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Уфимский Государственный Авиационный Технический Университет

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Линейная алгебра

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				1	−2	−6	5
				−1	3	8	−7
5.				−1	3	8	−7
5.	A =			−5	5	20	−15	.
				−5	5	20	−15
				1	4	6	−7
				1	4	6	−7
	a1 = (1; 1;−1; 1),
	r		= (1; λ; 2; 1),
6.	ar2		= (1; λ; 2; 1),
	a3		= (0; 1−λ; λ −10; 0),
	ar4 = (3; 2λ +1; 3; 5).
	6x			+ x		− 3x		+ 9x	4	+ 5x	= 0,
			1		2		3		4	5
7. 6x1				+ 5x2		− 3x3		+ 9x4		+ 7x5 = 0,
		2x		+ 4x		− x		+ 3x		+ 2x	=	0,
			1		2		3	4		5
	4x			+ 7x		− 2x		+ 6x		+ 5x	=	0.
			1		2		3		4	5

8. a = (−1;−2;1), b = (3;2; 7); x = (0;2;−3).

Вариант 8

x1 − x2 +3x3 = −13,

1.− x1 + 2 x2 −4x3 = 20,2x1 −3x2 +8x3 = −37;

x1 +3x2 −12x3 −2x4 =12,

2.−3x1 −8x2 +33x3 + x4 = −18,−2x1 −4x2 +19x3 + 2x4 = −12.

x1 +3x2 −12 x3 −3x4 = 24,

3.−2x1 −5x2 + 21x3 + 2x4 = −42,− x1 −5x2 +19x3 +5x4 = −37,−2x1 −6x2 +24x3 +λx4 = −48.

4.а) V ={xr R3 | 3x1 + 2x2 − x3 = 0};

б) W ={xr R3 | −x1 − x2 +7x3 ≤ −8}.

				1	4		3	−9
				1	5		4
5.				1	5		4	−11
5.	A =			2	2		0	−6	.
				2	2		0	−6
				−4	1		5	2
				−4	1		5	2
	a1 = (1; 1;−1; 1),
	r		= (1; λ; 2; 1),
6.	ar2		= (1; λ; 2; 1),
	a3		= (0; 1−λ; λ −11; 0),
	ar4 = (3; 2λ +1; 3; 5).
	5x			+ 6x		2		+ x		+ 10x		4	+ 7x	= 0,
			1			2			3			4	5
7. 5x1				+ x2				+ 2x3		+ 5x4			+ 4x5		= 0,
		4x		+ 3x		2		+ x		+ 7x	4		+ 5x		= 0,
			1			2			3		4		5
	3x			+ 2x		2		+ x		+ 4x	4		+ 3x		= 0.
			1			2			3		4		5
8.	a = (0;−2;1), b = (−1;3; 6); x = (−2;3;−3).

Вариант 9

x1 −3x2 −9x3 = −21,

1.2x1 −5 x2 −16x3 = −37,−2x1 +5x2 +17x3 = 39;

x1 + 2x2 +5x3 − x4 = −8,

2.−3x1 −5x2 −12x3 − 2x4 = 9,= −1.2x4x1 −

		x − x +x				+3x	=1,
		1	2		3	4
3.		2x1 − x2 +		4x3 −4x4 = −23,
3.		−3x	+6x	+ 4x +			3x		=		3,
		−3x	+6x	+ 4x +			3x		=		3,
		1	2			3	4
	−5x		+12x	2	+11x		+λx				= −18.
		1		2		3			4
4. а) V ={xr R3 \| x +5x								2	−3x			3	= 0};
						1		2				3
	б) W ={xr R3 \| 2x + 2x									2	+ x		3	= −9}.
						1				2			3

				1	2	−6	0
				0	1	−2
5.				0	1	−2	−1
5.	A =			3	−5	4	11	.
				3	−5	4	11
				1	1	−4	1
				1	1	−4	1
	a1 = (1; 1;−1; 1),
	r		= (1; λ; 2; 1),
6.	ar2		= (1; λ; 2; 1),
	a3		= (0; 1−λ; λ −12; 0),
	ar4 = (3; 2λ +1; 3; 5).
	13x				− 4x	2	− x		− 4x	4	− 6x	= 0,
			1			2		3		4	5
7. 11x1					− 2x2		+ x3		− 2x4		− 3x5	= 0,
		5x			+ 4x	2	+ 7x		+ 4x	4	+ 6x	=	0,
			1			2		3		4	5
		7x			+ 2x	2	+ 5x		+ 2x	4	+ 3x	= 0.
			1			2		3		4	5
8.	a = (−3;−3;1), b = (−3;2;−3); x = (0;1;−2).

Вариант 10

x1 +3x2 +9x3 = −1,

1.− x1 −2 x2 −7x3 = 2,

3x1 +10x2 +30x3 = −3;

=10,

2.3x1 −8x2 +5x3 − x4 = 23,x1 −5x2 +5x3 −3x4 =16.

x1 + 2x2 −3 x3 +3x4 = −6,

3.−6x1 −8x3 + 4x3 + λx4 = −2,− x1 −5x2 +13x3 +5x4 = 33,

3x1 + 7x2 −12x3 +5x4 = −26.

4.а) V ={xr R3 | −x1 −4x2 +3x3 = 0};

б) W ={xr R3 | 3x1 − x2 +5x3 ≥ −10}.3x2 + 2x3x1 −

					1	1		1		0
					−3 −2			−	1	1
5.					−3 −2			−	1	1
5.	A =				1	−2		−	5	−3	.
					1	−2		−	5	−3
					1	−1		−	3	−2
					1	−1		−	3	−2
	a1 =			(2; 1;−1; 1),
	r			(1;		λ −1; 2; 1),
6.	ar2		=	(1;		λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ				−3; 0),
	ar4 =			(4; 2λ −1; 3; 5).
	15x					+ 2x	2		+ 4x			− 3x		4	+ 9x	= 0,
				1			2				3			4	5
7.		3x1				+ 20x2			+ 5x3			− 2x4			+ 6x5 = 0,
7.		3x				+ 6x			+ 2x			− x			+ 3x	= 0,
		3x				+ 6x	2		+ 2x			− x	4		+ 3x	= 0,
			1				2				3		4		5
		9x				+ 4x	2		− 3x			− 2x		4	+ 6x	=	0.
			1				2				3			4	5

8. a = (3; 0;1), b = (2;−3;−6); x = (−1;0;0).

Вариант 11

x1 +3x2 −5x3 = −26,

1.3x1 +10 x2 −16x3 = −85,3x1 +12x2 −17x3 = −95;

x1 + 2x2 +3x3 −2x4 =10,

2.2x1 +5x2 +9x3 −2x4 = 21,2x1 +5x2 +10x3 + x4 =17.

x1 − x2 +x3 + 4x4 = 7,

3.−4x2 −10x3 +λx4 =10,2x1 + x2 +9x3 −3x4 =1,x1 + 3x3 −3x4 = −1.

4.а) V ={xr R3 | 4x1 − x2 +5x3 = 0};

б) W ={xr R3 | x1 +5x2 − x3 ≤ −11}.

					1	−3		7	−8
					0	1		−2	2
5.					0	1		−2	2
5.	A =				1	2		−3	2	.
					1	2		−3	2
					−5	5		−15	20
					−5	5		−15	20
	a1 =			(2; 1;−1; 1),
	r			(1; λ −1; 2; 1),
6.	ar2		=	(1; λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ −4; 0),
	ar4 =			(4; 2λ −1; 3; 5).
		6x			+ 5x		2	+ 7x		+ 5x	4	+ 3x	= 0,
			1				2		3		4	5
7. 14x1					+ 5x2			+ 3x3		+ 9x4		− x5	= 0,
		4x			+ 5x		2	+ 8x		+ 4x	4	+ 4x	=	0,
			1				2		3		4	5
		8x			+ 5x		2	+ 4x		+ 7x	4	+ 2x	=	0.
			1				2		3		4	5
8.	a = (−2;−3;1), b = (−3;1;−3); x = (0;−1;2).

Вариант 12

x1 −2x2 −9x3 = −42,

1.x1 − x2 −6x3 = −26,− x1 +5x2 +19x3 = 94;

x1 +3x2 + 2x4 = 6,

2.3x1 +10x2 + x3 =16,

−2x1 −4x2 +3x3 = 3.

		x1 −3x2														= 6,
	−3x			+10x					2	+			x	4		= −16,
3.		1							2					4
3.		− x	+ x			−4x						− x				= −7,
		− x	+ x		2	−4x					3	− x			4	= −7,
		1			2						3				4
	5x −14x						2	+λx					3			= 29.
		1					2						3
4. а) V ={xr R3 \| 7x + x																	2	− x			3	= 0};
													1				2				3
	б) W ={xr R3 \| −x + 2x																		2	+3x			3	= −12}.
														1					2				3

					1	1		0	1
					−3	4		−1 −5
5.					−3	4		−1 −5
5.	A =				3	−5		2	7	.
					3	−5		2	7
					−1	−3		4	7
					−1	−3		4	7
	a1 =			(2; 1;−1; 1),
	r			(1; λ −1; 2; 1),
6.	ar2		=	(1; λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ				−5; 0),
	ar4 =			(4; 2λ −1; 3; 5).
		5x			+ 7x		2		+ 4x		+ 6x	4	+ 6x	= 0,
			1				2			3		4	5
7. 15x1					+ 30x2				+ 7x3		+ 8x4		+ 3x5	= 0,
		9x			+ 6x		2		+ 5x		+ 8x	4	+ 9x	=	0,
			1				2			3		4	5
		6x			+ 9x		2		+ 3x		+ 4x	4	+ 3x	= 0.
			1				2			3		4	5

8. a = (−2; 0;1), b = (3;2; 6); x = (−2;−3;2).

Вариант 13

x1 −2x2 +8x3 = −27,

1.−3x1 +7 x2 −27x3 = 91,− x1 = 3;− x3

	x1 + x2 −2x3 −2x4				=14,
2. − x1 +3x3 −				x4 = −6,
−3x		− x	+9x	−3x	= −19.
	1	2	3	4

x1 − x2 +x3 + x4 = −5,

3.−4x1 +3x2 −5x3 +λx4 = 21,− x1 +3x2 + 2x3 −4x4 = 4,2x1 − x2 +3x3 + 2x4 = −11.

4.а) V ={xr R3 | −3x1 + x2 −4x3 = 0};

б) W ={xr R3 | 2x1 +5x2 − x3 ≥ −13}.

								1			−1			4						4
								1			0			2						2
5.	A =							1			0			2						2
5.	A =					−4				−5				2						2			.
						−4				−5				2						2
						−3					3			−12
						−3					3			−12				−12
	a1 =				(2; 1;−1; 1),
	r				(1; λ −1; 2; 1),
6.	ar2		=		(1; λ −1; 2; 1),
	a	3	=		(1; 2 −λ; λ −6; 0),
	ar4 =				(4; 2λ −1; 3; 5).
	x							+			x	2		+ 3x									− 2x				4	+ 3x	= 0,
			1									2						3									4	5
7.	2x1 +									2x2				+ 4x3									− x4					+ 3x5	= 0,
7.		3x		+						3x				+		5x							− 2x					+ 3x	= 0,
		3x		+						3x			2	+		5x							− 2x				4	+ 3x	= 0,
			1										2					3									4	5
	2x				+					2x			2	+		8x							− 3x				4	+ 9x	=	0.
			1										2					3									4	5
8.	a = (2; 0;1),b = (−3;−3;6); x = (3;2;3).
																		Вариант 14
	x −3x													= −8,
1.			1					2
1.	3x1 −8 x2 + x3 = −20,
	2x				−8x					− x				= −21;
			1						2				3
					x1 −3x2 −6x3 − x4													= 21,
2. −2x1 +7x2 +15x3 −2x4 = −60,
			3x				−8x			2	−14x +					3x					= 62.
					1					2				3				4
			x			+3x				+3 x + x					4		= −1,
			1						2					3	4
3.			x1 + 4x2 +5x3 −5x4															= 20,
3.		−3x						−	11x				−	12x	+3x							= −18,
		−3x						−	11x				−	12x	+3x							= −18,
						1					2			3							4
			7x					+	27x				+31x				+	λx					=		77.
						1					2			3							4
4. а) V ={xr R3 \| 2x −3x																				2		+ x			3	= 0};
														1						2					3
	б) W ={xr R3 \| 4x + x																		2		+ x			3		≤ −14}.
														1					2					3

					1	−3			−4	8
					2	−5			−6	14
5.					2	−5			−6	14
5.	A =				3	−2			2	10	.
					3	−2			2	10
						−1			−8	−4
			−3			−1			−8	−4
	a1 =			(2; 1;−1; 1),
	r			(1; λ −1; 2; 1),
6.	ar2		=	(1; λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ					−7; 0),
	ar4 =			(4; 2λ −1; 3; 5).
		2x			+ x		2		+ x		− 2x		4	+ 4x		= 0,
			1				2			3			4	5
7. 13x1					+ 8x2				+ 4x3		− 3x4			+ 6x5		= 0,
		5x			+ 4x			2	+ 2x		− 3x		4	+ 6x	= 0,
			1					2		3			4	5
		3x			+ 2x			2	+ x		− x	4		+ 2x		=	0.
			1					2		3		4		5
8. a = (−2;−3;1), b = (−3;2; 0); x = (0;−1;−3).
										Вариант 15
			x1 − x2						= −2,
1. −2x1 +3 x2 + 2x3 = 9,
			3x		−5x −3x = −14;
				1		2			3
			x1 −2x2 +3x3 −3x4 = 3,
2. − x1 +3x2 −4x3										= −12,
	−2x				+	6x −7x				= −21.
				1		2			3

x1 +3x2 +x3 +5x4 = −4,

3.−2x1 −9x2 −3x3 +λx4 = −23,x1 + x2 + 3x4 = −7,− = −13.8x2 −2x33x1 −

4. а) V ={xr R3 | x1 −4x2 + 2x3 = 0}; б) W ={xr R3 | 3x1 +5x2 − x3 = −15}.

					1	2			−6	5
					0	1			−2	2
5.					0	1			−2	2
5.	A =				5	5			−20	15	.
					5	5			−20	15
					−4	−5			18
					−4	−5			18	−14
	a1 =			(2; 1;−1; 1),
	r			(1; λ −1; 2; 1),
6.	ar2		=	(1; λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ −8; 0),
	ar4 =			(4; 2λ −1; 3; 5).
	2x				+ 3x			2	+ x		− 6x	4	+ 9x	= 0,
			1					2	3			4	5
7.	x1								− 2x3		+ 2x4		+ 3x5	= 0,
7.		2x			+ x				+ 4x		+ 2x		+ 3x	= 0,
		2x			+ x		2		+ 4x		+ 2x	4	+ 3x	= 0,
			1				2			3		4	5
	3x				+ 2x			2	+ 5x		+ 4x	4	+ 6x	=	0.
			1					2		3		4	5

8. a = (3; 2;1), b = (−1;2;−1); x = (−3;−2;−3).

Вариант 16

x1 +3x2 + 3x3 = 53,

1.3x1 +10 x2 + 9x3 =171,

2x1 +3x2 + x3 = 66;

x1 −3x2 −7x3 −3x4 = 22,

2.−2x1 +7x2 +16x3 −2x4 = −33,x1 − + =17.x44x2 −8x3

		x	−3x	−9 x −4x = −7,
		1	2		3	4
3.	− x1 + 4x2 +11x3 +					3x4 = 9,
3.		2x	−3x	−11x		= −7,
		2x	−3x	−11x		= −7,
		1	2		3
	7x		−16x		−51x +	λx = −37.
		1		2	3	4
4. а) V ={xr R3 \| x1 −5x2 − x3 = 0};
	б) W ={xr R3 \| 3x + 4x						2	+ x	3	≥1}.
					1		2		3

					1	0		−2	−1
					1	1		−3	−3
5.					1	1		−3	−3
5.	A =				−1	−1		3	3	.
					−1	−1		3	3
					−4	5		3	−6
					−4	5		3	−6
	a1 =			(2; 1;−1; 1),
	r			(1; λ −1; 2; 1),
6.	ar2		=	(1; λ −1; 2; 1),
	a	3	=	(1; 2 −λ; λ −9; 0),
	ar4 =			(4; 2λ −1; 3; 5).
	2x				+ 3x		2	+ 7x		+ x	4		+ 2x	= 0,
			1				2		3		4		5
7.	x1				+ 2x2			+ 3x3		+ 2x4			+ 4x5 = 0,
7.		3x			+ 2x			+ x		+ 2x			+ 4x	=	0,
		3x			+ 2x		2	+ x		+ 2x		4	+ 4x	=	0,
			1				2		3			4	5
	4x				+ 3x		2	+ 2x		+ 3x		4	+ 6x	=	0.
			1				2		3			4	5
8.	a = (2;1;1), b = (3;−2;−4); x = (−1;2;3).

Вариант 17

x1 −3x2 −4x3 = −18,

1.2x1 −5 x2 −6x3 = −29,− =8;x2 + x3x1 +

	x1 −3x2 −6x3 + x4 = −7,
2. −3x1 +10x2 + 21x3					= 23,
	x	−5x	−11x +3x	4	= −17.
	1	2	3	4

x1 −2x2 −5 x3 + 2x4 =14,

3.− x1 +3x2 −5x3 −4x4 = 4,x1 − x4 =1,= −2.2x43x2 +λx3 +3x1 −

4. а) V ={xr R3 | 2x1 +3x2 − x3 = 0};

б) W ={xr R3 | −2x1 − x2 +5x3 ≤ 2}.

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