ìe¢		= e	1	+ e	2	+ (8 9)e								,		ìe′		= e + e					2	- 9e		,
		1												3				1		1						3
	ïe¢		= -8e - e					2		,							ïe¢		= (9 10)e - e									2	,
4.27.	í	2			1											4.28.	í	2								1
	ï	¢	= -e1 + e2 + e3 ,														ï	e¢	= -e				+ e			+ e		,
	î	e3															î					1			2
																		3								3
		x = {9,			9,			2}.										x = {3,					-10, 10}.
	ìe¢		= e	1	+ e	2	+ (9 10)e									,	ìe′		= e	1	+ e			2	+10e			3	,
		1													3			1
	ïe¢		= -9e - e					2		,							ïe¢		= (10 9)e - e									2	,
4.29.	í	2			1											4.30.	í	2								1
	ï	¢	= -e1 + e2 + e3 ,														ï	e¢	= -e + e							+ e		,
	î	e3															î								2
																		3			1					3
		x = {10, 10,									7}.							x = {1,					9,			18}.
	ìe′		= e	1	+ e	2	+11e					,
		1									3
	ïe¢		= (11 10)e						1		- e		2	,
4.31.	í	2
	ï	¢	= -e1 + e2 + e3,
	îe3
		x = {1,			10,					10}.

Задача 5. Пусть x = (x1,					x2 , x3 ). Являются ли линейными следующие
преобразования:
Ax = (6x1 - 5x2 - 4x3 , -3x1 - 2x2 - x3 , x2 + 2x3 ),
5.1. Bx = (6	- 5x2 - 4x3 ,			3x1 - 2x2 - x3,			x2 + 2),
Cx = (x	4 , 3x - 2x - x , x + 2x						).
3		1	2	3	2	3
Ax = (5x1 - 4x2 - 3x3, 2x1 - x2 , x2 + 2),
5.2. Bx = (5x - 4x			- 3x ,	0,	x4	+ 2x	),
	1	2	3		2	3

Cx = (5x1 - 4x2 - 3x3, 2x1 - x2 , x2 + 2x3 ).

254

Ax = (4x − 3x − 2x , x , x + 2x4						+ 3x	),
1	2	3	1	1	2	3
5.3. Bx = (4x1 − 3x2 − 2x3,			x1,	x1 + 2x2 + 3x3 ),
Cx = (4x1 − 3x2 − 2x3, x1, x1 + 2x2 + 3).
Ax = (3x1 + 2x2 + x3, x3, 2x1 − 3x2 − 4x3 ),
5.4. Bx = (3x1 + 2x2 + x3,			1,	2x1 − 3x2 − 4),
Cx = (3x + 2x + x , x , 2x4					− 3x − 4x		).
1	2	3	3	1	2	3
Ax = (x1, x1 − 2x2 − 3, 4x1 − 5x2 − 6),
5.5. Bx = (x ,	x − 2x − 3x ,			4x4	− 5x − 6x		),
1	1	2	3	1	2	3
Cx = (x1, x1 − 2x2 − 3x3, 4x1 − 5x2 − 6x3 ).
Ax = (2x + x , x − 2x , 3x − 4x2						− 5x	),
1	2	2	3	1	2	3
5.6. Bx = (2x1 + x2 ,		x2 − 2x3 ,		3x1 − 4x2 − 5x3 ),
Cx = (2x1 + x2 , x2 − 2x3 , 3x1 − 4x2 − 5).
Ax = (x1, x1 + 2x2 + 3x3 , 4x1 + 5x2 + 6x3 ),
5.7. Bx = (x1,	x1 + 2x2 + 3, 4x1 + 5x2 + 6),
Cx = (x , x + 2x + 3x , 4x4					+ 5x + 6x		).
1	1	2	3	1	2	3
Ax = (3x1 − 2x2 − x3, 1, x1 + 2x2 + 3),
5.8. Bx = (3x − 2x		− x ,	0,	x3 + 2x + 3x ),
1	2	3		1	2	3
Cx = (3x1 − 2x2 − x3, x3 , x1 + 2x2 + 3x3 ).
Ax = (2x − x , x , x + 2x + 3x4 ),
1	2	3	1	2	3
5.9. Bx = (2x1 − x2 ,		x3,	x1 + 2x2 + 3x3 ),
Cx = (2x1 − x2 , 1, x1 + 2x2 + 3).

255

Ax = (x3, 2x1 + 3x2 + 4x3, 5x1 + 6x2 + 7x3 ),

5.10. Bx = (x3 ,	2x1 + 3x2 + 4,					5x1 + 6x2 + 7),
Cx = (x , 0, 5x4				+ 6x + 7x ).
3			1		2		3
Ax = (6x1 − 5x2 − 4x3, 3x1 − 2x2 − x3 , 0),
5.11. Bx = (6x1 − 5x2 − 4,					3x1 − 2x2 − x3,				0),
Cx = (6x − 5x − 4x , 3x − 2x − x2 , 0).
1	2			3		1		2	3
Ax = (5x − 4x		2	− 3x , 2x − x					, x2 ),
1				3		1	2		3
5.12. Bx = (5x1 − 4x2 − 3x3 ,					2x1 − x2 , 1),
Cx = (5x1 − 4x2 − 3x3 , 2x1 − x2 , x3 ).
Ax = (4x − 3x − 2x , x						2 , x + 2x ),
1	2			3	1		2		3
5.13. Bx = (4x1 − 3x2 − 2x3,					x1,		x2 + 2x3 ),
Cx = (4x1 − 3x2 − 2, x1, x2 + 2).
Ax = (3x1 + 2x2 + x3, 0, x1 − 2x2 − 3x3 ),
5.14. Bx = (3x1 + 2x2 +1,				0,		x1 − 2x2 − 3),
Cx = (3x + 2x + x , 0, x2							− 2x − 3x ).
1	2		3			1		2	3
Ax = (x1, x2 − 2x3 , 3x1 − 4x2 − 5),
5.15. Bx = (x ,	x2 ,		3x − 4x			− 5),
1	2		1		2
Cx = (x1, x2 − 2x3 , 3x1 − 4x2 − 5x3 ).
Ax = (2x + x , x2				, 2x − 3x − 4x ),
1	2		3		1		2		3
5.16. Bx = (2x1 + x2 ,			x3 ,		2x1 − 3x2 − 4x3 ),

Cx = (2x1 + x2 , x3 , 2x1 − 3x2 − 4).

256

Ax = (x1, x2 + 2x3 , 3x1 + 4x2 + 5x3 ),
5.17. Bx = (x1,	x2 + 2x3 ,		3x1 + 4x2 + 5),
Cx = (x , x2 + 2x , 3x + 4x + 5x							).
1	2	3	1	2	3
Ax = (3x1 − 2x2 −1, 0, x1 + 2x2 + 3x3 ),
5.18. Bx = (3x2	− 2x	− x ,	0,	0),
1	2	3
Cx = (3x1 − 2x2 − x3, 0, x1 + 2x2 + 3x3 ).
Ax = (2x2	− x , x , 2x + 3x ),
1	2	3	2	3
5.19. Bx = (2x1 − x2 ,		x3 ,	2x2 + 3x3 ),
Cx = (2x1 − x2 , x3 , 2x2 + 3).
Ax = (0, x1 + 2x2 + 3x3 , 4x1 + 5x2 + 6x3 ),
5.20. Bx = (0,	x1 + 2x2 + 3x3 , 4x1 + 5x2 + 6),
Cx = (0, x2 + 2x + 3x , 4x + 5x + 6x ).
	1	2	3	1	2		3
Ax = (6x1 − 5x2 − 4x3 , 3x1 − 2x2 − x3 , x2 ),
5.21. Bx = (6x1 − 5x2 − 4,			3x1 − 2x2 − x3,				x2 ),
Cx = (6x − 5x − 4x3, 3x − 2x − x , 0).
1	2	3	1	2		3
Ax = (5x1 − 4x2 − 3, 2x1 − x2 , x1 + 2x2 + 3x3 ),
5.22. Bx = (5x − 4x		− 3x3 ,	2x − x ,		x + 2x			+ 3x	),
1	2	3	1	2	1		2	3
Cx = (5x1 − 4x2 − 3x3, 2x1 − x2 , x1 + 2x2 + 3x3 ).
Ax = (4x − 3x3		− 2x , x + x , 0),
1	2	3	1	3
5.23. Bx = (4x1 − 3x2 − 2x3,			x1 + x3,		2x1 + 3x2 + 4x3 ),
Cx = (4x1 − 3x2 − 2, x1 + x3 , 2x1 + 3x2 + 4x3 ).

257

Ax = (3x1 + 4x2 + 5x3 , 6x1 + 7x2 + 8x3, 9x1 + x3 ),

5.24. Bx = (3x1 + 4x2 + 5,			6x1 + 7x2 + 8, 9x1 + x3 ),
Cx = (3x + 4x + 5x3			, 6x + 7x + 8x , 0).
1	2	3			1	2	3
Ax = (2x1 + 3x2 + 4, 5x1 + 6x2 + 7, 8x1 + x3 ),
5.25. Bx = (2x + 3x		+ 4x3	,	5x + 6x			+ 7x ,	0),
1	2	3			1	2	3
Cx = (2x1 + 3x2 + 4x3, 5x1 + 6x2 + 7x3, 8x1 + x3 ).
Ax = (x3 + x , 2x + 3x + 4x , 0),
1	3	1		2		3
5.26. Bx = (x1 + x3,		2x1 + 3x2 + 4x3,					5x1 + 6x2 + 7x3 ),
Cx = (x1 +1, 2x1 + 3x2 + 4, 5x1 + 6x2 + 7x3 ).
Ax = (3x1 − 2x2 − x3, x2 + 2x3 , 3x1 + 4x2 + 5x3 ),
5.27. Bx = (3x1 − 2x2 −1,			x2 + 2,			3x1 + 4x2 + 5x3 ),
Cx = (3x − 2x − x3 , x + 2x , 0).
1	2	3		2		3
Ax = (2x1 − x2 , x1 + 2x2 + 3, 4x1 + 5x2 + 6x3 ),
5.28. Bx = (2x − x3,		x + 2x			+ 3x ,		0),
1	2	1	2			3
Cx = (2x1 − x2 , x1 + 2x2 + 3x3 , 4x1 + 5x2 + 6x3 ).
Ax = (x3 + 2x + 3x , 4x + 5x + 6x , 7x + 8x										),
1	2	3			1	2	3	1	2
5.29. Bx = (x1 + 2x2 + 3x3,			4x1 + 5x2 + 6x3,					7x1 + 8x2 ),
Cx = (x1 + 2x2 + 3, 4x1 + 5x2 + 6, 7x1 + 8x2 ).
Ax = (x2 + 2x3 , 3x1 + 4x2 + 5x3, 6x1 + 7x2 + 8x3 ),
5.30. Bx = (x2 + 2,		3x1 + 4x2 + 5,				6x1 + 7x2 + 8x3 ),
Cx = (x3 + 2x , 3x + 4x + 5x , 6x + 7x + 8x										).
2	3	1			2	3	1	2	3

258