Тр номер 1(математика)
.pdf5x1 +3x2 + x3 − x4 = −6,
10.27. а) 2x1 + x2 + 2x3 −6x4 = −3,−2x1 + 4x2 +7x3 −5x4 = 22,
x1 + x2 −4x3 −2x4 =16,2x1 + x2 +6x3 +8x4 = 6,
3x1 + 2x2 +7x3 +7x4 = 5, б) 4x1 +3x2 +8x3 +6x4 = 4,
5x1 + 4x2 +9x3 +5x4 = 3,2x1 +5x2 −2x3 + x4 = 2,
в) 2x1 −2x2 + x3 −5x4 = −6,−2x1 +3x2 + 2x3 + 4x4 = 7,
2x1 +6x2 + x3 =1.
3x1 − x2 −2x3 −8x4 =12,
10.28. а) −3x1 + x2 −4x3 + x4 = −1,
3x1 +5x2 +3x3 +3x4 =14,2x1 −2x2 − x3 −5x4 = 6,x1 +5x2 −12x3 −3x4 =15,
2x1 −7x2 + 21x3 +5x4 = −27, б) 3x1 −2x2 +9x3 + 2x4 = −12,
7x1 + 20x3 +13x4 = −1,5x1 + 4x2 +7x3 +3x4 = 6,
9x1 +5x2 +14x3 + 2x4 =1, в) 3x1 + 2x2 + 4x3 + 2x4 = 5,
x1 +5x2 + x3 +3x4 = 0.
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x1 −3x2 +6x3 + 4x4 = −8, |
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+ 2x2 + x3 −4x4 =1, |
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10.29. а) 5x1 |
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2x1 −2x2 − x3 + x4 = −5, |
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= −7, |
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2x1 − x2 −2x4 = 2, |
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−3x2 + x3 −2x4 = 4, |
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− x2 + x3 −4x4 = 3, |
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x1 |
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+ x2 + x3 −8x4 = 3, |
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3x1 + x2 + 4x3 +5x4 = 3, |
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+ 2x2 +3x3 + 2x4 = −1, |
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−2x2 + 2x3 +3x4 = 2, |
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− x2 + x3 +5x4 = 4. |
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2x1 +3x2 − x3 + x4 =1, |
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+5x2 −2x3 + 4x4 = 2, |
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10.30. а) |
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+ 2x2 + x3 + x4 =1, |
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3x1 |
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+5x2 − x3 + 2x4 = 2, |
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3x1 + 2x2 + x3 +3x4 = −1, |
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+3x2 + x3 + 2x4 =1, |
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+ x2 +3x4 = −2, |
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= −5, |
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2x1 +3x2 −5x3 +3x4 = −3, |
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+7x2 −4x3 + 2x4 = 3, |
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−4x1 +5x2 +7x3 −6x4 = 5, |
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= 2. |
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3x1 + x2 + x3 +5x4 = 4,
10.31. а) x1 +5x2 + 2x3 + x4 = −2,
2x1 + x2 +3x3 + 4x4 =15,2x1 −3x2 −2x3 + x4 = 0,
x1 −4x2 + x3 +3x4 =8,
2x1 −8x2 + x3 + 4x4 =13, б) 3x1 −12x2 + 2x3 +7x4 = 21,
−2x1 +8x2 +3x3 + 4x4 = −1,2x1 +3x2 − x3 + 2x4 = 5,
в) −3x1 +5x2 + 4x3 +7x4 =1,3x1 −2x2 − x3 +5x4 = −2,
x1 +3x2 + 2x3 +11x4 = 0.
Задача 11. Найти общее решение однородной СЛАУ и фундаментальную систему решений.
x + x −3x −4x = 0,
11.1.4x1 +5x2 −2x3 − x4 = 0,3x1 + 4x2 + x3 +3x4 = 0.x1 −5x2 + 2x3 −16x4 = 0,
11.2.3x1 + x2 −8x3 + 2x4 = 0,2x1 −2x2 −3x3 −7x4 = 0.x1 −4x2 + 2x3 +3x4 = 0,
11.3.2x1 −7x2 + 4x3 + x4 = 0,x1 −3x2 + 2x3 −2x4 = 0.1 2 3 4
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3x1 +5x2 +3x3 + 2x4 = 0, 11.4. 5x1 +7x2 +6x3 + 4x4 = 0,
7x1 +9x2 +9x3 +6x4 = 0.x1 −3x 2 + 4x3 +3x 4 = 0, 11.5. 3x1 −8x 2 + x3 + 2x 4 = 0,2x1 −5x 2 −3x3 − x 4 = 0.3x1 +5x2 −4x3 + 2x4 = 0, 11.6. 2x1 + 4x2 −6x3 +3x4 = 0,
11x1 +17x2 −8x3 + 4x4 = 0.
x1 + x2 −3x3 −4x4 = 0,
11.7. 4x1 +5x2 −2x3 − x4 = 0,3x1 + 4x2 + x3 +3x4 = 0.
3x1 −2x2 + x3 − x4 = 0,
11.8. x1 + x2 −4x3 +5x4 = 0,5x1 −5x2 +6x3 −7x4 = 0.
x1 − x2 + 4x3 +3x4 = 0,
11.9. 3x1 −2x2 + x3 + 2x4 = 0,2x1 − x2 −3x3 − x4 = 0.
3x − x −2x = 0,
11.10.2x1 + x2 −2x3 − x4 = 0,x1 +3x2 −2x3 −2x4 = 0.x1 −5x2 +3x3 + 4x4 = 0,
11.11.2x1 −9x2 + 2x3 + x4 = 0,x1 −4x2 − x3 −3x4 = 0.1 2 3
x + 2x +3x − x = 0,
11.12.x1 − x2 + x3 + 2x4 = 0,x1 +5x2 +5x3 −4x4 = 0.x1 − x2 +3x3 + 4x4 = 0,
11.13.2x1 − x2 + 2x3 + x4 = 0,4x1 −3x2 +8x3 +9x4 = 0.x1 + x2 −2x3 − x4 = 0,
11.14.3x1 − x2 + x3 + 4x4 = 0,x1 +5x2 −9x3 −8x4 = 0.x1 + x2 + 4x3 + 2x4 = 0,
11.15.3x1 + 4x2 + x3 +3x4 = 0,2x1 +3x2 −3x3 + x4 = 0.5x1 −3x2 + 2x3 + 4x4 = 0,
11.16.4x1 −2x2 +3x3 +7x4 = 0,8x1 −6x2 − x3 −5x4 = 0.x1 + 4x2 −2x3 −3x4 = 0,
11.17.2x1 +9x2 − x3 −4x4 = 0,x1 +5x2 + x3 − x4 = 0.x1 +3x2 − x3 + 2x4 = 0,
11.18.2x1 − x2 +3x3 +5x4 = 0,x1 +10x2 −6x3 + x4 = 0 .x1 −2x2 + 2x3 +3x4 = 0,
11.19.3x1 −5x2 + x3 + 4x4 = 0,2x1 −3x2 − x3 + x4 = 0.1 2 3 4
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x −2x + x = 0,
11.20.3x1 − x2 −2x3 = 0,2x1 + x2 −2x3 − x4 = 0.x1 +3x2 − x3 −2x4 = 0,
11.21.2x1 +7x2 −4x3 −3x4 = 0,x1 + 4x2 −3x3 − x4 = 0.2x1 − x2 +3x3 −2x4 = 0,
11.22.4x1 −2x2 +5x3 + x4 = 0,2x1 − x2 + x3 +8x4 = 0.x1 −2x2 + 2x3 +3x4 = 0,
11.23.2x1 −3x2 + x3 + 4x4 = 0,3x1 −5x2 +3x3 +7x4 = 0.x1 +3x2 +5x3 − x4 = 0,
11.24.2x1 − x2 −3x3 + 4x4 = 0,5x1 + x2 − x3 +7x4 = 0.x1 + 2x2 −2x3 −3x4 = 0,
11.25.2x1 +5x2 − x3 −4x4 = 0,x1 +3x2 + x3 − x4 = 0.3x1 + 2x2 + 2x3 + 2x4 = 0,
11.26.2x1 +3x2 + 2x3 +5x4 = 0,9x1 + x2 + 4x3 −5x4 = 0.2x1 − x2 +3x3 + 4x4 = 0,
11.27.x1 −3x2 + 4x3 +7x4 = 0,− x1 + 2x2 −3x3 −5x4 = 0.1 2 4
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11.28.5x1 −3x2 + 2x3 +3x4 = 0,x1 −3x2 −5x3 = 0.
x1 + x2 −2x3 + 2x4 = 0,
11.29.3x1 +5x2 +6x3 −4x4 = 0,4x1 +5x2 −2x3 +3x4 = 0.5x1 −3x2 + 2x3 + 4x4 = 0,
11.30.4x1 −2x2 +3x3 +7x4 = 0,8x1 −6x2 − x3 −5x4 = 0.9x1 + 4x2 −5x3 + x4 = 0,
11.31.2x1 + x2 − x3 = 0,x1 + 2x2 + x3 −3x4 = 0.3x1 − x2 +3x3 + 2x4
Задача 12. Линейные операторы.
Пусть x = (x1, x2 , x3 ) R3 . Проверить, являются ли
следующие операторы линейными, в случае линейности записать матрицу оператора.
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, x1 − x2 + x3 ); |
12.1. а) Ax = (x1 + x2 ,2x1 − x2 +3x3 |
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+ x2 +1, x1 − x2 + x3 ,2 − x2 + x3 ). |
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б) Bx = (x1 |
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+ 2x2 + x3 ); |
12.2. а) Ax = (3 − x1 + x3 , x2 − x3 , x1 |
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− x3 ). |
б) Bx = (2x1 − x2 + x3 , x3 ,−3x2 |
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− x3 ,1,−2x2 + x3 ); |
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12.3. а) Ax = (3x1 |
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− x2 + x3 , x2 |
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12.4. а) Ax = (− x3 ,2x1 |
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б) Bx = (2, x1 |
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, x2 −2x3 , x1 + x2 + x3 ); |
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12.5. а) Ax = (x1 |
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+ x3 ,2x1 − x2 + x3 ,3x1 −2x3 ). |
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б) Bx = (x2 |
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−2x2 ,0,−x1 + 2x2 − x3 ); |
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12.6. а) Ax = (x1 |
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б) Bx = (−2x1 + x3 , x1 +1,3x2 −2x3 ). |
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12.7. а) Ax = (− x1 + x2 , x2 |
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б) Bx = (−3x3 , x1 + 2x2 ,−x1 + x2 + x3 ). |
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− x2 + x3 , x2 ,−3x3 ); |
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12.8. а) Ax = (2x1 |
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+1, x1 − x2 , x1 + x3 −1). |
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б) Bx = (2x2 |
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− x3 ,0, x2 −1); |
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12.9. а) Ax = (2x1 |
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−2x3 , x1 −3x2 + x3 , x1 − x2 ). |
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−2x3 , x1 + 2x3 ,−x1 + x2 − x3 ); |
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12.10. а) Ax = (x2 |
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, x1 +3x3 ,−x2 + x3 ). |
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б) Bx = (x2 |
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12.11. а) Ax = (−4x3 , x2 +3x3 ,4x1 − x2 + 2x3 ); |
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+ 2x2 ,−x2 + 2, x1 + x2 − x3 ). |
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б) Bx = (x1 |
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12.12. а) Ax = (2x1,3,2x1 − x2 + x3 ); |
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−2x2 + x3 , x2 − x3 , x1 + 2x3 ). |
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12.13. а) Ax = (3x1 − x2 +1, x2 − x3 , x1 + 2x2 ); |
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б) Bx = (−2x2 + x3 ,−x2 ,2x1 − x2 + x3 ). |
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−2x2 + x3 , x2 + x3 ,0); |
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12.14. а) Ax = (x1 |
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− x3 , x2 − x3 +1,−x1 ). |
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б) Bx = (x1 |
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− x2 ,1,2x1 + x2 − x3 ); |
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12.15. а) Ax = (x1 |
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б) Bx = (2x2 − x3 , x1 + x2 , x1 − x2 − x3 ). |
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12.16. а) Ax = (2x1 + x2 ,0, x3 +1); |
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− x2 + x3 ,2x1,2x2 − x3 ). |
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б) Bx = (x1 |
12.17.а)
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12.18.а)
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12.19.а)
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12.20.а)
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12.27.а)
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12.28.а)
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−2x3 ,−x1 + x3 ,2x2 + x3 ); |
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− x2 , x2 + 2, x3 ). |
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− x2 + x3 ,3x3 , x1 + x2 ); |
Ax = (2x1 |
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− x2 +1, x2 − x3 , x1 ). |
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Bx = (x1 |
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+ x2 − x3 ,1, x2 + x3 ); |
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Ax = (x1 |
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−2x3 , x1 + 2x2 − x3 ,2x1 −3x3 ). |
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− x3 , x2 −1, x1 + 2x3 ); |
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+ x3 , x1 −2x2 ,2x1 − x2 + x3 ). |
Bx = (2x2 |
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+ 2x2 ,−2x2 + x3 ,0); |
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Ax = (x1 |
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− x3 , x1 −1, x2 −3x3 ). |
Bx = (2x2 |
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+ 2x3 , x1 + x2 ,−2x2 + x3 ); |
Ax = (2x2 |
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Bx = (1,2x1 − x2 , x3 ). |
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− x2 , x3 + 2, x1 − x2 + x3 ); |
Ax = (2x1 |
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, x1 − x2 + x3 , x2 −2x3 ). |
Bx = (3x3 |
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− x3 , x1 + x2 +1, x2 − x3 ); |
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− x2 ,3x2 − x3 ,2x1 − x3 ). |
Bx = (2x1 |
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+ 2x2 ,0,−x1 +3x2 + 2x3 ); |
Ax = (2x1 |
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− x2 ,2,2x2 − x3 ). |
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Ax = (− x1 +3x2 ,2x1 − x2 + 2x3 ,3x2 − x3 ); |
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+3x2 +1, x1 + 2x2 , x1 − x3 ). |
Bx = (2x1 |
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+ x2 ,2x2 + x3 , x1 −1); |
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−3x2 + x3 ,3x2 − x3 , x1 + 2x3 ). |
Bx = (2x1 |
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+ 2x3 , x1 − x2 ,0); |
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(4x1 + x2 , x1 − x2 +3,0). |
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(2x1 + x3 ,1, x1 + 2x2 ); |
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12.29. а) Ax |
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(4x1 −2x2 + x3 , x2 , x1 −3x3 ). |
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(3x2 − x3 , x1, x1 −2x2 +3x3 ); |
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12.30. а) Ax |
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(x1 −1, x2 + 2x3 ,2x2 − x3 ). |
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(x2 −3x3 , x2 +1, x1 − x2 − x3 ); |
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12.31. а) Ax |
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(3x1 − x2 ,2x1 − x2 + x3 , x2 −3x3 ). |
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Задача 13. Линейный оператор в базисе (e1, e2 , e3 ) задан |
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матрицей |
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матрицу этого оператора в базисе |
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(e1′, e2′, e3′). |
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e1′ = −e1 + e3 , e2/ = 2e2 + e3 , e3′ = e1 −e2 . |
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13.2. A = |
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e1′ = e1 − e3 , e2′ = 2e1 − e2 , e3′ = −2e2 + e3 . |
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13.3. A = |
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e1′ = −e1 + e2 −2e3 , e2′ = 2e1 + e3 , e3′ = e1 +3e2 .
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e1′ = 3e2 + e3 , e2′ = −e1 − e2 + 2e3 , e3′ = e1 − e3 . |
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13.5. A = |
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e1′ = e1 − e2 + e3 , e2′ = −2e2 + e3 , e3′ = −e1 + e3 . |
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13.6. A = |
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e1′ = 2e1 + e2 + e3 , e2′ = e1 + 2e3 , e3′ = −e2 − e3 . |
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13.7. A = |
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e1′ = −2e1 − e2 + e3 , e2′ = −2e3 , e3′ = e1 + e2 + e3 . |
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13.8. A = |
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e1′ = e1 −2e2 + 2e3 , e2′ = e2 − e3 , e3′ = e1 + e3 . |
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0 |
1 |
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13.9. A = |
−3 , |
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1 |
1 |
−1 |
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e1′ = 2e2 −2e3 , e2′ = e1 − e2 , e3′ = 2e1 + e2 − e3 .
51 |
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52 |
0 |
3 |
−1 |
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2 |
1 |
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13.10. A = |
0 , |
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1 |
1 |
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−1 |
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e1′ = 2e1 − e3 , e2′ = e1 − e2 , e3′ = −e1 + e2 + 2e3 . |
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2 |
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1 |
−1 |
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−1 |
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3 |
0 |
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13.11. A = |
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, |
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1 |
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0 |
1 |
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e1′ = e2 + 2e3 , e2′ = −e1 + e3 , e3′ = 2e1 + e2 − e3 . |
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3 |
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−1 |
2 |
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0 |
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1 |
− |
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13.12. A = |
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1 , |
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−1 |
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0 |
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2 |
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e1′ = e1 −2e2 , e2′ = 2e2 −e3 , e3′ = −e1 − e2 + e3 . |
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−1 |
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0 |
1 |
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2 |
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1 |
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13.13. A = |
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−1 , |
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0 |
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−2 |
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3 |
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e1′ = −2e1 + e2 , e2′ = e1 − e3 , e3′ = −e1 + 2e2 + e3 . |
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1 |
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2 |
0 |
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0 |
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3 |
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13.14. A = |
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−1 , |
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−2 |
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1 |
1 |
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e1′ = −e1 + 2e2 + e3 , e2′ = e1 + 2e2 , e3′ = −e2 + e3 . |
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0 |
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1 |
−1 |
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−1 |
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2 |
1 |
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13.15. A = |
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, |
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1 |
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0 |
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−1 |
e1′ = e3 , e2′ = −e1 + 2e2 + e3 , e3′ = e1 − e2 + 2e3 .
1 |
−1 |
2 |
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0 |
0 |
1 |
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13.16. A = |
, |
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2 |
1 |
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−1 |
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e1′ = e1 + e2 − e3 , e2′ = −e2 + 2e3 , e3′ = e2 − e3 . |
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1 |
0 |
0 |
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−2 |
1 |
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13.17. A = |
−1 , |
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−1 |
2 |
1 |
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e1′ = −2e2 + e3 , e2′ = −e2 + e3 , e3′ = e1 − e2 + 2e3 . |
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2 |
0 |
−1 |
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−1 |
1 |
0 |
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13.18. A = |
, |
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1 |
−1 |
1 |
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e1′ = −e1 −2e3 , e2′ = e1 − e2 + e3 , e3′ = 2e1 + e3 . |
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1 |
−1 |
1 |
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2 |
1 |
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13.19. A = |
−1 , |
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0 |
0 |
1 |
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e1′ = 2e2 , e2′ = −e1 + e2 − e3 , e3′ = e1 −2e2 − e3 . |
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−1 |
1 |
−1 |
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2 |
2 |
1 |
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13.20. A = |
, |
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1 |
0 |
0 |
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e1′ = e1 − e2 + e3 , e2′ = −e2 , e3′ = 2e1 + e2 − e3 . |
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1 |
−1 |
1 |
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−1 |
0 |
0 |
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13.21. A = |
, |
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2 |
−1 |
2 |
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e1′ = e1 + e2 −2e3 , e2′ = 2e1 + e2 − e3 , e3′ = e2 .
53 |
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54 |
0 |
2 |
1 |
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0 |
−1 |
1 |
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13.22. A = |
, |
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−1 |
1 |
2 |
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e1′ = −e1 + e3 , e2′ = −2e1 + e2 − e3 , e3′ = e2 − e3 . |
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−1 |
2 |
0 |
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1 |
−1 |
0 |
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13.23. A = |
, |
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2 |
1 |
1 |
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e1′ = 2e1 − e3 , e2′ = −e1 + e2 + 2e3 , e3′ = −e2 + e3 . |
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1 |
−1 |
1 |
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2 |
1 |
0 |
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13.24. A = |
, |
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−1 |
1 |
0 |
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e1′ = e1 + 2e2 − e3 , e2′ = −e1 + e3 , e3′ = e2 + 2e3 . |
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2 |
1 |
0 |
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1 |
1 |
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13.25. A = |
−1 , |
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1 |
−1 |
0 |
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e1′ = −e2 + e3 , e2′ = e1 + e2 − e3 , e3′ = 2e1 − e2 . |
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0 |
2 |
−1 |
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1 |
−1 |
0 |
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13.26. A = |
, |
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−1 |
2 |
1 |
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e1′ = 2e1 − e2 + 2e3 , e2′ = −e1 + e3 , e3′ = e1 −2e3 . |
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2 |
0 |
1 |
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−1 |
1 |
0 |
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13.27. A = |
, |
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1 |
−1 |
1 |
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e1′ = 2e1 − e2 , e2′ = −e1 + e2 , e3′ = e1 + e2 −2e3 .
55
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−1 |
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1 |
−1 |
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0 |
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−2 |
1 |
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13.28. A = |
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, |
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1 |
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0 |
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−1 |
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e1′ = e2 − e3 , e2′ = 2e1 − e2 + 2e3 , e3′ = e1 + e3 . |
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2 |
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−1 |
0 |
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−1 |
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1 |
1 |
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13.29. A = |
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, |
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0 |
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−1 |
2 |
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e1′ = e1 −2e2 − e3 , e2′ = −e1 + e2 + e3 , e3′ = e3 . |
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1 |
−2 |
−1 |
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0 |
1 |
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13.30. A = |
−1 , |
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1 |
0 |
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−1 |
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e1′ = e1 − e2 + e3 , e2′ = −2e1 + e2 , e3′ = −e2 + e3 . |
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0 |
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1 |
−1 |
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−1 |
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−1 |
1 |
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13.31. A = |
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, |
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0 |
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2 |
1 |
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e1′ = 2e1 − e2 , e2′ = e1 − e2 , e3′ = −e1 + e2 + e3 . |
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Задача 14. Найти собственные числа и собственные |
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векторы матрицы. |
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4 −3 3 |
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7 −6 6 |
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1 |
2 |
1 |
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2 |
3 |
2 |
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14.1. |
. |
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14.2. |
. |
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1 |
1 |
2 |
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2 |
2 |
3 |
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7 −6 6 |
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5 −4 |
4 |
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4 |
−1 |
4 |
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2 |
1 |
2 |
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14.3. |
. |
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14.4. |
. |
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4 |
−2 5 |
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2 0 3 |
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56
6 |
−2 |
−1 |
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−1 |
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5 |
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14.5. |
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−1 . |
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1 |
−2 |
4 |
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1 |
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2 |
0 |
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0 |
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2 |
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14.7. |
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. |
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−2 |
−2 |
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−1 |
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1 |
−2 |
−1 |
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−1 |
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1 |
1 |
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14.9. |
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. |
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1 |
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0 |
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−1 |
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−1 |
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−2 |
12 |
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4 |
3 |
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14.11. |
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. |
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0 |
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5 |
6 |
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2 |
19 |
30 |
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0 |
−5 |
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14.13. |
−12 . |
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0 |
2 |
5 |
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1 |
1 |
3 |
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1 |
5 |
1 |
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14.15. |
. |
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3 |
1 |
1 |
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1 |
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1 |
−3 |
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1 |
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1 |
3 |
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14.17. |
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. |
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−3 |
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3 |
3 |
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4 |
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−2 |
−1 |
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−1 |
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3 |
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14.6. |
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−1 . |
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1 |
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−2 |
4 |
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1 |
10 |
3 |
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2 |
1 |
2 |
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14.8. |
. |
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3 |
10 |
1 |
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5 |
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−7 |
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0 |
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−3 |
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1 |
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0 |
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14.10. |
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. |
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12 |
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6 |
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− |
3 |
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4 |
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0 |
5 |
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7 |
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−2 |
9 |
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14.12. |
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. |
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3 |
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0 |
6 |
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1 |
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0 |
2 |
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0 |
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1 |
0 |
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14.14. |
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. |
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2 |
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0 |
4 |
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3 |
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−1 |
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1 |
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−1 |
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5 |
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14.16. |
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−1 . |
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1 |
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−1 |
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3 |
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1 |
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0 |
1 |
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0 |
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1 |
0 |
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14.18. |
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. |
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1 |
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0 |
1 |
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1 |
−4 0 |
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2 |
2 |
1 |
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−4 |
1 |
0 |
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2 |
2 |
1 |
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14.19. |
.14.20. |
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. |
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0 |
0 |
1 |
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1 |
1 |
3 |
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1 |
1 |
−1 |
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1 |
1 |
1 |
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14.21. |
. |
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−1 |
1 |
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−1 |
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1 |
2 |
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0 |
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0 |
2 |
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0 |
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14.23. |
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. |
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−2 |
−2 |
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−1 |
0 7 4
14.25. 0 1 0 .
1 13 0
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2 |
−1 |
1 |
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1 |
2 |
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14.27. |
−1 . |
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1 |
−1 |
2 |
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2 |
0 |
1 |
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0 |
2 |
0 |
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14.29. |
. |
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1 |
0 |
2 |
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2 |
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−2 |
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0 |
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−2 |
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9 |
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2 |
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14.31. |
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. |
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0 |
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2 |
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2 |
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57
7 |
−12 |
−2 |
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3 |
−4 |
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0 |
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14.22. |
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. |
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−2 |
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0 |
−2 |
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1 |
0 |
0 |
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2 |
−2 |
0 |
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14.24. |
. |
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1 |
3 |
3 |
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5 |
2 |
−3 |
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4 |
5 |
−4 |
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14.26. |
. |
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6 |
4 |
−4 |
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6 |
−2 |
2 |
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−2 |
5 |
0 |
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14.28. |
. |
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2 |
0 |
7 |
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6 |
−2 |
−1 |
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|
−1 |
5 |
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14.30. |
−1 . |
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1 |
−2 |
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4 |
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