ALGEBRA
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Аксиома A1:
(x1, x2, . . . , xn) + (y1, y2, . . . , yn)=
=(x1 + y1, x2 + y2, . . . , xn + yn) =
=(y1 + x1, y2 + x2, . . . , yn + xn) =
=(y1, y2, . . . , yn) + (x1, x2, . . . , xn).
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Аксиома A1:
(1.1)
(x1, x2, . . . , xn) + (y1, y2, . . . , yn) =
=(x1 + y1, x2 + y2, . . . , xn + yn)=
=(y1 + x1, y2 + x2, . . . , yn + xn) =
=(y1, y2, . . . , yn) + (x1, x2, . . . , xn).
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Аксиома A1:
(1.1)
(x1, x2, . . . , xn) + (y1, y2, . . . , yn) =
C1.
=(x1 + y1, x2 + y2, . . . , xn + yn) =
=(y1 + x1, y2 + x2, . . . , yn + xn)=
=(y1, y2, . . . , yn) + (x1, x2, . . . , xn).
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Аксиома A1:
(1.1)
(x1, x2, . . . , xn) + (y1, y2, . . . , yn) =
C1.
= (x1 + y1, x2 + y2, . . . , xn + yn) =
(1.1)
= (y1 + x1, y2 + x2, . . . , yn + xn) =
= (y1, y2, . . . , yn) + (x1, x2, . . . , xn).
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Аксиома A2:
(1.1)
((x1, x2, . . . , xn) + (y1, y2, . . . , yn)) + (z1, z2, . . . , zn) =
(1.1)
= (x1 + y1, x2 + y2, . . . , xn + yn) + (z1, z2, . . . , zn) =
C2.
= ((x1 + y1) + z1, (x2 + y2) + z2, . . . , (xn + yn) + zn) =
(1.1)
= (x1 + (y1 + z1), x2 + (y2 + z2), . . . , xn + (yn + zn)) =
(1.1)
= (x1, x2, . . . , xn) + (y1 + z1, y2 + z2, . . . , yn + zn) =
= (x1, x2, . . . , xn) + ((y1, y2, . . . , yn) + (z1, z2, . . . , zn))
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Аксиома A3:
(1.1)
(x1, x2, . . . , xn) + (0, 0, . . . , 0) =
C3.
=(x1 + 0, x2 + 0, . . . , xn + 0) =
=(x1, x2, . . . , xn).
Упорядоченная n−ка (0, 0, . . . , 0) Rn есть нулевой элемент в Rn.
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Аксиома A4:
(1.1)
(x1, x2, . . . , xn) + (−x1, −x2, . . . , −xn) =
− − − C4.
= (x1 + ( x1), x2 + ( x2), . . . , xn + ( xn)) = = (0, 0, . . . , 0).
Упорядоченная n−ка (−x1, −x2, . . . , −xn) Rn есть элемент противоположный элементу (x1, x2, . . . , xn) Rn.
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Аксиома A5:
(1.2)
1 · (x1, x2, . . . , xn) =
· · · C7.
= (1 x1, 1 x2, . . . , 1 xn) =
= (x1, x2, . . . , xn).
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Аксиома A6:
(1.2)
α · (β · (x1, x2, . . . , xn)) =
= α · (β · x1, β · x2, . . . , β · xn) = (α · (β · x1), α · (β · x2), . . . , α · (β
(1.2)
=
· C6.
xn)) =
(1.2)
= ((α · β) · x1, (α · β) · x2, . . . , (α · β) · xn) = = (α · β) · (x1, x2, . . . , xn).
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Аксиома A7:
(α + β) · (x1, x2, . . . , xn) |
(1.2) |
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C9. |
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= ((α + β) · x1, (α + β) · x2, . . . , (α + β) · xn) = |
(1.1) |
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= (α · x1 + β · x1, α · x2 + β · x2, . . . , α · xn + β · xn) |
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= (α · x1, α · x2, . . . , α · xn) + (β · x1, β · x2, . . . , β · xn) |
(1.2) |
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= α · (x1, x2, . . . , xn) + β · (x1, x2, . . . , xn).
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