19.1
19.2 , .
19.3 . 2 .
20.1 ., , r, r , , ., , r, r , , .
20.2 ( ).
20.3 1). ; 2). () , ; 3). () (-), .
21.1 ( )
21.2 , .
26
. . .
1. L L, :
1) .
2) L (.. ):
. (V2 2 ), (V3 3 ), Rn ( ), ≤ n - (1,,2,,n).
1. .
{ }
2. .
( .1 )
: .
, , , :
2. :
{ }
3. :
λ = (λα1,,λαn). { }
3. L ( dim L) .
.
4. .
{ L (n + 1) { e } An,n+1 . .. rang An,n+1 ≤ n, , aţ
|
|
dimL = n }
, , , .
. V2, V3, Rn.
28
.
1. . V W , n m. , V W, : V > W, V W. = () = .
. , V W, , x1 u x2 V λ :
1. λ(x1 u x2) = λx1+ λx2 ( );
2. (λ) = λ ( ).
1. W , A, V W, .
2. W V, , V V, V.
2. . . , V W, .
, V W. + ,
( + ) = + . (5.1)
λ λ,
(λ)= λ(). (5.2)
, V W.
, = 0.
-
- = (-1).
.
L(V, W) , V W, .
3. L (V, V) . , V V, . . L(V, V).
( ) I, I = ( V).
L(V, V).
L(V, V) ,
() = (). (5.3)
, , , ≠ .
L(V, V):
1. λ() = (λ);
2. ( + ) = + ;
3. ( + ) = + ;
4. () = ().
1-4 (. 5.2)) (. 5.3)).
2. , (5.1), (5.2) (5.3),
|
|
(( + )) = ( + )() = () + () = () + () = ( + ). (5.4)
, ( + ) = + . 2 .
3.
4 , , (. (5.3)), , () () , , .
1. 4 ... L(V, V) , , n-
, An+m = AnAm.
L(V, V).
1. L(V, V) L(V, V),
= = I.
-1.
, V -1 = .
, -1 = 0, = 0, .. , = 0 , = 0.
, V V, x1 x2 y1 = Ax1 2 = x2.
V V, : V > V V V, . . V x V:
y = .
, , , , n x1,x2,...,xn V n Ax1,Ax2,...,Axn .
, x1,x2,...,xn V.
α1Ax1+ α2Ax2,...+ αnAxn V:
α1Ax1+ α2Ax2,...+ αnAxn = 0,
(. . 1 ) , A(α1x1+ α2x2,...+ αn xn) = 0.
V V , , α1x1+ α2x2,...+ αn xn = 0. x1,x2,...,xn . α1 = α2 =... = αn = 0. , Ax1,Ax2,...,Axn .
.
L(V, V) , , V V.
, . , V V.
, x1 x2, x2 - x1 ≠ 0 V = Ax1 = 2. (x2 - x1) = 0, , x1 - x2 = 0. , x2 - x1 ≠ 0. .
.
, V V.
V V , = . , , -1 = () = . , . .
29
, . , , ( ) .
. .
|
|
. . :
,
.
.
. ,
.
,
,
- - .
,
.
, , , , , , , , .
: , , , , . ,
30
.
.
2. V, = 0. ker .
ker A = 0, V V. , = 0 = 0, , x1 x2 1 = Ax1 2 = 2 ( y1 = 2, (x2 - x1) = 0, . . x1 = 2 x1 x2 ).
, ker A = 0 , .
3. V, = . im A ( im Im, ).
2. , ker = 0, im A = V, . ker A = 0 im A = V , .
3. , ker im A V. dim (ker ) dim (imA) .
.
5.1. dimV V n, L(V, V).
dim (im ) + dim (ker A) = n.
. ker V, V1 V, V1 V ker A. 2.10 dim V1 + dim (ker A) = n. , dim V1 = dim (im A).
dimV1 = , dim(im A) = q y1,y2,...,yq im A. V1 im , im V1 , = . V1 x1,x2,...,xq , k = k, = 1, 2,..., q. x1,x2,...,xq , α1x1 + α2x2+...+ αqxq = 0, A(α1x1 + α2x2+...+ αqxq) = α1y1 + α2y2+...+ αqyq = 0, y1,y2,...,yq , α1 = α2 =... = αq = 0, . . x1,x2,...,xq . , V1 q . , ≥ q (, = dim V1).
, > q. x1,x2,...,xq xq+1,xq+2,...,xp , x1,x2,...,xp V1. > q q = dim (im A), Ax1,Ax2,...,Axp , im A, , λ1,λ2,...,λp , λ1Ax1 + λ2Ax2 +...+ λpAxp = 0. , A(λ1x1 + λ2x2 +...+ λp xp) = 0. V1 im A , λ1x1 + λ2x2 +...+ λp xp = 0.
x1,x2,...,xp V1. λ1 = λ2 =... = λp = 0.
, λ1,λ2,...,λp . , > q . , = q.
.
, 5.1.
5.2. V1 V2 n- V, dimV1 +dimV2 = dim V. L(V, V), V1= im A V2 = ker .
. dim V1 = p, dim V2 = q. V 1, 2,..., n , 1, 2,..., n V2. V1 g1, g2,..., gp.
1, 2,..., n V :
|
|
Ae1=g1, Ae2 = g2,..., = gp,
Aep+1 = 0, +2 = 0,..., n = 0.
, = x1e1 + x2e2 +... + xpep + xp+1ep+1 +... +xnen,
= x1g1 + x2g2 +... + xpgp. , . .
.
, rang rang A = dim(im A).
5.1 2 .
5.1. L(V, V) -1 , rang A = dim V = n.
L(V, V). .
5.3. :
rang AB ≤ rang A, rang AB ≤ rang .
. . , im AB im A. dim(im AB) ≤ dim(im A), .. rang AB ≤ rang .
( , , , im AB im B , rang AB ≤ rang ): ker ker AB.
, dim (ker ) ≤ dim (ker AB).
, , dim V dim (ker AB) ≤ dimV dim (ker ), , 5.1, dim(im AB) ≤ dim(im B), .. rang AB ≤ rang .
.
.
5.4. L(V, V) n V. rang AB ≥ rang A + rang - n.
. 5.1
dim (im AB) + dim (ker AB) = n. (5.5)
rang AB = dim(im AB), (5.5)
rang AB = n - dim (ker AB). (5.6)
, 5.1,
dim (ker A) + dim (ker ) = 2n - (rang A + rang ), (5.7)
dim (ker AB) ≤ dim (ker A) + dim (ker B). (5.8)
, (5.6)
rang AB ≥ n (dim (ker A) + dim (ker)),
, (5.7), .
, (5.8).
dim (ker ) = q. (5.9)
5.3 dim (ker AB) ≥ q.
dim (ker AB) = p + q, > 0. (5.10)
ker ker AB, ker AB x1,x2,...,xp+q , xp+1,...,xp+q ker . x1,x2,...,xp+q Bx1,Bx2,...,Bxp ( , , x1,x2,...,xp, λk, = 0, = 1, 2,..., ). Bx1,Bx2,...,Bxp ker , .. ≤ dim (ker ). (5.9) (5.10) (5.8). .
5.3 5.4. rang = n (n V), rang AB = rang = rang .
|
|
rang AB ≤ rang ( 5.3),
rang AB ≥ rang ( 5.4 rang = n).
, rang AB = rang . rang = rang .
31
19.3 , , .
.
" " , .
-- , -- , . , ( ) , , , ( ).
19.1 2. 19.2 , .
19.7 -- , -- , , -- , , (. 19.5). 19.5 , , , . , , , , 1, , , , . . C a,
[an error occurred while processing this directive]
19.2 -- , -- . -- , .
.
19.8 -- , -- , , -- , (. 19.6). , , , , 1, , , 0.
19.9 -- 19.3. , , , , , 0.
, . -- , , -- . , .
19.4 - , , .
19.2 ( ) . , , . , , , , (, , ). - , .