Высшая математика. Том 1

- , ’ -

, , -

. 8:

’ , #

, rang A < n. 0$ -

’ , ’ .

& - ’ -

		p1				l1
	X1	p2		,	X 2	l2
		=				= .
		p				l	n
		n

65 ’ -

:

1. 1’ , , ’ . ,

λ p1

λX1 = λ p2 – ’ .

λ pn

2.6 ’ ’

.

& -

.

: ’ X1, X2, …, Xk

4$ . ! ! . "’" . 8: ’ X1, X2, …, Xk AX = 0 -

’ , $ X1, X2, …, Xk -

# - ’

.

: X1, X2, …, Xk – 5 ’

A X = 0 . . X = C1 X1 + C2 X3 + + Ck X k ,C1,C2, ,Ck – , * "’ ! !-

AX = 0 .

+5 ’ , $ -

’

’ C1,C2, ,Ck . 8 , - 5-

C1,C2, ,Ck ’ ’

.

# :	X1, X2, …, Xk – 5 ’ -
	A X = 0 . & k – ’ 5-

. . rang A + k = n, n – . 8 -

, 5 ’ k ’ ,

, k = n – rang A.

91

2.19. 0$ $ "’"

! ! - "

m n :

a11x1+ a12x2 + a13x3+... +				a1n xn =b1,
a	x + a	x + a	x +... + a	x	=b ,
	21 1 22 2 23 3			2n n	,	2	.
						=b .
a	x + a	x + a	x +...+ a x
	m1 1	m2 2	m3 3	mn n		m

"A X = B , A

m×n.

c d – ’ A X = B . .

g=c–d ’ , ’

A X = 0 .

c – ’ A X = B , g – -

’ A X = 0 . . d=c+g – ’

.

. X0 – ’ -

A X = B , z – ’ A X = 0 .

. X=X0+z * "’ ! !.

+ ’ 5-

’ X1, X2, …, Xk 5 -

’ :

X = X0 + C1 X1 + C2 X3 + + Ck Xk .

8, ’ m n

’

’ .

8 , $ - ’ -

’ -

5 C1,C2, ,Ck .

# . 6 A X = B

’ , ’ , ’ .

2.20. "’ $" ! ! - * ) " $!!

m n A X = B . &-

’ , $ , -

.

:

•;

•, ;

•, -.

92

9, $ ’ , -

, # . & ? -

. – -

, $ -

, , , , , -

, -

.

! , $ 5 a11

. 0$ , -

5 x1 5. ' , , $

, a11 ≠ 0 .

a11x1

+ a12x2

+ ...

+ a1m xm

=

b1,

a21x1

+

a22x2

+ ...

+

a2m xm

=

b2,

...

...

...,

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

a

x

+

a

x

+ ...

+

a

x

=

b

n1 1

n2

2

nm m

n

a11

2 1: , ,

, i

( ).

ai1

a11x1

+

a12x2

+

...

+

a1m xm

=

b1,

a11

a11

a11

a11

a

x

+

a x

+

...

+

a

x

=

b

a

a

21 1

22 2

a

2m m

2

a

...

21

...

...

21

... ... ...

...

21

...

21 ,

...,

a11

a11

a11

a11

an1x1

+

an2x2

+

...

+

anm xm

=

bn

a

a

a

a

n1

n1

n1

n1

& 5 x1 -

a11.

a11x1

+ a12x2

+

...

+

a1m xm

=

b1,

a x

+

a′

x

+

...

+

a′

x

=

b′ ,

11 1

22

2

2m m

2

...

...

...,

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

′

′

′

a11x1

+

+

...

+

=

an2x2

anm xm

bn

2 2: 9 , ,

. - , 5 x1

, .

a x		+ a	x	+	...	+	a	x	=	b ,
	11 1	12	2				1m m			1
		a′′	x	+	...	+	a′′	x	=	b′′,
		22	2				2m m			2
				... ... ...				...	...	...,
... ... ...
		a′′	x	+	...	+	a′′	x	=	b′′
		n2	2				nm m			n

93

2 3: & 1-2 , -

. .

2 1-3 – ?. &

, :

1

0$ - k, -

, -

, , $ -

. !, k- :

0		x1 + 0		x2	+ + 0		xn	′′
								= bk .

x1, x2, , xk . 1-

:

a x		+ a x		+	a x		= b			a	a	a		b
	11 1	12	2		13	3		1		11	12	13		1
		a22x2		+	a23x3		=	b2		0	a22 a23			b2		.
					0		=	b3		0	0	0		b3

" , ,

’ .

/$" ' 1. & ’ ?

, .

/$" ' 2. !

.

	2x1 + 3x2 − x3 + x4 = 5
			− x2	+ 2x3	+ x4	= 1
1’ 64;1	3x1							.
		x +	2x + 3x + 4x
						= 6
		1	2	3	4
	6x + 4x			+ 4x	+ 6x		=	1
		1	2	3	4

,2 1: ! 5, $ -

, $ 5, $ ,

.

2	3	−1	1		5					6	9	−3	3		15
3	−1	2	1		1					6	−2 4		2		2				.
1	2	3	4		6					6	12	18	24		36
6	4	4	6		1					6	4	4	6		1

2 2: 9 , ,

. - , 5 x1 -

, .

94

6	9	−3	3		15					2	3	−3	3		5
0	−11 7		−1		−13					0	−11 7		−1		−13
0	3	21	21		21					0	1	7	7		7
0	−5	7	3		−14					0	−5	7	3		−14

2 3: & 1-2 , -

.								−1																−1
				2			3		1		5									2			3		1					5
				0			55	−35	5		65									0		55		−35	5					65								.
				0			55	385	385		385									0			0	420	380					320
				0			55	−77	−33		154									0			0	−42	−38					89
2 4: & 1-2 , -
.
							2	3	−1	1						5						2	3 −1 1				5
							0	11	−7	1						13						0 11		−7 1			13							.
							0	0	42	38						32						0	0	42	38			32
							0	0	42	38						−89						0	0	0	0		−121

- , 5 x3 4 4
.
2 x1	+3 x2	− x3			+ x4			= 5,
	11 x2	−7 x3			+ x4			= 13,
		42 x3			+38 x4			= 32,
		0		x3	+0		x4	= −121

- , –

. " ’ . 1

, rang A = 3. 1

, , . - . - ,

rang A < rang A*, ’ ./

2

0$ 5-

, . 1 -

:

a x + a x +					a x = b					a	a	a		b
	11 1	12	2		13	3		1		11	12	13		1
		a22x2		+	a23x3		=	b2		0	a22 a23			b2
					a33x3		=	b3		0	0 a33			b3

" ’ ,

, .

!.

95

x3 = b3 ;

a33

x1 = 1 (b1 − a12 x2 a11

x2 = 1 (b2

a22

− a13 x3 ) = 1 a11

− a23 x3 ) =			1								b
						b2 − a23						3		;
																	(2.2)
				a22							a33				b
			1						b
b	− a					b	− a			3				− a	3	.
								23 a
1	12 a					2								13 a
			22							33					33

" rang A =3,

rang A* =3, rang A = rang A*.

.

>, ,

, $ -

( 2.6). 9 , $ -

.

. 2.6

		1
		1				",
	2	, %
						.
		(1	(2		(3
	(1	20	25		30	2360
	(2	45	50		50	4540
	(3	35	25		20	2410

, & x1, x2, x3 , $ -

, . 9 ,

:

	0,20 x1	+ 0,25 x2	+ 0,30 x3	= 2360
	0,45 x1	+ 0,50 x2	+ 0,50 x3	= 4540
0,35 x1		+ 0,25 x2	+ 0,20 x3	= 2410.

1’ ? Maxima. " A. !

? 5 row. & ( )

L1=row(A,1), L2=row(A,2) L3=row(A,3). 2 , $ -

$ 5 ,

fpprintprec, $ 5, 5. & ?:

2 1: L2 0,20 L1.

0,45

, - .

96

1. 2.12. 7 Maxima

97

2 2: L3 0,20 L1. 0,35

- A1, 5 x1 ,

, . & A1 L*1=row(A1,1), L*2=row(A1,2) L*3=row(A1,3).

2 3: L*3 0,0278 L*2. 0,107

- A2 ,

5 5 3 ( . 2.12).

, $

Maxima.

1 : .

6 ’ , $

, . ! 4, 5, 6

5 2.2. - , , $ , -

, , 1=2296 ., 2=4070 .,

3=2945 . /

3

0$ 5

,

. 1 :

a x + a x + a x = b									a	a	a		b
	11	1	12	2	13	3	1		11	12	13		1
		a22 x2			+ a23 x3		= b2		0	a22	a23		b2
					0		= 0		0	0	0		0
-										,

– . + ’

:

x3 = C ;

x1 = 1

a11

x =	1	(b − a			x ) =					1	(b − a C );
				23
2	2					3				a22	2			23
	a22
(b − a x − a C ) =				1								1				b
							b		− a					b − a		3		− a C .
			a												23 a
1 12 2	13							1		12 a				2				13
				11							22				33

+ %, ’ -

. - , ’ . - ’ -, ’ 1. 1

, , , ( 2),

( ), rang A = rang A* < n.

", n,

. " , $ -

98

, xi , x j , , xp . 9-

, $ p=rang . 8 . "

5 , -

.

., -

, ’

A X = B . ' ’ , -

- C1,C2, ,Cn− p , -

, 5-. . , -

C1,C2, ,Cn− p ( , -

Ck ). ’ .

0$ , ’ -

. 2 5 C1 ,

’ , ’ 5 ’ , - 5 C2 – ’ . .

7 ’ -

. ! , ,

1, – . - -

, ’ 5 . -

1, – ,

’ 5 . .

) : « $ ,

A* ? ; -

». 6 , $ ,

, , ,

, $ -

’ .

& 1. + ’ :

2x2	+ x3 − 2x4 + 4x5 + x6 = 2,
8x2	+ 4x3		− 8x4	+ 13x5 +		2x6 = 14,
6x2	+ 3x3		− 6x4	+ 6x5 − x6 = 18,
x1, , x6 – .
, 9 :
0		2	1	−2	4	1			2
A* = 0		8	4	−8	13 2				14 .
0		6	3	−6	6	−1			18

! , ?. !

99

	,								−	8	,
										2
	,	−	6	. - :
			2
				0	2	1	−2		4		1		2
		A* = 0 0 0 0						−3			−2		6 .
				0 0 0 0				−6			−4		12		6
	! ,													−			. - -
	:														3
				0	2	1	−2		4		1		2
		A* = 0 0				0 0		−3			−2		6 .
				0 0		0 0 0 0							0

& ? . 1 -

, , , ( 2), -

( n=6), rang A = rang A* < n. - , ’ .

- - 5 ’ k

’ (k = n – rang A=6–2=4), ’ 4. 9 :

{2x2 + x3 − 2x4 + 4x5 + x6 = 2, −3x5 − 2x6 = 6.

"x5 ,

-

p=rang =2. + -

: x2 -

x5 . 8 x1, x3, x4, x6 ( x1

, 5 ). )-:

									{		2x2		+ 4x5			= 2 − x3 + 2x4 − x6 ,
													−3x			= 6 + 2x .
															5			6
	" 5 ,
	.
	, ,					x1 = C1 ,						x3 = C2 , x4 = C3 ,								x6 = C4 . . :
							{			2x2		+ 4x5				= 2 − C2 + 2C3 − C4 ,
													−3x			= 6 + 2C			.
															5			4
	+: x		= −2 −					2	C ,
		5		1			3				4		1				4				1			1
	x = −2x + 1−				C	2	+ C −							C	4	= 4 +		C + 1−				C + C −			C =
																								2
	2	5	2								3	2				3		4		2		2	3		4

100