F [x] , - u, v F [x] , 1 = fu+ gv.

106 5.

. (f, g) = 1, u, v, -, 5.2.2. , - u, v , 1 = fu+ gv, d f, g fu + gv = 1. , d -.

5.2.2 ( ). f, g, h F [x]. .

1. (f, g) = (f, h) = 1, (f, gh) = 1.2. (f, g) = 1 f | (gh), f | h.3. (f, g) = 1, f | h g | h, (fg) | h.. .

(f, g) = 1, a, b F [x] , fa + gb = 1. h = h(fa) + h(gb). , c, d F [x] , fc + hd = 1. h. fc+(hfa+hgb)d = f(c+had)+(gh)cd = 1. u = c+ha v = cd, fu + (gh)v = 1. , 5.2.3 f gh .

.

5.2.2. . 2 3 5.2.2. , -

, , - .

5.2.5. f F [x] , , , f = uv, u, v F [x],, deg u = 0, deg v = 0. - f .

. - , , . , , .

. x2 + 1 Q[x] R[x], C[x]: x2 + 1 = (x+ i)(x i). x2 2 Q[x], R[x]: x2 2 = (x + 2)(x 2). , , .

5.2. 107

5.2.3. f F [x] . - .

1. af , a - .

2. g F [x] f | g, (f, g) = 1.3. g F [x] , g 6= 0 deg g < deg f , (f, g) = 1. 5.2.3. 5.2.3.

5.2.4. f F [x], f 6= 0. a F p1, p2, . . . , pr -, 1, ,

f = ap1p2 . . . pr. (3)

(3) -.

. a f . deg f = 0, f = a, r = 0, . - , deg f > 0.

(3). f -, p1 = 1af - 1. f = ap1 . - f . f -, u, v F [x] , f = uv deg u < deg f , deg v < deg f . u = bp1 . . . ps,v = cps+1 . . . pr. f = (bc)p1 . . . pr.

(3). f = ap1p2 . . . pr = bq1q2 . . . qs f . , a = b, - f . F [x] , a(p1 . . . pr q1 . . . qs) = 0 p1 . . . pr = q1 . . . qs. r 6 s. r = 0. - , r > 0, r.

pr | q1 . . . qs, j {1, . . . , s} , pr | qj . - , , pr | qs,. . qs = upr. qs , , u 0. , pr qs - , u = 1 pr = qs. , pr(p1 . . . pr1

108 5.

q1 . . . qs1) = 0, p1 . . . pr1 = q1 . . . qs1, pr 6= 0. - .

. - .

5.3.

5.3.1. f = anxn+ . . .+ a1x+ a0 F [x], F . f f() = ann + . . .++a1+ a0 F .

5.3.1. f, g F [x], F . 1. (f + g)() = f() + g().2. (fg)() = f()g().

5.3.2. F - f F [x], f() = 0.

5.3.1 (). f F [x], F . - f f x . -, f , (x ) | f .

. f x. f = q(x ) + r, r = 0 deg r = 0. , . f() = r. .

5.3.2.1. f F [x] n n

.2. i = 1, . . . , n i F , i 6= j, -

i 6= j. f, g F [x], f, g n f(i) = g(i) i = 1, . . . , n. f = g.

3. i = 1, . . . , n i, i F , i 6= j, i 6= j. f n, i = 1, . . . , n f(i) = i. -

5.3. 109

f

f(x) =ni=1

i(x 1) . . . (x i1)(x i+1) . . . (x n)

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

., (1), .

. 1. 1, . . . , s - f . , i 6= j (xi, xj) = 1. -, , . , . , , . , g = (x1) . . . (xs) | f . , s = deg g f - n = deg f .

2. h = f g. deg h 6 max{deg f,deg g} < n h(i) == f(i) g(i) = 0 i = 1, . . . , n. h 6= 0, . 1 .

3. i (1), f(i) = i i = 1, . . . , n. f . 2 .

. F , , - . 2 , f F( 5.1.1) f . - , f, g F [x] , F f() = g().

5.3.2. (1), , 1, 2, 3 1, 4, 9 .

f , f x , x . . .

5.3.3. f F [x], r N0. F f r, (x )r | f (x )r+1 - f . 1 , , 1, f .

. 0, , .

110 5.

, r, , f r , . -, .

5.3.3. f F [x] n n . , f n , F - , . . f = a(x 1)r1 . . . (x s)rs , a, i F s

i=1 ri = n.. -

. 1 5.3.2. 1, . . . , s - f r1, . . . , rs . , ((x i)ri , (x j)rj ) = 1 i 6= j. , ( 5.2.4), d (x)m, d = c(x)k, c F 0 6 k 6 m. - (x i)ri (x j)rj (x i)ki (x j)kj , - , d .

, g = (x 1)r1 . . . (x s)rs - f . deg g 6 deg f . , deg g = deg f f = ag, a , . . .

- . , . - .

5.3.4. f(x) =n

k=0 akxk -

n F . f ,

f (x) =n

k=1

kakxk1,

f .. , ka,

k N a F , F , k - a F .

5.3.3. f, g F [x], , F , k N. - .

5.3. 111

1. (f + g) = f + g.2. (fg) = f g + fg.3. ((x )k) = k(x )k1. 5.3.5. f F [x]. k

f (0) = f , k- f :f (k) = (f (k1)).

, - ., f(x) = x2 F = Z2, , f (x) = 2x = 0x = 0. , ka 6= 0 k a F . -, . .

5.3.6. F . p , p1 = 0 ( p 0), -, F . , F 0. charF .

. Z2 2. Q, R C .

5.3.4. p , p .

. , .

5.3.5. charF = 0, a F k N, ka = 0 , a = 0. , b F k N b/k = b (k1)1 F .

f(x) = a0+a1x+. . .+anxn x = y+, ( ) y = x .,

f(x) = b0 + b1(x ) + . . .+ bn(x )n, (2)

f x . 5.3.4 ( ). charF = 0, F ,

f F [x] deg f = n. f x

112 5.

f(x) =n

k=0

f (k)()k!

(x )k. (3)

.. (2) k -

x = . f (k)() = k!bk. (3).

( ). charF = 0,f F [x] r N. .

1. F r f , f (k)() = 0 k = 0, . . . , r1 f (r)() 6= 0. , r 1 f f .

2. f - g = f(f,f ) , g . , f , (f, f ) = 1.

. . 1 , f (r = 1), f ( f ).

. 1. f , , - (2). .

2. f r, (x )r1 | (f, f ) (x )r - (f, f ). , g. , f = g(f, f ), g f .

5.3.6. f(x) = x5 3x4 6x3 + 10x2++21x+ 9 g = f(f,f ) f(x) = 0.

, , - . , .

5.3.7. . F , f F [x] r, i {0, 1, . . . , r1} f (i)() = 0. , (f, f ) = 1, f .

- . -

5.3. 113

, . .

5.3.5. charF = 0, s N i = 1, . . . , s ri N0, i, i0, . . . , iri F , m 6= k m 6= k. f -,

si=1(ri + 1), , i = 1, . . . , s

j = 0, . . . , ri f (j)(i) = ij.. f f =

si=1 fi,

fi = digi di, gi F [x] i = 1, . . . , s. di(x) =k 6=i(x j)rk+1, gi .. , 0, . . . , r F d F [x] ,

d() 6= 0. g , r,, (dg)(j)() = j j = 0, . . . , r.

. , - u F [x] t = 1, . . . , n, n N, v F [x] ,

(u(x )n)(t) = n!(n t)!u(x )

nt + v(x )nt+1. (4)

(4) t. t = 1

(u(x )n) = n(x )n1u+ u(x )n

, v = u, . v1 ,

(u(x )n)(t1) = n!(n t+ 1)!u(x )

nt+1 + v1(x )nt+2.

,

(u(x )n)(t) =(

n!(n t+ 1)!u(x )

nt+1)

+(v1(x )nt+2

)=

=n!

(n t)!u(x )nt + v(x )nt+1,

.

114 5.

r. r = 0 - g ( g 0) (dg)() = 0. , g = 0d() , gd() = g()d() = (dg)() = 0.

r > 0 , r, . , h , r, , j = 0, . . . , r 1 (dh)(j)() = j . F g = h+ (x )r (dg)(j)() = j j = 0, . . . , r 1. ,

(dg)(j) = (dh)(j)+(d(x)r)(j) = (dh)(j)+(x)w (. (4)).

(dg)(j)() = (dh)(j)()+0 = j j < r. , (dg)(r) = r.

(dg)(r) = (dh)(r) + (d(x )r)(r) = (dh)(r) + dr!(x )0 + v(x ).

(dg)(r)() = (dh)(r)() + d()r! = r,

,

=r (dh)(r)()

d()r!

(dg)(r)() = r. -, , r! 6= 0 , charF = 0, d() 6= 0 .

. di , di(i) 6= 0 i = 1, . . . , s. -, - gi , ri, , f

(j)i (i) =

= (digi)(j)(i) = ij , i = 1, . . . , s, j = 0, . . . , ri. (x i)ri+1 | dk k 6= i, f (j)k (i) = 0

k 6= i j = 0, . . . , ri. , i = 1, . . . , s j = 0, . . . , ri

f (j)(i) =s

k=1

f(j)k (i) = f

(j)i (i) = ij .

5.4. 115

,

deg f 6 maxi=1,...,s

{deg fi} s

i=1(ri + 1); .

5.3.8. - , f , : f(1) = 1,f (1) = 2, f(2) = 4.

5.4.

-. , , , , - , . , -, , .

5.4.1. n , 1, R R[x1, . . . , xn1] R n 1 . R n - R[x1, . . . , xn] = R[x1, . . . , xn1][xn], . . - xn R[x1, . . . , xn1].

. R[x1, . . . , xn] 5.1.1. , , - R R[x1, . . . , xn].

5.4.2. axk11 xk22 . . . x

knn , a R, -

( ). a - , n- (k1, k2, . . . , kn) , - ki xi, k1+k2+. . .+kn .

116 5.

F . , f F [x1, . . . , xn]

:

f(x1, x2, . . . , xn) =

k1,k2,...,kn

ak1k2...knxk11 x

k22 . . . x

kn , (1)

k1, k2 . . . , kn , ak1k2...kn F . , - (k1, k2 . . . , kn) . -, n - . - , , . (k1, k2, . . . , kn) , (l1, l2, . . . , ln) -, i {1, 2, . . . , n} , k1 = l1, . . . , ki1 = li1 ki > li. u, v , u v , u v. .

. :

x21x2 2x1x2x3 + 3x1x33 2x22x3 + 1.

, 3x1x33 , x21x2, 2x1x2x3, - .

- F .

5.4.1. F , u, v, w, z F [x1, . . . , xn]. .

1. u v, v w, u v.2. u v w 6= 0, uw vw.3. u v, w z, uw vz. 5.4.1. 5.4.1.

. - F [x1, . . . , xn] -. , F [x1, . . . , xn] .

5.4. 117

5.4.3. f F [x1, x2, . . . , xn] , Sn f(x1, x2, . . . , xn) = f(x1, x2, . . . , xn), . . f .

. 1.

sk =

16i1 kn. i, ki < ki+1. = (i, i + 1) u = axk11 . . . x

kii x

ki+1i+1 . . . x

knn v = ax

k11 . . . x

ki+1i x

kii+1 . . . x

knn ,

v. -

118 5.

, f , - v, u .

f , . . (0, . . . , 0), . (l1, . . . , ln), - l1 > l2 > . . . > ln , (k1, . . . , kn) - f , f . , , , f , .

h = ask1k21 . . . skn1knn1 s

knn . -

sk , 5.4.2 h - x1, . . . , xn. v - h. i {1, 2, . . . , n} v xi ki, xi si, . . . , sn - s1, . . . , si1. -, v h u, - f . f1 = f h - , - f . g1 F [y1, . . . , yn] , f1(x1, . . . , xn) = g1(s1, . . . , sn). , - g = ayk1k21 . . . y

kn1knn1 y

knn + g1(y1, . . . , yn) .

. - - .

5.4.3. , x21 + . . . + x2n = s21 2s2 x31 +. . .+ x3n = s

31 3s1s2 + 3s3.

5.5.

- . , , , , . , f(x) = x2 2 Q[x] , g = x2 + 1 R. f R, g C, . . , - . , .

5.5. 119

5.5.1. K F , F K, . . F K , F , F , K.

, x2 2 x2+1, , .

5.5.1. F , f F [x] deg f > 0. K F , K f() = 0.

. f() , f , F , K . , - () K. , K F f F [x], , f K. , f() f , f K[x].

. 5.2.4 f F -, : f = ap1p2 . . . ps. - p1 K F , f() = ap1() . . . ps() = 0 K . , , f F 1. deg f = n

f(x) = a0 + a1x+ . . .+ an1xn1 + xn. (1)

, , deg f > 1. f = a0+x, a0 f F , , K = F .

, . K F t, n. ,

K = {g(t) F [t] | deg g < deg f}.

K , - + , . + - . g, h K

120 5.

g(t)h(t) f(t), . . g h = r, r - gh = fq + r deg r < deg f . , K - . , deg(g + h) 6max{deg g, deg h} < deg f deg r < deg f -. , r , - K.

, K . , K,+ , , -

. . g, h, u K

gu = fq1+r1, hu = fq2+r2 (g+h)u = fq+r. gu+hu = r1+r2, (g + h) u = r. , F [t] . , g, h, u F [t] gu+hu = (g+h)u. , (g+h)u w, w = f(q1+q2)+(r1+r2) == fq + r. r1 + r2 = r, , g u + h u = (g + h) u. .

K - F [t]. - . K, F [t], - , F . , K .

g(t) K g(t) 6= 0. f(t) F deg g < deg f , . 3 5.2.3 (g, f) = 1. -, u, v F [t] , gu + fv = 1 deg u < deg f . u K g u = 1, gu = f(v) + 1., u K g. , K .

a, b F . K. F K, a + b. ab f , , ab. - a b = ab F . , F K, ,K F .

K, t = 0+1 t++0 t2 . . .+0 tn1 F [t]. , = t K, , deg f > 1. f() K. t (1),

5.5. 121

f() = a0 + a1 t+ a2 t2 + . . .+ an1 tn1 + tn, k = 1, . . . n tk k t K, . .

tk = t . . . t k

.

aktk, , f k < n aktk, deg aktk 0, K1 F 1 K1 , f(1) = 0. K1 f - f = (x1)f1. f1 n1. , K K1 - 2, . . . , n K , f1 = a(x2) . . . (xn) K[x]. (2) f K. , K K1 F .

. x2 + 1 R[x] C, .

5.5.2. , f(x) = x4 5x2 + 6 Q[x] Q[

2] = {a + b2 | a, b Q}, -

. Q, f .

- .

2 ( ). , f(x) = anxn + an1xn1 + . . . + a1x + a0 F [x] f(x) = an(x1)(x2) . . . (xn) K F . k = 1, . . . , n

sk(1, . . . , n) = (1)k ankan

, (3)

sk k- n -. , g - n F , g(1, . . . , n) F .

. (3) . - 5.4.1 - .

. -. , 1, . . . , n F , - F F .

. i i x2+1 R, i2+(i)2 = 2 R. ,

5.5. 123

x2 + 1 , . , 1, 2, 5.4.3 21+

22 = s

21(1, 2)2s2(1, 2). ,

s1(1, 2) = 0/1 = 0, s2(1, 2) = 1/1 = 1, .

- - - . , x2 2 = 0 , - x2 + 1 . , - , ? , , , , C , , - .

5.5.2. F -, F [x] F.

, XVIII , . - . , , - , , , - , -. -. , -, . , - , R.

. , - , , . [2] [5].

124 5.

5.5.2 ( ). f C[x] deg f > 0. C , f() = 0.

. , , - f 1, . .

f = a0 + a1x+ . . .+ an1xn1 + xn. (4)

- .

1. f R[x] deg f . - R , f() = 0.

. A = max{|a0|, |a1|, . . . , |an1|}. R || > 1 +A.

|a0 + a1 + . . .+ an1n1| 6 A(1 + ||+ . . .+ ||n1) =

= A||n 1|| 1 0, f() > 0, f() < 0. f(x) , [, ], .

., (5) , ai, C. , - -. - -.

5.5.3.1. , C, |+ | 6 ||+ || || = ||||.2. f C[x] (4), C f() = 0, || 6 1 +

A, A = max{|a0|, |a1|, . . . , |an1|}. , f 1 + A .

5.5. 125

. , - .

2. f R[x] deg f > 0. C, f() = 0.

. f (4) n = 2km, k N0 m . k, , . , , R[x], 2k, .

R C, , f C[x]. - - 5.5.1 K C , f(x) = (x 1) . . . (x n), i K i = 1, . . . , n. . i, j {1, . . . , n} , i < j,

ij = ij + (i + j). (6)

ij K

n(n 1)2

=2km(2km 1)

2= 2k1l,

l . , , - i ij .

g(x) K[x] ij , .

g(x) =i

126 5.

1, . . . , n, . f , - R. , g R[x].

, g 2k1l 2k, (, , f). , g . , - R i, j {1, . . . , n}, i < j, ij C. , (i, j), R . 1 2 , (i, j) a b, - {

a = ij + 1(i + j),b = ij + 2(i + j),

(8)

. - (8) , (12)(i+j) = ab. ,

i + j =a b1 2

. ij - . , i, j , , C. - .

. f C[x]. f(x) = a0 + a1x+ . . .+ an1xn1 + xn,

,

h(x) = f(x)f(x) = b0 + b1x+ . . .+ b2n1x2n1 + b2nx2n.

h

bk =i+j=k

aiaj .

bk =

i+j=k

aiaj = bk,

bk R k = 0, . . . , 2n. , h R[x] 2 C , h() = f()f() = 0.

5.6. 127

f() 6= 0, f() = 0. f() = f() = 0 = 0. , , f . .

( - ). 1. f C[x] , 0, C .

2. f R[x] , 0, R 2.

. 1. . f C, f1 C f = (x)f1. f1 - f , , . , f .

2. f . f - , , , . 1, - f1 R, f . , , f . , f . , f(x) = f(x). -, f , f() == f() = f() = 0 = 0. - f . (x , x ) = 1, f g(x) = (x)(x) = x2