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Уфимский Государственный Авиационный Технический Университет

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Линейная алгебра

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				1	2			6	1
				0	1			2	1
5.				0	1			2	1
5.	A =			5	5	20			0	.
				5	5	20			0
				5	0	10			−5
				5	0	10			−5
	a1 = (2; 1;−1; 1),
	r		= (1; λ −1; 2; 1),
6.	ar2		= (1; λ −1; 2; 1),
	a3		= (1; 2 −λ; λ −10; 0),
	ar4 = (4; 2λ −1; 3; 5).
	2x				− x		2		+ 3x		+ 4x	4		− x	= 0,
			1				2			3		4		5
7.	x1				+ 2x2				− 3x3		+ x4			+ 2x5	= 0,
7.		5x			− 5x				+ 12x		+ 11x			− 5x	= 0,
		5x			− 5x			2	+ 12x		+ 11x		4	− 5x	= 0,
			1					2		3			4	5
	x				− 3x			2	+ 6x		+ 3x	4		− 3x	= 0.
			1					2		3		4		5
8.	a = (−1;−3;1), b = (−2;1;1); x = (−2;0;3).

Вариант 18

x1 − 2x2 −7x3 = −5,

1.3x1 −5 x2 −19x3 = −13,−3x1 +8x2 + 26x3 =19;

	x1 +3x2 −5x3									= −6,
2. x1 + 4x2 −6x3 + 2x4 = −15,
	− x − x +				4x		− x = 3.
			1	2	3				4
			x	+3x	+x			−2x		4	=		1,
			1	2	3					4
3.	−		3x1 −8x2		−2x3 −3x4								=16,
3.		−	3x	−8x	− x			−	4x		= 21,
		−	3x	−8x	− x			−	4x		= 21,
			1	2	3					4
	−		7x	−19x −		5x			+	λx			4	= 31.
			1		2			3					4
4. а) V ={xr R3 \| 4x + x												2	+ 2x			3	= 0};
								1				2				3
	б) W ={xr R3 \| x − x										2		+5x		3		= 3}.
								1			2				3

−1

−3

−5

−7

A =

−1

−2

−3

−5

−4

a1 = (2; 1;−1; 1),

= (1; λ −1; 2; 1),

ar2

= (1; 2 −λ; λ −11; 0),

ar4 = (4; 2λ −1; 3; 5).

− 2x

+ x

− x

= 0,

2x1 −

+ 7x3

− 3x4

+ 5x5

= 0,

− 2x

+ 5x

− 7x

−

+ 7x

− 5x

+ 8x

= 0.

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

Вариант 19

x1 −3x2 −7x3 = 43,

1.3x1 −8 x2 −18x3 =112,

−2x1 + 4x2 +9x3 = −57;

x1 −3x2 +7x3 + x4 = −12,

2.x1 −2x2 +5x3 +3x4 = −7,x1 −5x2 +12x3 +3x4 = −17.

	x1 + x2 −x3								= 2,
			−5x2	+ 2x3 +		λx4			= −22,
3.	−3x1		−5x2	+ 2x3 +		λx4			= −22,
3.		−2x	−3x	+ 2x +		4x			= −11,
		−2x	−3x	+ 2x +		4x		4	= −11,
		1	2		3			4
		x	+ 2x	2	−	5x			4	=11.
		1		2					4
4. а) V ={xr R3 \| x + x							2	− 4x				3	= 0};
					1		2					3
	б) W ={xr R3 \| 2x +5x									2	+ x			3	≥ 4}.
					1					2				3

				1	4		3	7
				5	21		16	37
5.				5	21		16	37
5.	A =			0	−2		−2	−4	.
				0	−2		−2	−4
				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),
	a3		= (1; 2 −λ; λ −12; 0),
	ar4 = (4; 2λ −1; 3; 5).
	x				+ 2x	2	− 3x			+ 2x		4		= 0,
			1			2		3				4
7.	x1				− x2		− 3x3 − 4x4						− 3x5	= 0,
7.		2x			+ 3x		+ x			− 5x			+ 2x	=	0,
		2x			+ 3x	2	+ x			− 5x		4	+ 2x	=	0,
			1			2		3				4	5
	x				− 2x	2	− 2x			−	3x	4	− 5x	= 0.
			1			2		3				4	5
8.	a = (−2;−2;1), b = (3;1; 8);										x = (2;1;2).

Вариант 20

x1 + 2x2 +6x3 = −17

1.−3x1 −5 x2 −16x3 = 46,−3x1 −8x2 −21x3 = 57;

x1 −3x2 −10x3 + 2x4 = −11,

2.x1 − = −8,

3x1 − − 2x4 = −30.2x2 −8x2 7x326x3 −

x1 −2x2 −7 x3 + x4 = 9,

3.4x1 −6x2 −24x3 +λx4 = 20,

x1 − x2 −4x3 + x4 = 5,=1.4x4x1 − x2 −5x3 −

4. а) V ={xr R3 | −x1 − x2 +7x3 = 0};

б) W ={xr R3 | 3x1 − x2 + 4x3 ≤ 5}.

−3

−4

−14

−18

A =

−5

−1

−12

−4

−6

a1 = (3; 1;−1; 1),

ar2 = (1; λ −2; 2; 1),

a3 = (2; 3 −λ; λ −3; 0),

ar4 = (5; 2λ −3; 3; 5).

− 2x

+ 3x

− 4x

+ 2x

= 0,

+ 2x2

− x3

− x5

= 0,

− x

+ 2x

− 3x

= 0,

− x3

+ x4

− 2x5

= 0.

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

Вариант 21

x1 −3x2 + 7x3 = 39,

1.−3x1 +10 x2 −23x3 = −127,

12x2 + 28x33x1 − =152;

x1 −3x2 +8x3 = −14,

2.−3x1 +10x2 −26x3 = 45,

− x1 + 2x2 −5x3 =10.

x1 −3x2 −4 x3 − x4 = 0,

3.2x1 −7x2 −11x3 +λx4 = 2,−2x1 + 7x2 +11x3 −3x4 = −2,

2x2 5x4−2x3 +x1 − = 7.

4. а) V ={xr R3 | 7x1 + x2 + 2x3 = 0};

б) W ={xr R3 | −4x1 + x2 − x3 = 6}.

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

Вариант 22

x1 −2x2 + x3 = −7,

1.−3x1 +7 x2 −4x3 = 27,−2x1 + 6x2 −3x3 = 23;

x1 − 2x2 +7x3 + 2x4 = −24,

2.− x1 +3x2 −9x3 + x4 = 34,4x +

3 = −12.3x4x1 +

x1 + x2 −3 x3 + 4x4 = 0,

3.8x1 +9x2 −24x3 +λx4 =15,3x1 +6x2 −11x3 +5x4 = 7,

−2x1 − x2 +5x3 −3x4 = −4.

4.а) V ={xr R3 | −2x1 − x2 −5x3 = 0}; б) W ={xr R3 | −4x1 + x2 + 2x3 ≥ 7}.

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

Вариант 23

x1 − x2 −4x3 = −5,

1.−3x1 + 4 x2 +13x3 =12,3x1 − =16;2x2 −10x3

x1 + 2x2 −4x3 − x4 = 4,

2.− x1 − x2 +3x3 −2x4 = −6,2x1 +3x2 −6x3 −2x4 = 5.

x1 −3x2 −6x3										= 4,

3. 9x1		−23x2 −52x3			+λx4			= 36,
	−3x +8x +			18x		+3x			= −8,
		1	2	3		4
		x	−2x	−5x		+5x			=8.
		1	2	3		4
4. а) V ={xr R3 \| −x +5x							2			+ 2x	3	= 0};
					1		2				3
б) W ={xr R3 \| 3x −4x								3		= 8}.
						1		3

					1	5			6	8
					−2	−9			−11
5.					−2	−9			−11	−14
5.	A =				−4	1			−3	10	.
					−4	1			−3	10
					−2	5			3	14
					−2	5			3	14
	a1 =			(3; 1;−1; 1),
	r			(1; λ −2; 2; 1),
6.	ar2		=	(1; λ −2; 2; 1),
	a	3	=	(2; 3 −λ; λ					−6; 0),
	ar4 =			(5; 2λ −3; 3; 5).
	2x				+	x	2		− x		− x	4		+ x	= 0,
			1				2			3		4		5
7.	x1				−	x2			+ x3		+ x4			− 2x5 = 0,
7.		3x			+	3x			− 3x		− 3x			+ 4x	=	0,
		3x			+	3x		2	− 3x		− 3x		4	+ 4x	=	0,
			1					2		3			4	5
	4x				+	5x		2	− 5x		− 5x		4	+ 7x	=	0.
			1					2		3			4	5
8.	a = (2;−2;1),b = (3;3; 0);										x = (−3;3;−3).

Вариант 24

x1 + 2x2 −3x3 = −4,

1.− x1 − x2 + x3 = 0,

3x1 +8x2 −12x3 = −19;

x1 −2x2 +9x3 − x4 = 27,

2.3x1 −5x2 + 24x3 +3x4 = 67,3x1 −4x2 + 22x3 + 2x4 = 63.

x1 − 2x2 +5 x3 +5x4 =15,

3.−2x1 +5x2 −2x3 −12x4 = −6,− x1 + x2 + 4x3 −2x4 =12,−4x1 + 6x2 +λx3 −14x4 = −6.

4.а) V ={xr R3 | 7x1 − x2 +8x3 = 0};

б) W ={xr R3 | −5x1 + 2x2 − x3 ≥ 9}.

				1	5			−8	12
				−5	−24			38	−58
5.				−5	−24			38	−58
5.	A =			3	1			4	8	.
				3	1			4	8
				2	5			−6	14
				2	5			−6	14
	a1 = (3; 1;−1; 1),
	r		= (1; λ −2; 2; 1),
6.	ar2		= (1; λ −2; 2; 1),
	a	= (2; 3 −λ; λ						−7; 0),
		3
	ar4		= (5; 2λ −31; 3; 5).
	2x			−	2x	2		+	x	−
			1			2			3
7.	x1			+	2x2			− x3		+
7.		4x		−	10x			+	5x	−
		4x		−	10x		2	+	5x	−
			1				2		3
	2x			−	14x		2	+	7x	−
			1				2		3

8. a = (−1;−1;1), b = (−1;−2;−3);

x4	+	x5	=	0,
x4	−	2x5	=	0,
5x4	+	7x5	=	0,
7x4	+	11x5	=	0.

x = (0;3;−2).

Вариант 25

x1 −3x2 +12x3 = −44,

1.2x1 −5 x2 + 21x3 = −75,−2x1 + 4x2 −17x3 = 59;

x1 +3x2 +10x3 −3x4 = 5,

2.−3x1 −8x2 −27x3 −3x4 = −47,

x1 + 2x3 = 3.

		x	+ 2x		+	3 x	−4x	= 4,
		1		2		3	4
3.	−3x1 −5x2 +						9x4	= −28,
3.		− x −4x			+ x +		11x = −41,
		− x −4x			+ x +		11x = −41,
		1	2			3	4
		x	+ 7x		+	λx	−21x		= 62.
		1	2			3	4
4. а) V ={xr R3 \| 4x + 2x									2	− x			3	= 0};
							1		2				3
	б) W ={xr R3 \| −2x +3x										2	+5x			3	=10}.
							1				2				3

				1		−3		5	7
				2		−5		9	12
5.				2		−5		9	12
5.	A =			0		2		−2	−4	.
				0		2		−2	−4
				0		0		0	0
				0		0		0	0
	a1 = (3; 1;−1; 1),
	r		= (1; λ −2; 2; 1),
6.	ar2		= (1; λ −2; 2; 1),
	a3 = (2; 3 −λ; λ −8; 0),
	ar4 = (5; 2λ −3; 3; 5).
	x				+	3x	2	+	2x
			1				2		3
7.	x1				−	2x2		+	x3
7.		x			−	4x		+	x
		x			−	4x	2	+	x
			1				2		3
	3x				−	3x	2	+	4x
			1				2		3

8. a = (2;1;1), b = (1;−1;−1);

−	2x4	+	x5	=	0,
−	x4	−	x5	=	0,
+	x4	−	x5	=	0,
−	2x4	−	x5	=	0.

x = (2;−3;−3).

Вариант 26

x1 − x2 − x3 = −1,

1.2x1 − x2 + x3 = −6,− x1 −2x2 −7x3 =13;

x1 +3x2 + 4x3 −3x4 =11,

2.3x1 +10x2 +13x3 + 2x4 =12,−3x1 −11x2 −13x3 + 2x4 = −18.

x1 + 2x2 −x3 +7x4 = −25,

3.−2x1 −3x2 +5x3 −11x4 = 46,

− x1 + x2 −2x3 +3x4 = −14,6x1 +5x2 +λx3 +19x4 = −68.

4.а) V ={xr R3 | 2x1 −2x2 +3x3 = 0}; б) W ={xr R3 | −x1 +3x2 −5x3 ≤11}.

			1	4			5	6
			5	21			26	31
5.			5	21			26	31
5.	A =		−2	−2			−4	−6	.
			−2	−2			−4	−6
			−4	5			1	−3
			−4	5			1	−3
	a1 = (3; 1;−1; 1),
	r	= (1; λ −2; 2; 1),
6.	ar2	= (1; λ −2; 2; 1),
	a = (2; 3 −λ; λ						−9; 0),
	3
	ar4 = (5; 2λ −3; 3; 5).
	x		+ 2x			2	+ x		− 3x	4		+ 2x	= 0,
		1				2		3		4		5
7.	2x1		+ x2				+ x3		+ x4			− 3x5	= 0,
7.		x	+ x				+ 2x		+ 2x			− 2x	= 0,
		x	+ x		2		+ 2x		+ 2x	4		− 2x	= 0,
		1			2			3		4		5
	2x		+ 3x			2	− 5x		− 17x		4	+ 10x	=	0.
		1				2		3			4	5
8.	a = (0;−3;1), b = (−2; 0;0); x = (1;3;1).

Вариант 27

x1 −2x2 −5x3 =8,

1.2x1 −3x2 −9x3 =12,

− x1 +3x2 +7x3 = −11;

		x1 −3x2 +						2x4 = 0,
2.	−2x1 +7x2 − x3										= −14,
		2x		−4x			− x	−2x			4		=14.
		1				2	3				4
		x	− x		2	+	6 x	+ 2x					= −12,
		1			2		3			4
3.	3x1 −3x2 +17x3 +λx4 = −45,
		−2x		+ 4x			−17x			+		2x		= 45,
		1				2		3					4
		x	+					3x		+ 2x					= −6.
		1						3					4
4. а) V ={xr R3 \| 5x1 + x2 −3x3 = 0};
	б) W ={xr R3 \| −x + 2x													2	+ 4x	3	≥12}.
									1					2		3

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