Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://matematika.phys.msu.ru/files/stud_gen/6/Lin_Alg_N02.pdf Äàòà èçìåíåíèÿ: Mon Mar 17 07:52:27 2014 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 13:19:18 2016 Êîäèðîâêà:

. .


¬ ; ( ); ( ); = ( . . . , . . . ); ; ; ; A|x ; {x : A} , A; M (A) A ; xA x A ; ; BA B A ; BA B A (B A B = A); P (A) A; {x1 , . . . , xr } , x1 , . . . , xr ({u : u = x1 · · · u = xr }); (x1 , . . . , xr ) , x1 , . . . , xr ; A B A, B ; A B A, B ; A \ B A, B ; A1 â · · · â Ar A1 , . . . , Ar ; r A A; D(F ) F ; R(F ) F ; {}x : A , A, ({}x : A = F : F , D(F ) = {x : A}, xA F (x) = ). F: A B F A B ( F , D(F ) A, R(F ) B ); fun(A, B ) F , F : A B ; F : A = B F A B ( F , D(F ) = A, R(F ) B ); Fun(A, B ) F , F : A = B ; D(F, A) A F ; F [A] A F ;


2

F |A F A; F2 F1 F2 , F1 ; F -1 F ; Z ; Z+ = {k : k Z k 0}; N = {k : k Z k 1}; Z = Z {-, +}; Z+ = {k Z k 0}; N = {k : k Z k 1}; Q ; Q+ = {x : x Q x 0}; Q = Q {-, +}; Q+ = {x : x Q x 0}; R ; R+ {x : x R x 0}; R R {-, +}; R+ {x : x R x 0}; C ; S (M ) M ; S0 = S ( ) ; Sr = S {1, . . . , r} ; ker(F ) F ( x : x D(F ) F (x) = ).


2.

3

2.
2.1.
( ). : K; Q L. , r N, e r-, e1 , . . . , er Q, e1 , Q L( e 1 , . . . , e r ) . : Q = L N N, x1 , . . . , xN L, Q = {x1 , . . . , xN }. K {C, R, Q}; L e Q r, : . . . , er , ;Q L;

. : K {C, R, Q}; L K. : N N, x1 , . . . , xN L, x1 , . . . , xN . , (x1 , . . . , xN ) {x1 , . . . , xN } N . : N N, x1 , . . . , xN L, x1 , . . . , xN . , (x1 , . . . , xN ) L(x1 , . . . , xN ) N . : Q L, e Q r. : Q L(e1 , . . . , er ), e1 , . . . , er L, e1 , . . . , er . : Q L(e1 , . . . , er ), e L(e1 , . . . , er ) r. : Q L, e Q r. : e1 , . . . , er Q, Q L(e1 , . . . , er ). , Q = L(e1 , . . . , er ). . : K {C, R, Q}; L K; N N, x 1 , . . . , x N L. 1. e {x1 , . . . , xN } r. e L(x1 , . . . , xN ) r. 2. : e L(x1 , . . . , xN ) r; e1 , . . . , er {x1 , . . . , xN }. e {x1 , . . . , xN } r. 1. r-, L( e 1 , . e {x1 , . . . , xN } r, : r e1 , . . . , er {x1 , . . . , xN }, e1 , . . . , er . . . , er ). : e1 , . . . , er {x1 , . . . , xN } L(x1 , . . . , xN ). 1 , . . . , N K, u 1 {x1 , . . . , xN } L(e1 , . . . , er ), k , . . m xk = k em . : N, e , {x1 , . . . , xN } u L(x1 , . . . , xN ). = k xk . k = 1, N . r . , k K, -

m m u = k xk = k (k em ) = (k k )em L(e1 , . . . , er ).

, e L(x1 , . . . , xN ) r. 2. e L(x1 , . . . , xN ) r, : r N, e r-, e1 , . . . , er L(x1 , . . . , xN ), e1 , . . . , er , L( x 1 , . . . , x N ) L( e 1 , . . . , e r ) . , e1 , . . . , er {x1 , . . . , xN }. : {x1 , . . . , xN } L(x1 , . . . , xN ) L(e1 , . . . , er ). , e {x1 , . . . , xN } r.

2.2.
( ). : K {C, R, Q}; L K; e L N .


4

2.

x KN ~ : x ~

x L. , x ~ x e, : k , x = x ek . ~ x L. , x, ~ x e. x L. [x](e) x e.

. : K {C, R, Q}; L K; e L N . 1. x, y L. [x + y ](e) = [x](e) + [y ](e). 2. : K, x L. [x](e) = [x](e). ~ ~ 3. : [](e) = ( KN ). m m 4. : [ek ] (e) = k k , m = 1, r. . 1. : [x](e) + [y ](e) KN , x + y = [ x] k ( e ) e k + [ y ] k ( e ) e k = [ x] k ( e ) + [ y ] k ( e ) e k = [ x] ( e ) + [ y ] ( e ) e k . [x + y ](e) = [x](e) + [y ](e). 2. : [x](e) KN , x = [ x] k ( e ) e
k k

= [ x] k ( e ) e k = [ x] ( e ) e k .

k

[x](e) = [x](e). ~ ~ ~ 3. : KN , = k ek . [](e) = . m m 4. k = 1, N . , ek = [ek ] (e)em . , ek = k em . : m [ek ]m (e) = k m = 1, N . . : K {C, R, Q}; L K; r N, x1 , . . . , xr L, e L N . x1 , . . . , xr , [x1 ](e), . . . , [xr ](e) . . x1 , . . . , xr . 1 , . . . , r K, : k xk = , k = 1, r(k = 0). k xk = , : ~ k [ xk ] ( e ) = [ k xk ] ( e ) = [ ] ( e ) = . k = 1, r(k = 0), [x1 ](e), . . . , [xr ](e) . [x1 ](e), . . . , [xr ](e) . ~ 1 , . . . , r K, : k [xk ](e) = , k = 1, r(k = 0). k ~, : [ xk ] ( e ) = k xk = [ k xk ] ( e )
m

e m = k [ xk ] ( e )

m

~ em = m em = .

k = 1, r(k = 0), x1 , . . . , x

r

.

( ). : K {C, R, Q}; L K; e L N . : he (x) = [x](e) x L. : he , D(he ) = L, N -1 k N R(he ) = K ; he (x) = x ek x K . , he ~ ~ ~ ( ) L, e. x L. , he (x) x he .


2.2.

5

. : K {C, R, Q}; N N, x1 , . . . , xN , x1 , . . . , xN , Q = {x1 , . . . , xN }. k = 1, N . : ek (x) = 0 : x Q, x = xk ; ek (x) = 1 x = xk . 1. : e Fun(Q, K) N ( e = (e1 , . . . , eN )). 2. : Q = K. : []k (e) = (xk ) k = 1, N . . 1. : N N, e N -, e1 , . . . , eN : Q = K. . x1 , . . . , xN , : m ek (xm ) = k k , m = 1, N . . 1 , . . . , N K. m = 1, N . :
m ( k e k ) ( xm ) = k e k ( xm ) = k k = m .

. : Q = K. x Q. m = 1, N , x = xm . :
N N

( xk ) e
k =1 N

k

( x) =
k =1

( xk ) e

k

( xm ) = ( xm ) = ( x) .


k =1

( xk ) e k = . Fun(Q, K)).

: 1 , . . . , N K, k ek = ( m = 1, N . :

( k e k ) ( xm ) = ( xm ) , m = 0. , e1 , . . . , e
N

.
N

: Q = K. =
k =1

(xk )ek . , L(e1 , . . . , eN ). ,
N k =1

e

Fun(Q, K) N . 2. , = []k (e)ek . , = (xk )ek . : []k (e) = (xk )

k = 1, N . ( Fun(Q, K)). : K {C, R, Q}; N N, x1 , . . . , xN , x1 , . . . , xN , Q = {x1 , . . . , xN }. k = 1, N . : ek (x) = 0 : x Q, x = xk ; ek (x) = 1 x = xk . e Fun(Q, K) N ( e = (e1 , . . . , eN )). , e Fun(Q, K). ( K). K {C, R, Q}. , e = 1. e K 1. , e K. x K. [x](e) = x. ( KN ). : K {C, R, Q}; N N. : e1 = (1, 0, . . . , 0)T , . . . , eN = (0, . . . , 0, 1)T . e KN N ( e = (e1 , . . . , eN )). , e KN . x KN . : [x]k (e) = xk k = 1, N . , [x](e) = x.


6

2.
N2 âN

( : 10 0 0 . . e1 = . . . . 0 0 00 01 0 0 . . e2 = . . . . 0 0 00 0 0 . = . 1 . 0 0 0 0 . = . . 0 0

K

1

). : K {C, R, Q}; N1 , N2 N. 0 0 . . . 0 0 . . .

0 ··· 0 ··· . . ··· . 0 ··· 0 ··· 0 ··· 0 ··· . . ··· . 0 ··· 0 ··· ··· 0 0 . . .

0 0 . . .

00 00 0 0 . . . 0 0 . . .

00 00 0 0 . . .

, 0 0 0 0 . . , . 0 0 0 0 . . . 0 0 . . .

e

N2 N1 -

0 ··· 0 ··· . . ··· . 0 0 ··· 0 0 ··· 0 ··· 0 ··· . . ··· . 0 0 ··· 0 0 ··· 0 0 . . .

00 01 0 0 . . . 0 0 . . .

e

N2 N

1

00 00

, 0 0 0 0 . . . . 0 1

e , N2 N1 [A] (e)

KN2 âN1 N2 N1 ( e = (e1 , . . . , eN2 N1 )). e KN2 âN1 . 1 A KN2 âN1 . : [A]1 (e) = A1 , [A]2 (e) = A2 , . . . , [A]N2 N1 -1 (e) = AN2 -1 , 1 N1 = AN2 . N1 - . : O, A E 1 , O = A. OA E 1 1. : O, A1 , A2 E 2 , l E 2 , - - - - O, A1 , A2 l. OA1 , OA2 E 2 2. : O, A1 , A2 , A3 E 3 , E 3 , - - - - - - O, A1 , A2 , A3 . OA1 , OA2 , OA3 E 3 3.

2.3.
( µ (Q)). : K {C, R, Q}; L K; Q L. µ (Q) k , : k N, x1 , . . . , xk Q, : x1 , . . . , xk . ( ). : K {C, R, Q}; L K; Q L. -


2.3.

7

:

µ (Q) = . , µ (Q) = , r Nk r Nk µ (Q)(k > r , rank(Q)

r an k ( Q) = 0. µ (Q)(k r). , rank(Q) = max µ (Q) . ). , rank(Q) = +. Q.

( ). : K {C, R, Q}; L K; Q L. , dim(Q) = rank(Q). , dim(Q) Q. . : K {C, R, Q}; L K; Q L. : k N, k µ (Q). : k N, x1 , . . . , xk Q / : x1 , . . . , xk . : k N, x1 , . . . , xk Q : x1 , . . . , xk . : k N, k µ (Q). / . : K {C, R, Q}; L K; Q L. r µ (Q). : r N, k = 1, r k µ (Q) . : r N, r µ (Q). : r N, k (k Z k r) k µ (Q) . / / . : K {C, R, Q}; L 1 µ (Q). k N k µ (Q) . , / / : r µ (Q), r + 1 µ (Q). : r µ (Q), k (k / : r µ (Q), k µ (Q)(k r). : r µ (Q), k µ (Q)(k r). : r N, k = 1, r r). : r N, µ (Q) = {1, . . . , r}. r Nk µ (Q)(k > r). µ (Q) = N. K; Q L. µ ( Q) = . Zk r + 1) k µ ( Q) . / k µ ( Q) , k µ ( Q) ( k

. : K {C, R, Q}; L K; Q L. rank(Q) = 0. µ (Q) = . : r N, rank(Q) = r. : r µ (Q), k µ (Q)(k r). : r N, µ (Q) = {1, . . . , r}. rank(Q) = +. r Nk µ (Q)(k > r). , µ (Q) = N. 1 µ (Q). µ (Q) = . , rank(Q) = 0. / : r µ (Q), r + 1 µ (Q). : r µ (Q), k µ (Q)(k r). : / r N, rank(Q) = r. r Nk µ (Q)(k > r). rank(Q) = +. . : K {C, R, Q}; L K. : Q L, rank(Q) = 0. : Q , x Q(x = ). , Q = Q = {}. Q = Q = {}. : Q , x Q(x = ). : Q L, r an k ( Q) = 0. : Q L, Q . rank(Q) card(Q). : N N, x1 , . . . , xN L. : rank {x1 , . . . , xN } card {x1 , . . . , xN } N. : N N, x1 , . . . , xN L, x1 , . . . , xN . rank {x1 , . . . , xN } = N . . : K {C, R, Q}; L Q1 Q2 . rank(Q1 ) rank(Q2 ). K; Q2 L,

. : r1 = rank(Q1 ), r2 = rank(Q2 ). r1 , r2 Z+ . , r2 < r1 . : r1 N, r2 Z+ .


8

2.

: r2 + 1 N, r2 + 1 r1 = rank(Q1 ), x1 , . . . , xr2 +1 , : x1 , . . . , xr2 +1 Q1 , x1 , . . . , xr2 +1 . : x1 , . . . , xr2 +1 Q1 , Q1 Q2 , x1 , . . . , xr2 +1 Q2 . r2 + 1 > r2 = rank(Q2 ), x1 , . . . , xr2 +1 ( : x1 , . . . , xr2 +1 ). , r1 r2 .

2.4.
. : K {C, R, Q}; L K; Q L. . : r N, rank(Q) = r, x1 , . . . , xr Q, x1 , . . . , xr (x1 , . . . , xr ) Q r. . : r N, (x1 , . . . , xr ) r-, x1 , . . . , xr Q, x1 , . . . , xr . u Q. : x1 , . . . , xr Q, r + 1 > r = rank(Q), x1 , . . . , xr , u . x1 , . . . , xr , u L(x1 , . . . , xr ). , (x1 , . . . , xr ) Q r. . : K {C, R, Q}; L K; Q L. : r N, rank(Q) = r. e1 , . . . , er , : (e1 , . . . , er ) Q r. . : r N, rank(Q) = r, e1 , . . . , er , . : e1 , . . . , er Q, e1 , . . . , er rank(Q) = r, (e1 , . . . , er ) Q r. . : K {C, R, Q}; L K; Q1 , Q2 L, Q1 Q2 , dim(Q1 ) = dim(Q2 ), dim(Q2 ) = +. Q1 = Q2 . . , N = dim(Q2 ). : N Z+ , dim(Q1 ), dim(Q2 ) = N . N = 0. dim(Q1 ), dim(Q2 ) = N , Q1 , Q2 = {}. Q1 = Q2 . N = 0. N N. dim(Q1 ) = N , e1 , . . . , eN , : e1 , . . . , eN Q1 , e1 , . . . , eN . dim(Q1 ) = N , (e1 , . . . , eN ) Q1 N . Q1 = L(e1 , . . . , eN ). : e1 , . . . , eN Q1 , Q1 Q2 , e1 , . . . , eN Q2 . : e1 , . . . , eN , dim(Q2 ) = N , (e1 , . . . , eN ) Q2 N . Q2 = L(e1 , . . . , eN ). , Q1 = Q2 . . : K {C, R, Q}; N1 , N2 N, A KN2 âN1 . : r N, i1 , . . . , ir = 1, N1 , i1 < · · · < ir , j1 , . . . , jr = 1, N2 , j1 < · · · < jr . :
j1 ,...,jr i1 ,...,ir
1 Aj1 i . (A) = . . r Aj1 i 1 · Ajr i . .. .. .. r · Ajr i



1 , j1 ,,......,,ijrr (A) A r. i : r1 N, i1 , . . . , ir1 = 1, N1 , i1 < · · · < ir1 , j1 , . . . , jr1 = 1, N2 , j1 < · · · < jr1 ; r2 N, k1 , . . . , kr2 = 1, N1 , k1 < · · · < kr2 , m1 , . . . , mr2 = 1, N2 , m1 < · · · < mr2 . ,


2.4.
m ,...,m j ,...,j

9

1 1 k11...,kr r2 (A) i1 ,...,irr1 (A), : r1 < r2 , i1 , . . . , ir1 {k1 , . . . , kr2 }, , 2 j1 , . . . , jr1 {m1 , . . . , mr2 }. : r N, i1 , . . . , ir = 1, N1 , Ai1 , . . . , Air {A1 , . . . , AN1 } r. A. , Ai1 , . . . , Air {A1 , . . . , AN2 } r. : r N, j1 , . . . , jr = 1, N2 , Aj1 , . . . , Ajr A. , Aj1 , . . . , Ajr : r N, i1 , . . . , ir = 1, N1 , i1 < · · · < ir , j1 , . . . , jr = 1, N2 , j1 < · · · < jr , Ai1 , . . . , Air {A1 , . . . , AN1 } r; Aj1 , . . . , Ajr j1 ,...,jr 1 N2 {A , . . . , A } r. , i1 ,...,ir (A) A.

( ). : K {C, R, Q}; N1 , N2 N, A KN2 âN1 . : r N, i1 , . . . , ir = 1, N1 , i1 < · · · < ir , j1 , . . . , jr = 1, N2 , j1 < · · · < jr , 1 j1 ,,......,,ijrr (A) = 0. i 1 A r + 1, j1 ,,......,,ijrr (A), i ( ). {A1 , . . . , AN1 } r; (Aj1 , . . . , Ajr ) : (Ai1 , . . . , Air ) 1 N2 {A , . . . , A } r. . : r N, (Ai1 , . . {A1 , . . . , AN1 }. 1 : = j1 ,,......,,ijrr (A), i j 1 Ai1 ~ . A= . . r Aj1 i . , Ai r ) r-, Ai1 , . . . , A
i
r



1 · Ajr i . . . . . .. r · Ajr i

~ ~ , Ai1 , . . . , Air . A1 , . . . , Ar ~1 , . . . , Ar ) = 0 ( ~ . : = det(A : = 0). , Ai1 , . . . , Air . : i = 1, N1 , j = 1, N2 . : j1 1 Ai1 · Ajr Aj1 i i . . . . . . . . . ... B ( i, j ) = j r . jr Ai1 · Air Ajr i j Ai1 · Ajr Aj i i : i {i1 , . . . , ir }, j {j1 , . . . , jr }. det B (i, j ) ( ) / / A r + 1, . , det B (i, j ) = 0. i {i1 , . . . , ir }. B (i, j ) B (i, j ). , det B (i, j ) = 0. j {j1 , . . . , jr }. B (i, j ) B (i, j ). , det B (i, j ) = 0. , det B (i, j ) = 0. : d et B ( i , j ) = = ( - 1)
(r +1)+1



r +1 1

B ( i, j ) B + ( - 1)

r +1 1

( i, j ) + · · · + ( - 1 )
r +1 r +1

(r +1)+r r +1 r +1



r +1 r

B ( i, j ) B

r +1 r

( i, j ) +

(r +1)+(r +1)

B ( i, j ) B

( i, j ) =


10
(r +1)+1 r +1 1

2.
j B (i, j ) Ai1 + · · · + (-1) (r +1)+r r +1 r j B (i, j ) Air + Aj . i

= ( - 1)





det B (i, j ) = 0, : ( - 1)
(r +1)+1



r +1 1

j B (i, j ) Ai1 + · · · + (-1)

(r +1)+r



r +1 r

B (i, j ) Ajr + Aj = 0. i i

= 0, : Aj i = - ( - 1)
(r +1)+1



r +1 1

B ( i, j )
-(-1)
r +1 k

A + ··· +

r +1 k

j i1

- ( - 1)

(r +1)+r



r +1 r

B ( i, j )

j Air .

k = 1, r. , , C k (i) =
j (Ai )j = Ai = k =1 -(-1)
(r +1)+k

(r +1)+k

(B (i,j ))

j = 1, N2 .
r k =1



(B (i,j ))

. Aj = i
r

C k (i)Ajk . : i
r j
k

r

r

C k (i)Ajk = i
k =1

C k (i)(Aik )j =
k =1

C k (i)Ai

=
k =1 r

C k (i)Ai

j
k

.

j = 1, N2 , Ai =
k =1

C k (i)Aik .

Ai L(Ai1 , . . . , Air ). , (Ai1 , . . . , Air ) {A1 , . . . , AN1 } r. . ~ ~ . : K {C, R, Q}; N1 , N2 N, A KN2 âN1 , A = ( N2 âN1 ). r = 1, min {N1 , N2 } , K i1 , . . . , ir = 1, N1 , j1 , . . . , jr = 1, N2 , : 1 i1 < · · · < ir , j1 < · · · < jr , j1 ,,......,,ijrr (A) = 0, A r + 1, i j1 ,...,jr i1 ,...,ir (A), ( ). . : K {C, R, Q}; L K; Q L. e Q r. : r N, rank(Q) = r. . e Q r, : r N, e1 , . . . , er Q, e1 , . . . , er . e Q r, : Q L(e1 , . . . , er ), e L(e1 , . . . , er ) r. x L(e1 , . . . , er ). [x](e) x e. x1 , . . . , xr+1 Q. : xj = [xi ]j (e) : i = 1, r + 1, j = 1, r. : ~i r â(r +1) xK ~ , xi = [xi ](e) i = 1, N . ~ ~ ~ x = ( Krâ(r+1) ). x1 , . . . , xr+1 = ~ ~ ~ ~ ( ~ Kr ). , x1 , . . . , xr+1 ~ ~ . x = . r0 = 1, r, i1 , . . . , ir0 = 1, r + 1, ~~ j1 , . . . , jr0 = 1, r, : i1 < · · · < ir0 , j1 < · · · < j1 ,...,j 0 j1 ,...,j 0 ~ ~ ~ jr0 , i1 ,...,irr0 (x) = 0, x r0 + 1, i1 ,...,irr0 (x), ( ). , x1 , . . . , xr+1 ~ ~ L(xi1 , . . . , xir0 ). : r0 ~ ~ r < r + 1, x, . . . , xr+1 ~ ~ . , x, . . . , xr+1 ~ ~ . x1 , . . . , xr+1 . , rank(Q) = r.




11

. : K {C, R, Q}; L K; N N, x1 , . . . , xN L. dim L(x1 , . . . , xN ) = rank {x1 , . . . , xN } . N , . , r = rank {x1 , . . . , xN } . r Z+ . r r Z+ . r = 0. rank {x1 , . . . , xN } = r, x1 , . . . , xN = . : dim L(x1 , . . . , xN ) = dim {} = 0 = r. r = 0. r N. rank {x1 , . . . , xN } = r, e1 , . . . , er , : (e1 , . . . , er ) {x1 , . . . , xN } r. (e1 , . . . , er ) L(x1 , . . . , xN ) r. , d i m L( x 1 , . . . , x N ) = r . . : K {C, R, Q}; L K; Q L. : N N, x1 , . . . , xN Q, x1 , . . . , xN , N < rank(Q). u Q, : x1 , . . . , xN , u . . , u Q : x1 , . . . , xN , u . : N N, (x1 , . . . , xN ) N -, x1 , . . . , xN Q, x1 , . . . , xN . u Q. x1 , . . . , xN , u . x1 , . . . , xN , u L(x1 , . . . , xN ). , (x1 , . . . , xN ) Q N . rank(Q) = N ( : N < rank(Q)). , u Q, : x1 , . . . , xN , u . . : K {C, R, Q}; L K; Q L. : N1 N, x1 , . . . , xN1 Q, x1 , . . . , xN1 , N2 N, rank(Q). xN1 +1 , . . . , xN2 Q, N1 < N2 : x1 , . . . , xN2 . . : K {C, N2 N, rank(Q2 ) = N2 ; Q1 Q2 , : (e1 , . . . , eN1 ) eN1 +1 , . . . , eN2 , R, Q}; L K; Q2 L, N1 N, rank(Q1 ) = N1 . Q1 , N1 < N2 . : (e1 , . . . , eN2 ) Q2 .

. : e1 , . . . , eN1 Q1 , Q1 Q2 , e1 , . . . , eN1 Q2 . : e 1 , . . . , e N1 , N2 N, N1 < N2 = rank(Q2 ), eN1 +1 , . . . , eN2 , : eN1 +1 , . . . , eN2 Q2 , e1 , . . . , eN2 . : e1 , . . . , eN1 Q2 , rank(Q2 ) = N2 , (e1 , . . . , eN2 ) Q2 .


[1] . . . [2] . ., . . . [3] . . .


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[4] . ., . . . [5] . ., . ., . . . [6] . ., . . : . II, 1, 2.