Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный минерально-сырьевой университет «Горный»

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

attachments_22-12-2011_21-07-09 / razdel 1

.pdf

Скачиваний:

Добавлен:

02.04.2015

Размер:

465.65 Кб

Скачать

☆

<<< < Предыдущая 1 2 34 / 54 5 > Следующая >>>

	1	0	0	0
	1	0	0	0
	0	1	0	0
E =	0	0	1	0	.
	... ... ... ... ...
	0	0	0	1

4)@, 9 , " ) -

" O :

	0	0	0
	0	0	0
O =	0	0	0	.
O =				.
	0	0	0

0 ! ( , -

Omn .

F " E , ) " -

(, 1 9 , " O ,

( ( 9 . * , :

1., " ,

2.# ,

3.& ,

4.& .

& " -

". 0 " XIX , ")

, .

!# '# # '# #'

1. !# '#.

. * A B " # ',

) 9

aik = bik (i = 1,2,...,m; k =1,2,...,n).

, 9			A = B.	(1.45)

$ , (1.45) ( m n -
:	a11	= b11, a12	= b12,...,amn = bmn.

2. < '#

F " " " " ( " , ) !

. )'' A + B

A B , ) ! -

, " C , 9 -

) ! 9 A B , (

i = 1, 2,...,m

cik = aik + bik

(1.46)

k = 1, 2,...n

* " A B ( "

C = A + B.

a11

a12 ...

a1n

b11

b12

...

b1n

% , A =

a21

a22 ...

a2n

;

B =

b21

b22

...

b2n

...

... ... ...

...

am1

am2 ...

amm

bm1

bm2

...

bmn

	a11	+ b11	a12	+ b12	...	a1n + b1n
	a11	+ b11	a12	+ b12	...	a1n + b1n
C = A + B =	a21	+ b21	a22	+ b22	...	a2n + b2n	.
C = A + B =		...		...	...	...	.
	am1	+ bm1	am2	+ bm2	...	amn + bmn

8 '.		2				3				5
	3		4	+	1		2	=	4		6	.
	3		4		1		2		4		6
	1	0	2		0	1	5		1	1	7

# # < " ( " ). 1. , (

A + B = B + A .

2. # (

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

3. # A O

A :

A + O = A.

% , " ( "

3. C' < '# # 4

. 8 . ' '# A # 4 " -

B , 9 9 -

) 9 A , (

bik = aik		(i =1,2,...,m;		k =1,2,...,n).
* " " A ( " "
% ,		B = A.
% ,
	a11	a12 ...	a1n
	a11	a12 ...	a1n
A =	a21	a22 ...	a2n		,
A =	...	... ...	...		,
	am1	am2 ...	amm

	a11	a12		... a1n
	a11	a12		... a1n
B = A =	a21	a22		... a2n	.	(1.47)
B = A =	...	...		... ...	.	(1.47)
	am1 am2 ... amm
8 '. C
		A =	2	4
			2	4
			3	1
			1	0

= 3.

! &. ,

B = 3A = 3	2	4	=	6	12
	2	4		6	12
	3	1		9	3	.
	1	0		3	0

# # )' < " '# # 4

1. E ( ( -

	( + ) A = A + A,	(1.48)
	( A + B) = A + B,	(1.49)
, - .
2. # ( ( ":
	( A) =( ) A.	(1.50)
3.	1 A = A.	(1.51)
4.	Omn = Omn ,	(1.52)

Omn " .

Omn = Omn ,

(1.53)

* " ( ( A B - )(

A B = A + ( 1)B.

(1.54)

C' < '#

F " " ! ( " ! , -

" .

. 8 . ' '# A #.' # m × n # '# ) B

#.' # n × p " C m × p , 9

cik 9 i - A -

) 9 k - B , (

cik = ai1b1k + ai2b2k + + aimbmk (i = 1, 2,..., m;

k = 1, 2,..., p)

(1.55)

* " " A B ( " C = A B

: !.

B(n × p)

b1k

A(m ×

c(m ×

b2k

bnk

cik

ai1

ai2

ain

& ( , . . '-

') #.

A -

" 9 " (

B . , AB "

A B , BA - A -

B .

G " # 4 , "

! AB = BA . C,

A =		3	5	B =		1 5		.
A =		3	5	B =		1 5		.
		1	2			1 2
,					2	5
	AB = BA =						.

					1	1

# # )' < " '#

1. # ( ( -

:
( AB) = ( A)B = A( B),	(1.56)
(AB)C=A(BC).	(1.57)
2. E ( ( ":
( A + B)C = AC + BC.	(1.58)

3.% " ! ( -

! , " ! ", (

( AB) = BT AT .

(1.59)

4.F ( " ! A B -

9 ! :

D( AB) = D( A) D(B).	(1.60)
5 ., ( A m ×	n

En Em A , (

AEn = A; EmA = A.	(1.61)
, " A n - "
AEn = En A = A.	(1.62)
% # #" '# #
? , " a 0 )	b ,

a b =1. J b " " a . E "" 9

, ) ,

( , "

E .

. ? " A " % # ', )-

" X , " ) " "
AX = XA = E.	(1.63)
" " X , " ) " (1.63),	" %-

# A " A 1 .

@ (, A ) (

" " .

" (1.63) A X ! " . # (, -

A - " " X , X - " " A , (

( A 1) 1 = A . (1.64)

", ! "! " .

5 '# 1.3. * " , A , - ! , ( " " .

? A , ( D( A) ", -

< %. 0 D( A) = 0 , -

" < %.

&" A " n ) (

" , " (

A11

A21

An1

A11

A21

An1

D( A)

A12

A22

An2

A 1 =

A12

A22

An2

D( A)

(1.65).

A1n

A2n

Ann

A1n

A2n

Ann

D( A)

! & ' ( ) # '- '#

, ( n ! n x1, x2,..., xn :

	a x + a x + + a x = b ,
	11 1	12 2	1n n	1
a21x1 + a22x2 + + a2nxn = b2,
		........................		(1.66)
		........................		(1.66)
a x		+ a x	+ + a x	= b .
	n1 1	n2 2	nn n	n

: ( " ! ! :

	a11x1 + a12x2 + + a1nxn			b1
	a11x1 + a12x2 + + a1nxn			b1
	a21x1 + a22x2 + + a2nxn		=	b2	.	(1.67)
	..............................				.	(1.67)
	an1x1 + an2x2 + + annxn			bn
:		A (1.66) –
" n ( n × n )

a11

a12

a1n

A =

a21

a22

a2n

(1.68)

an1 an2

ann

: X ! B -

! .

X =

B =

(1.69)

% " " (

(1.67) (

AX , " ( -

B . % , (1.67), , (, -

n ! (1.66) n

(

AX = B .

(1.70)

@ (1.70) ( (1.66) -

X . E " 9 ,

(, "" X , !-

" ! ! x1, x2 ,..., xn.

*, ( (1.66) ", 9 ? (1.66)

. %, " (1.70)	A 1 ,

A 1AX = A 1B,
, ", A 1A = E, EX = X , (
X = A 1B .	(1.71)

@ (1.66) . # (1.71) 9 ?, (, ( -

. @ " -

( " (1.71) , " , -

( A 1 .

# ( ! ! -

! K [1].

			! & .# #4
3# #4# 1. & (
	2	3 0		B =	1	2	2	3
					1	2	2	3
A =					0	1 1 2			.
	3	1 1			1	3	0	1	.
					1	3	0	1

! & . % A , B - ,

A B , 9 -

A B C , ") " ! !

9 C

c11 = ( 2)( 1) + 3 0 + 0 1 = 2, c12 = ( 2)2 + 3 1+ 0( 3) = 1, c13 = ( 2)( 2) + 3( 1) + 0 0 = 1, c14 = ( 2)3 + 3 2 + 0 1 = 0,

c21 = 3( 1) + ( 1)0 +1 1 = 2, c22 = 3 2 + ( 1)1 + 1( 3) = 2,

c23 = 3( 2) + ( 1)( 1) +1 0 = 5, c24 = 3 3 + ( 1)2 +1 1 = 8.

% ,	2	1	1	0
A B =	2	1	1	0	.
	2	2	5	8

$ , ( (

AB A B . , B A

, B A .

2.3# #4# ( " AB BA ,


				A =		2	1		B =			1	3	.


						0 1						1	2
! & .			2	1			1 3				2 + 1 6 2						3		4
A B =									=						=					,

			0	1			1 2				0 1	0 + 2					1		2
B A =		1 3			2		1	=		2 + 0 1 + 3						=		2	2		.
B A =		1 3			2		1	=		2 + 0 1 + 3						=		2	2		.
	1 2				0		1			2 + 0			1 + 2					2	3

G, " AB																									BA ,
AB BA.
3# #4# 3. C "
				A =										1	2						1
														1	2						1
														2	1						2		.
													1		0						1
! &. ( T
	1	2	1		1										0						0	1+1		3	4



D( A) =	2 1 2			=	2 3 4																	= 1( 1)		2	2	= 2.
	1	0	1		1										2						2
	1	0	1		1										2						2
F ( ", (, "
		A 1 , " (1.65). # -
" 9 A :
	A			1+1		1								2						= 1 0 = 1;
				1+1		1								2
			= ( 1)
11								0							1
								0							1
	A			1+2										2		2					= (2 2) = 0;
				1+2										2		2
			= ( 1)
12												1			1
												1			1
	A			1+3										2	1					= 0 +1 = 1;
				1+3										2	1
			= ( 1)
13											1				0
											1				0
	A		= ( 1)2+1							2 1							= (2 0) = 2;
	A		= ( 1)2+1							2 1							= (2 0) = 2;
21										0					1
										0					1
	A		= ( 1)2+2										1 1					= 1+ 1 = 2;
	A		= ( 1)2+2										1 1					= 1+ 1 = 2;
22													1		1
													1		1
	A		= ( 1)2+3								1 2								= (0 + 2) = 2;
	A		= ( 1)2+3								1 2								= (0 + 2) = 2;
23												1			0
												1			0
	A		= ( 1)3+1				2									1			= 4 1 = 5;
	A		= ( 1)3+1				2									1			= 4 1 = 5;
31								1						2
								1						2
	A		= ( 1)3+2								1 1									= ( 2 2) = 4;
	A		= ( 1)3+2								1 1									= ( 2 2) = 4;
32											2				2
											2				2
	A		= ( 1)3+3						1 2								= 1 4 = 3;
	A		= ( 1)3+3						1 2								= 1 4 = 3;
33									2						1
									2						1

, (1.65)

		1	2	5			1	1	5
A 1 =	1	1	2	5	=	2		1	2
	1	0 2		4		0 1			2	.
	2	0 2		4		0 1			2	.
	2	1	2	3			1	1	3
						2		1	2
						2			2

, ( ( (, ( " -

(1.63). * " 9 A A 1 :
			1		2	1		1	1				5
								1					5
A A 1 =							2						2	=
A A 1 =			2 1 2				0		1				2	=
			1		0	1		1	1				3
							2		1				2
							2						2
	1	+ 0	+	1	1 + 2 1					5	+ 4				3		1	0	0
	1	+ 0	+	1	1 + 2 1					5	+ 4				3
	2		2						2					2
=		1 + 0 1 2 + 1+ 2 5 + 2 + 3														=	0	1	0	= E.
	1	+ 0	+	1		1 + 0 1				5		+ 0			3		0	0	1
	2	+ 0	+			1 + 0 1			2			+ 0		2
	2		2						2					2

% , A 1 .

4.3# #4# * A, B,C . C P = BC 2 A 1

B =	1 3 2	, C =	2	3	, A =	2	3
			2	3
			1	0				.
	0 2 1		2	1		1	2
			2	1

! &. C B C BC = K.

: , B -

C . @ K = BC (

, ( ( ". C 9 " (1.55):

k11 = ( 1) 2 + 3 1 + ( 2)( 2) = 5, k12 = ( 1) 3 + 3 0 + ( 2)( 1) = 1, k21 = 0 2 + 2 1 + 1( 2) = 0,

k22 = 0 3 + 2 0 + 1( 1) = 1.

% ,

<<< < Предыдущая 1 2 34 / 54 5 > Следующая >>>

Соседние файлы в папке attachments_22-12-2011_21-07-09

#
02.04.2015494.5 Кб10kontrol 1.pdf
#
02.04.2015465.65 Кб10razdel 1.pdf
#
02.04.2015254.75 Кб10razdel 2.pdf
#
02.04.2015467.57 Кб10razdel 3.pdf
#
02.04.2015738.49 Кб11razdel 4.pdf
#
02.04.20151.03 Mб11semestr1_1.pdf
#
02.04.20151.18 Mб9semestr1_2.pdf