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Национальный исследовательский ядерный университет (МИФИ)

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Линейная алгебра

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<<< < Предыдущая 12 / 42 3 4 > Следующая >>>

a b + a b

= C.

11 11

12 21

11 12

12 22

a21

a22

b21

b22

a21b11

+ a22b21

a21b12

+ a22b22

% :

a11b12

a11b11

a11b12

a12b21

a12b22

a12b21

det C =

a11b11

a21b11

a21b12

a22b21

a22b22

a21b11

a21b12

a22b21

a12b22 = a22b22

= a a

b11

b12

+ a a

b11

b12

+ a a

b21

b22

+ a a

b21

b22

11 22

12 22

= a a

b11

b12

+ a a

b21

b22

= a a

b11

b12

− a a

b11

b12

= det A det B .

det( AB) = det A det B.

(1.3)

/):

( .$. «+

- »/.

& - (1.3) .

% A−1 :

A A−1 = A−1 A = E , det( A A−1 ) = det A det A−1 = det E = 1 .

. . det A ≠ 0 det A−1 ≠ 0

det A det A−1 = 1

det A =

det A−1

A =

A A−1 (% * A A−1 = A−1 A = E ),

( n×n )

det

A ≠ 0 det A =

det A−1

. / A , A –

( A−1 det A ≠ 0 ).

% . " A *++, A−1

det A ≠ ±1 ,

*++ .

. * A – , det A . " det A ≠ ±1 ,

1
det A =		– .
	det A−1
(, A−1 .

§4.

& :		j1	j2	jk
	i
k -	1
, ,	i2
k k
.
	ik
. % A (n × n) , aij				–

A . /, aij

((n −1) × (n −1)) , A i -	j - .
: M ij – , aij ( M ij – !).
% Aij % (aij ) .	Aij = (−1)	i + j	M ij .
	Aij = (−1)		M ij .

& ! :

•

% i - :

ai1 Ai1 + ai 2 Ai 2 + + ain Ain

= det A .

/ i - . &

. % det A /.

•

% j - :

a1 j A1 j

+ a2 j A2 j

+ + anj Anj = det A .

j -

. &

. % det A /.

/ /

$ . " i - :

det A = a M

− a M

+ a M

+ + (−1)n+1 a

& .

ai1

ai 2

ain − i -

ai1

ai 2

ain

A = (a ) :

→

a j1

a j 2

a jn − j

i j

ai 2

ain

( n×n)

ai1

A i - j - ,

. . % A . % det A = 0 ( ).

& det A j - : a

+ a

j 2

+ + a

= det A

= 0 .

j 2

, . . Ajk = Ajk , ai1 Aj1 + ai 2 Aj 2 + + ain Ajn = 0 .

* ,

0 . % .

					n
1.	ai1 Aj1 + ai 2 Aj 2 + + ain Ajn	= δij	det A.	+ :	aik Ajk	= δij	det A		.
					k =1


					n
2.	a1i A1 j + a2i A2 j + + ani Anj	= δij	det A.	+ :	aki Akj	= δij	det A	.
					k =1

2 .

. A =				A = (a ) , % det A ≠ 0 , A−1 , % +:
				( n×n )	ij
				( n×n )

					A	A	A
					11	21	n1
	A	−1	=	1	A12	A22	An 2	, det A ≠ 0	(1.4)
	A		=	det A				, det A ≠ 0	(1.4)
				det A

					A1n	A2n	Ann

//2 ,

det A . % .//

							1
.		(1.4)				B , bij = Aij			.
.		(1.4)				B , bij = Aij		det A	.
1.	% A B = C .		Ajk =	1 aik Ajk =		1	δij det A = δij ,
cij	= aik bkj = aik		Ajk =	1 aik Ajk =		1	δij det A = δij ,
	n	n			n
	k =1	k =1	det A	det A k =1		det A
cij = δij , C = E , A B = E .
2.	% B A = D .

n	n	Aki		1		n	1
dij = bik akj =		Aki	akj =	1		akj Aki =	1	det A δij		= δij ,
dij = bik akj =			akj =	det		akj Aki =	det A	det A δij		= δij ,
k =1	k =1	det A		det	A k =1		det A
dij		= δij , D = E = B A .
$ 1 2 , AB = BA = E . * B = A−1 .
. / A , A –
.
.
« » – §3,									A−1 det A ≠ 0.
« » – - .
#.	A = A B =				B , . . A B = E .
		( n×n)			( n×n)
A – A−1 = B ( . . AA−1 = A−1 A = E ).
. & - AB = E .

det AB = det A det B = det E = 1 , det A ≠ 0 ,	A –
. * A−1 , . . AA−1 = A−1 A = E . %

B = E B = ( A−1 A) B = A−1 ( AB) = A−1E = A−1 . %, B = A−1 .
(2 × 2) .	a	b	, det A = ad − bc	≠ 0 .
(2 × 2) .	A =		, det A = ad − bc	≠ 0 .
	c	d

a	b	'	d		−c	%	1	d		−b
		→				→				.

c	d	. %		−b	a	det A	ad − bc		−c	a

a b

d −b

ad − bc

0 1 0

= E .

c d ad − bc −c a

ad − bc 0

ad − bc 0 1

(2 × 2) :

A−1

−b

ad − bc −c

4 −1

7 −4

3 4

7 −4 1

−5 3

5 7

−5 3 0

41. " 3× 3 , A ( . .																AB = BA ).
: K = {B : AB = BA} = K ( A) – A .
		0			1	0		a	b	c
			0		0	1					. % AB = BA , :
. A =			0		0	1	. %	B = d	e	f	. % AB = BA , :
			0		0	0			h
			0		0	0		g	h	j
	0 1 0 a b c			a b c				0 1 0		d e f				0 a b

	0 0 1	d e f			= d e f			0 0 1	,	g h j			=	0	d e .
	0 0 0							0 0 0			0 0 0			0
	0 0 0	g h j		g h j				0 0 0			0 0 0			0	g h
			d = g = h = 0,								a b c
												0	a
% : j = e = a,								$ : B =				0	a		b , a, b, c .
					= b.							0 0				0 0 1
			f		= b.							0 0			a	0 0 1
																	0	1	0
' &. % A =																	0	1	0	.
42. .																	1	0	0
42. .																	1	0	0

3	1	1	1		1	1	1	1		1	1	1	1

1	3	1	1	= 6	1	3	1	1	= 6	0	2	0	0	= 6 (1 2 2 2) = 48 .
1	1	3	1		1	1	3	1		0	0	2	0
1	1	1	3		1	1	1	3		0	0	0	2

43.		.
	a	b b	b		1	1	1		1
	a	b b	b		1	1	1		1
	b a b		b		0	a − b	0	0
	b	b a	b	= (a + b(n −1))	0	0	a − b		0	= (a + b(n −1))(a − b)n−1 .

	b b b		a		0	0	0	a − b

44.	.
	1 1			−λ					1		1 1 1								1			1	1	1
1	1 1			−λ					1		1 1 1								1			1	1	1
1	1 −λ 1								1		1 −λ 1								0			0	−λ −1 0
					= (n −1− λ)										= (n −1− λ)										=
1	−λ 1			1					1		−λ 1 1								0		−λ −1 0			0
−λ 1 1				1					−λ 1 1 1										−λ −1			0	0	0

										n(n−1)											n(n+1)
	σ ( j )						n−1						n−1+1				n−1							n−1
	σ ( j )	(n −1− λ) (−λ −1)					n−1	= (−1)				(−1)	n−1+1	(λ +1	− n)(λ +1)		n−1	=		(−1)		(λ +1− n)(λ +1)		n−1	.
= (−1)		(n −1− λ) (−λ −1)						= (−1)			2	(−1)		(λ +1	− n)(λ +1)			=		(−1)	2	(λ +1− n)(λ +1)			.
											2										2
5 :
5 :
1		2	n −1			n										n(n −1)
j =							,	σ ( j) = (n −1) + (n − 2) + ... +1 =								n(n −1)				.
j =		n −1	2				,	σ ( j) = (n −1) + (n − 2) + ... +1 =												.
n		n −1	2			1											2

n ( n+1)

% αn

= (−1)

n = 2 αn = −1, n = 3 αn = 1, n = 4 αn = 1,

n = 5 αn = −1, n = 6 αn = −1, n = 7 αn = 1 . .

: −1; 1; 1; −1; −1; 1; 1; −1 ...

45.

xn−1

x − xn+1

x x2

xn−2

xn−1

x − xn+1

xn−1

x xn−3

xn−2

x − xn+1

= x(x − xn+1 )n−1 = xn (1 − xn )n−1.

= 0 x = 0 xn = 1.

46.

1 x x2

xn−3

xn−2

xn−1

1 x

xn−3

xn−2

xn−1

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

1 x

xn−3

xn−2

xn−1

	1	0			0		0				0				1	x xn−3		xn−2
																2	2	2
		x − x x (x − x ) xn−3								xn−2							xn−3	xn−2
=	1	x − x x (x − x ) xn−3						(x − x )		xn−2	(x − x )			=	1	x	xn−3	xn−2
=		2	1	2	2	1	2	2	1	2	2	1	=	=		3	3	3

	1	x − x x (x − x ) xn−3						(x − x )		xn−2	(x − x )				1	x	xn−3	xn−2
	1	x − x x (x − x ) xn−3						(x − x )		xn−2	(x − x )				1	x	xn−3	xn−2
		n	1	n	n	1	n	n	1	n	n	1				n	n	n

(x2 − x1 ) (x3 − x1 ) (xn − x1 ) = ( x2 − x1 ) (x3 − x1 ) (xn − x1 ) Vn−1 ( x2 , x3 , ..., xn ) .

% , :

V ( x		, x ) =	1	xn−1	= x − x	.
2	n−1	n	1	xn	n n−1
			1	xn

* Vn ( x1 , x2 , , xn ) = (x2 − x1 )( x3 − x1 ) (xn − x1 )(x3 − x2 ) (x3 − x2 ) (xn − xn−1 ),

* Vn ( x1 , x2 , , xn ) = ∏ (xi − x j ).

i > j

i, j =1,n

§5.

% ,

. & :

a x + a x + + a x = b ,
	11 1	12 2		1n n		1
a x + a			x + + a		x = b ,
	21 1		22 2		2n n	2	(5.1)

an1 x1 + an 2 x2 + + ann xn = bn .
! aij			– , bi – , x1 , , xn				– .

+ . " c1 , cn

- (5.1), x1 = c1 , x2 = c2 , , xn = cn (5.1)

( ). ( ,

- , - .

# (5.1):

a11	a12	a1n x1		b1
a21	a22	a2n x2		= b2 .


an1	an 2		ann xn	bn

:		A = (aij ) = (aij ) – ;
x			( n×n)
			( n×n)
1
x = x2	= ( x1	x2	xn )Τ = [x1	x2	xn ]	– - ;


xn
b
1
b = b2	= (b1	b2	bn )Τ = [b1	b2	bn ] – – .


bn
% (5.1):						Ax = b	(5.2)

. ( (5.2) $ det A ≠ 0 (

, ).

$ (

, ).

. (5.2) – $ (' n × n ),

det A ≠ 0 . % % b = [b1 b2 bn ]

x = [ x1	x2	xn ] , % x = A−1b (% * ).
.	* det A ≠ 0 , A−1 . –
x = [ x1 x2 xn ]		x = A−1b .

% (5.2): Ax = A( A−1b) = AA−1b = Eb = b – , x = A−1b – - .

% , - . % y = [ y1 y2 yn ] – - ,

b . * y = Ey = A−1 Ay = A−1 ( Ay) = A−1b = x y = x –

- .

% - . $

= x = A−1b =xn

	A11		A21	An1 b1
1 A21			A22	An 2 b2		.

det A
det A

		An1	An 2		Ann bn

& k - :

a11

a1n

xk =

( A1k b1 + A2 k b2 +

+ Ank bn )

a21

a2n

det A

an1 bn ann

A . % % !.

(1750 .). $ (5.2) %

n .

xk =

= 1, n

= det A ,

k – %, % '

% ' k - ( *++ [b1 b2 bn ] .

% +

, .

% 5 .

(n +1) ( n!(n −1) +

n!−1 ) = (n +1)!(n −1) + (n +1)!− (n +1) = =

= (n +

2)! 1

−

= (n + 2)!(1

+ o(1)) n → ∞.

n + 2 n!(n + 2)

" – (

6).	2x1 + x2 + x3 = 1,
	2x1 + x2 + x3 = 1,

. &- .	x1 + 2x2 + x3	= 2,
% +.		= 21.
% +.	x1 + x2 + 2x3	= 21.

		2	1	1		1	1	1		2	1	1		2	1	1
	=	1	2	1	1 =	2	2	1	2 =	1	2	1	3 =	1	2	2	(1 7".
		1	1	2		21	1	2		1	21	2		1	1	21
% .
x + x + x = 6,								x = −4,
	1 2			3				2	= 15,
0 + x			+ 0 = −4,					x	= 15,
	2							3
	0 + 0 + x3 = 15.							x1 = −5.

§6. ! (

" ! )

& + - . + ,

. ,

x1 + x3 = 1,

1.x1 + x2 + x4 = 2, (3 , 4 ).

x2 + x3 + x4 = 3.

& - :

1 0 1 0						1					1 0 1 0				1 1 0 1 0				1 1 0 1 0				1
												0 1 −1					0 1 −1 1					−1 1
	1 1 0 1					2		~				0 1 −1		1	1	~	0 1 −1 1		1	~	0 1	−1 1	1	~
							2%
	0	1	1		1	3	#					0	1 1	1	3		1	0 2 0	2		0 0	1 0
	0	1	1		1	3						0	1 1	1	3		1	0 2 0	2		0 0	1 0	1
	1		0	0	0		0
	1		0	0	0		0
								– .
~		0	1	0	1		2	– .
		0	0	1	0		1
		0	0	1	0		1
										x			= 0,
												1	= 2 − x4 ,
- :										x2			= 2 − x4 ,	- .
										x3 = 1.						x		= 0,
																1		= 2 − t,
																x		= 2 − t,
- :																2		t .
								x1 = 0,								x3 = 1,
								x1 = 0,
																		= t.
									= 0,							x4		= t.
								x	= 0,
' - :								2					( t = 2).
								x3 = 1,
									= 2.
								x4	= 2.

1 2 3 = 1,

2.2x1 + x2 − x3 = 2, (3 , 3 ).

4x1 + 3x2 + x3 = 5.x + x + x

1		1	1		1	1 1 1				1		1 1			1	1
				−1				−1	−3
&- .	2	1			2	~	0			0	~		0	1	3	0	.
	4	3	1		5		0	−1	−3	1			0	0 0		1

				0x1 + 0x2 + 0x3 = 1													( ).
& :

a x + a x + + a x = b ,
11 1 12 2	1n n	1
a x + a x + + a x = b ,
21 1 22 2	2n n	2 (m × n).	(6.1)

am1 x1 + am 2 x2 + + amn xn = bm .

&- (6.1) x1 , x2 , , xn ,

. , - , .

! ( : - ,

# 6.

& - (6.1):

		a		a	a	b
		a		a	a	b
			11	12	1n	1
A	=	a21		a22	a2n	b2	.
A	=						.


				am 2	amn
		am1		am 2	amn	bm

" - ( . §2).

. %

- . *

, .

. ( *, - .

. "

). - % & % ' (6.1) $

% % %

A′

! a1′j1

1 ≤ j1 < j2

			I1 I2	Ir
0		0 1 * 0 * * 0 *			*
	0	0	0 1 * *	0 *	*
						r


	0	0	0 0	1 * *			(6.2).
=
0		0 0 0 0 0 *

0		0	0 0 0 0 0			(m − r)


	0	0	0 0	0 0 0

= a′	= = a′	= 1 , ;
2 j2	rjr
< < jr ≤ n ;		« * » – , .

% (m − r) ,

r ≤ m r ≤ n .

. ( ,

, . !,

, .

									1		*		*	*
-% :										0	1		*	*	.
-% :										0	0		*		.
										0	0		*

									0		0		0	1		1	0	*	0	*
																0	1	*	0	*
-% % :																0	0	0		* .
& (6.2):																	0	0	1
			1					0				0				0	0	0	1	*
			0					1
( f		=	0	,	f		=	1	, …, f		=			– A′ .
( f		=		,	f		=		, …, f		=			– A′ .
	j1					j 2				jr		1	← r-e

			0					0

% x j1 , x j 2 , , x jr ' .

, , ' ( )

. ,

- . % .

! " "

	:
	/ A′					) , b′		≠ 0 . ( .
1)	/ A′		(0 0		b′	) , b′	+1	≠ 0 . ( .
					r +1	r	+1
2)	% b′	= = b′		= 0 ( ) r < n ( . .
	r +1		m
	) - .
3)	% b′	= = b′		= 0 ( ) r = n ( . .
	r +1		m
	) - .
&						.

§7. # . Rn

5, .

. : . ( ( ) . 1

. 6,

x, y, z, f , g, h, a, b, c, – ( f1 , f2 , , fm , );

λ, λ1 , , λm ,α1 , ,αn ,α , β , γ , – ≡ ( x1 , , xn , b1 , , bm , aij );

L, L1 , , Lm , – (

. ( L {x, y, z, } %,

L :
I).	( x + y : x + y L, x, y L .
II). ) λ x : λ x L,		x L, λ .
. :
1.	x + y = y + x	(	);
2.	(x + y) + z = z + ( y + z)	(	);
3.	θ : x +θ = θ + x = x x L	( ).
	% θ – , ;
4.	. x L (−x) L : x + (−x) = θ	( );

5.	1 x = x x L	( 1 );
6.	λ1 (λ2 x) = (λ1λ2 ) x λ1 , λ2	( );
7.	(λ1 + λ2 )x = λ1 x + λ2 x λ1 , λ2	( ).
8.	λ(x + y) = λ x + λ y x, y L λ	( ).
8.	λ(x + y) = λ x + λ y x, y L λ
$ 1-8 .
.
1.	) 0:

0 x = 0 x +θ = 0 x + (x + (−x)) = (0 x + x) + (−x) = x(0 +1) + (−x) = 1 x + (−x) = x + (−x) = θ .

* : 0 x = θ . 2. x + y = z x = z + (− y).

. z + (− y) = (x + y) + (− y) = x + ( y + (− y)) = x +θ = x .

# * + !.

1." θ 0

2.x + (− y) - x − y . * x + (− y) ≡ x − y.

1.L = C [a, b] – [a, b] ;

2.L = {P (t )} – t ;

3.L = M (n × n) – n × n .

0 . %

- – n .

. 3 n – . & n

:		x = ( x1 , x2 , , xn ) xk k =		1, n	.
( n . * :

							}.
			n = {( x1 , , xn ) : xk , k = 1, n				}.
# * + !.

1.	x = ( x1 , , xn ) = y = ( y1 , , yn ) xk = yk k =					1, n		.
2.	(0, 0, , 0) 0.
" x = ( x1 , , xn )		n – n . ' xk
k – x .
. % n :
1.	(

x + y = ( x1 , , xn ) + ( y1 , , yn ) ≡ ( x1 + y1 , , xn + yn ) ; 2. )

λx = λ ( x1 , , xn ) ≡ (λ x1 , , λ xn ) λ .

. % 8:

) λ ( x + y ) = λ (( x1 , , xn ) + ( y1 , , yn )) = λ ( x1 + y1 , , xn + yn ) = (λ ( x1 + y1 ) , , λ ( xn + yn )) = = (λ x1 + λ y2 , , λ xn + λ yn ) ;

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