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Рубрика:
(0, 1, 0, . . . , 0, −1) . . . (0, 0, 0, . . . , 1, −1)
a
1
, a
2
, a
4
x
1
−
x
3
− x
4
= 0 x
2
+ x
3
− x
4
= 0
a
1
, a
2
, a
3
, b
1
c
1
= a
1
+ a
2
+ a
3
= b
1
+ b
2
= (1, 2, 2, 1) c
2
= 2a
2
+
2a
3
= b
1
+ b
3
= (2, 2, 2, 2) M(−2, 1, 0, 3)
(5, 2, −4, −3), (0, 1, 1, 7)
a
1
, a
2
, a
5
x
1
−x
2
−2x
3
= 0
x
1
− x
3
+ 2x
4
= 0 2x
1
+ x
2
− x
5
= 0
a
1
, a
2
, a
3
, b
2
b
1
= −2a
1
+ a
2
+ a
3
b
3
= 5a
1
− a
2
− 2a
3
e
i
L
1
L
2
i
n−1
n
−1
n
L
2
L
1
1
n
(0, 1, −1, −2, −3) 1) ∅ (2, −1, 4, 5)
x
1
− 3x
2
− 2x
3
+ 3 = 0 3x
2
+ 2x
3
− x
4
− 4 = 0 x
1
+ x
2
− 3x
3
+
x
4
= 0 b
1
= (2, −2, −1, 0), b
2
= (1, 1, 0, −1)
y = 3a
1
− 2a
2
= (1, −1, −1, 5), z = (3, 0, −2, −1)
π
3
M
1
(
9
7
,
9
7
,
3
7
, −
5
7
)
√
3
2
(−
16
15
,
16
15
,
43
15
, −
42
15
)
√
1022
7
x
1
= 1, x
2
= λ + 1, x
3
= λ + 1, x
4
= λ + 1
6x
1
− 9x
2
− x
3
= 0, x
2
+ x
4
= 0 y = 2a
1
− a
2
=
(3, 1, −1, −2), z = (2, 1, −1, 4)
π
6
M
1
(1, −
1
2
, 2,
3
2
)
2
√
7
7
(1, −2, 2, 2)
√
15
√
465
6
√
670
10
M
0
(10, 6), P (
1
2
, 2) 2x − y − 12 = 0 x + y − 3 = 0
x = 0, y = 0 x − y = 0 2x − 2y − 3 = 0 4x − y = 0
ex
0
= −ex ey
0
= 5ey x
0
= 5x − 3y + 8 y
0
= −3x + 2y − 3
x
0
= x+8 y
0
= 4x−5y+14 x
0
= −x+2y−8 y
0
= 4x−3y+24
πab x
0
= x − y + 1 y
0
= x + y + 2
(2, 1) 2x + y − 3 = 0
ϕ
80. Ðàçìåðíîñòü ðàâíà 3.
(0, 1, 0, . . . , 0, −1), . . ., (0, 0, 0, . . . , 1, −1).
Áàçèñ îáðàçóþò, íàïðèìåð, âåêòîðû a1 , a2 , a4 . 81. Íàïðèìåð, x1 −
x3 − x4 = 0 , x2 + x3 − x4 = 0 . 82. Áàçèñ ñóììû ñîñòîèò, íà-
ïðèìåð, èç âåêòîðîâ a1 , a2 , a3 , b1 . Áàçèñ ïåðåñå÷åíèÿ, íàïðèìåð, èç
âåêòîðîâ c1 = a1 + a2 + a3 = b1 + b2 = (1, 2, 2, 1); c2 = 2a2 +
2a3 = b1 + b3 = (2, 2, 2, 2). 83. (-2,-5,-1,1,-1). 85. M (−2, 1, 0, 3).
87. (5, 2, −4, −3), (0, 1, 1, 7). 88. Ðàçìåðíîñòü ðàâíà 3. Áàçèñ îáðà-
çóþò, íàïðèìåð, âåêòîðû a1 , a2 , a5 . 89. Íàïðèìåð, x1 − x2 − 2x3 = 0,
x1 − x3 + 2x4 = 0, 2x1 + x2 − x5 = 0. 90. Áàçèñ ñóììû ñîñòî-
èò, íàïðèìåð, èç âåêòîðîâ a1 , a2 , a3 , b2 . Áàçèñ ïåðåñå÷åíèÿ, íàïðèìåð,
b1 = −2a1 + a2 + a3 ; b3 = 5a1 − a2 − 2a3 . 91. Ïðîåêöèÿ âåêòî-
n−1
ðà ei íà L1 ïàðàëëåëüíî L2 èìååò i-óþ êîîðäèíàòó n , à îñòàëüíûå
−1
n , ïðîåêöèÿ íà L2 ïàðàëëåëüíî L1 èìååò âñå êîîðäèíàòû ðàâ-
íûìè 1
n. 92. (0, 1, −1, −2, −3). 95. 1) ∅, 2) òî÷êà (2, −1, 4, 5).
96. x1 − 3x2 − 2x3 + 3 = 0, 3x2 + 2x3 − x4 − 4 = 0, x1 + x2 − 3x3 +
x4 = 0. 97. Íàïðèìåð: b1 = (2, −2, −1, 0), b2 = (1, 1, 0, −1).
98. y = 3a1 − 2a2 = (1, −1, −1, 5), z = (3, 0, −2, −1). 99. π3 .
√ √
100. M1 ( 97 , 97 , 37 , − 75 ). 101. 23 . 102. (− 16 , 16 43
,
15 15 15 , − 42
15 ). 103. 1022
7 .
104. x1 = 1, x2 = λ + 1, x3 = λ + 1, x4 = λ + 1. 105. Íà-
ïðèìåð: 6x1 − 9x2 − x3 = 0, x2 + x4 = 0. 106. y = 2a1 − a2 =
(3, 1, −1, −2), z = (2, 1, −1, 4). 107. π6 . 108. M1 (1, − 21 , 2, 32 ).
√ √ √ √
109. 7 . 110. (1, −2, 2, 2). 111. 15. 112. 1) 6 , 2) 10
2 7 465 670
.
113. M 0 (10, 6), P ( 12 , 2). 114. 1) 2x − y − 12 = 0, x + y − 3 = 0; 2)
x = 0, y = 0; 3) x − y = 0. 115. 2x − 2y − 3 = 0, 4x − y = 0.
116. xe 0 = −e x, ye 0 = 5e y . 117. x0 = 5x − 3y + 8, y 0 = −3x + 2y − 3.
118. 1) x0 = x+8, y 0 = 4x−5y+14; 2) x0 = −x+2y−8, y 0 = 4x−3y+24.
120. πab. 121. Ãèïåðáîëà. 123. x0 = x − y + 1, y 0 = x + y + 2.
124. (2, 1). 125. 2x + y − 3 = 0. 127. Îáðàçóåò. Ïîâîðîò ïëîñ-
êîñòè âîêðóã íà÷àëà êîîðäèíàò íà óãîë ϕ, ñîåäèíåííûé ñ ãîìîòåòèåé
58
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