Exact Coordinate Instances in Fixed Real Vector Spaces
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batch-20261013-advance-20261022-r04-real-vector-space-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-12, -3, 2], right=[-2, -8, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3, 2], [-2, -8, 0], [-14, -11, 2]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
2. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-11, -2, 1], right=[-1, -7, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2, 1], [-1, -7, 1], [-12, -9, 2]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
3. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-10, -1, 0], right=[0, -6, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([-10, -1, 0], [0, -6, 2], [-10, -7, 2])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1, 0], [0, -6, 2], [-10, -7, 2]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
4. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-9, 0, -1], right=[1, -5, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([-9, 0, -1], [1, -5, 0], [-8, -5, -1])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0, -1], [1, -5, 0], [-8, -5, -1]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
5. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-8, 1, -2], right=[2, -4, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([-8, 1, -2], [2, -4, 1], [-6, -3, -1])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1, -2], [2, -4, 1], [-6, -3, -1]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
6. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-7, 2, -3], right=[-2, -3, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2, -3], [-2, -3, 2], [-9, -1, -1]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
7. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-6, 3, -4], right=[-1, -2, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3, -4], [-1, -2, 0], [-7, 1, -4]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
8. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-5, -3, -5], right=[0, -1, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([-5, -3, -5], [0, -1, 1], [-5, -4, -4])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3, -5], [0, -1, 1], [-5, -4, -4]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
9. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-4, -2, -6], right=[1, 0, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([-4, -2, -6], [1, 0, 2], [-3, -2, -4])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2, -6], [1, 0, 2], [-3, -2, -4]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
10. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-3, -1, -7], right=[2, 1, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([-3, -1, -7], [2, 1, 0], [-1, 0, -7])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1, -7], [2, 1, 0], [-1, 0, -7]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
11. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-2, 0, -8], right=[-2, 2, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0, -8], [-2, 2, 1], [-4, 2, -7]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
12. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-1, 1, -9], right=[-1, 3, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1, -9], [-1, 3, 2], [-2, 4, -7]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
13. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[0, 2, -10], right=[0, 4, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([0, 2, -10], [0, 4, 0], [0, 6, -10])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2, -10], [0, 4, 0], [0, 6, -10]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
14. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[1, 3, -11], right=[1, 5, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([1, 3, -11], [1, 5, 1], [2, 8, -10])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3, -11], [1, 5, 1], [2, 8, -10]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
15. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[2, -3, -12], right=[2, 6, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([2, -3, -12], [2, 6, 2], [4, 3, -10])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3, -12], [2, 6, 2], [4, 3, -10]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
16. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[3, -2, -13], right=[-2, 7, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([3, -2, -13], [-2, 7, 0], [1, 5, -13])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -13], [-2, 7, 0], [1, 5, -13]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
17. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[4, -1, -14], right=[-1, 8, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([4, -1, -14], [-1, 8, 1], [3, 7, -13])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -14], [-1, 8, 1], [3, 7, -13]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
18. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[5, 0, -15], right=[0, 9, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([5, 0, -15], [0, 9, 2], [5, 9, -13])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -15], [0, 9, 2], [5, 9, -13]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
19. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[6, 1, -16], right=[1, 10, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([6, 1, -16], [1, 10, 0], [7, 11, -16])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -16], [1, 10, 0], [7, 11, -16]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
20. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[7, 2, -17], right=[2, 11, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([7, 2, -17], [2, 11, 1], [9, 13, -16])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -17], [2, 11, 1], [9, 13, -16]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
21. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[8, 3, -18], right=[-2, 12, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([8, 3, -18], [-2, 12, 2], [6, 15, -16])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3, -18], [-2, 12, 2], [6, 15, -16]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
22. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[9, -3, -19], right=[-1, 13, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([9, -3, -19], [-1, 13, 0], [8, 10, -19])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3, -19], [-1, 13, 0], [8, 10, -19]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
23. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[10, -2, -20], right=[0, 14, 1]. Give (left,right,the componentwise sum).
View answer and explanation
([10, -2, -20], [0, 14, 1], [10, 12, -19])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2, -20], [0, 14, 1], [10, 12, -19]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
24. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[11, -1, -21], right=[1, 15, 2]. Give (left,right,the componentwise sum).
View answer and explanation
([11, -1, -21], [1, 15, 2], [12, 14, -19])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1, -21], [1, 15, 2], [12, 14, -19]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
25. Coordinate Addition in a Fixed Real Space
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[12, 0, -22], right=[2, 16, 0]. Give (left,right,the componentwise sum).
View answer and explanation
([12, 0, -22], [2, 16, 0], [14, 16, -22])
Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0, -22], [2, 16, 0], [14, 16, -22]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-addition · candidate
26. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-12, -2, 3]. Give (scalar,vector,scaled vector).
View answer and explanation
(-3, [-12, -2, 3], [36, 6, -9])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-12, -2, 3], [36, 6, -9]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
27. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-11, -1, 2]. Give (scalar,vector,scaled vector).
View answer and explanation
(-2, [-11, -1, 2], [22, 2, -4])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-11, -1, 2], [22, 2, -4]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
28. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-10, 0, 1]. Give (scalar,vector,scaled vector).
View answer and explanation
(-1, [-10, 0, 1], [10, 0, -1])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-10, 0, 1], [10, 0, -1]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
29. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-9, 1, 0]. Give (scalar,vector,scaled vector).
View answer and explanation
(0, [-9, 1, 0], [0, 0, 0])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-9, 1, 0], [0, 0, 0]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
30. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-8, 2, -1]. Give (scalar,vector,scaled vector).
View answer and explanation
(1, [-8, 2, -1], [-8, 2, -1])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-8, 2, -1], [-8, 2, -1]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
31. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[-7, -2, -2]. Give (scalar,vector,scaled vector).
View answer and explanation
(2, [-7, -2, -2], [-14, -4, -4])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [-7, -2, -2], [-14, -4, -4]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
32. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[-6, -1, -3]. Give (scalar,vector,scaled vector).
View answer and explanation
(3, [-6, -1, -3], [-18, -3, -9])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [-6, -1, -3], [-18, -3, -9]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
33. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-5, 0, -4]. Give (scalar,vector,scaled vector).
View answer and explanation
(-3, [-5, 0, -4], [15, 0, 12])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-5, 0, -4], [15, 0, 12]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
34. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-4, 1, -5]. Give (scalar,vector,scaled vector).
View answer and explanation
(-2, [-4, 1, -5], [8, -2, 10])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-4, 1, -5], [8, -2, 10]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
35. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-3, 2, -6]. Give (scalar,vector,scaled vector).
View answer and explanation
(-1, [-3, 2, -6], [3, -2, 6])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-3, 2, -6], [3, -2, 6]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
36. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-2, -2, -7]. Give (scalar,vector,scaled vector).
View answer and explanation
(0, [-2, -2, -7], [0, 0, 0])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-2, -2, -7], [0, 0, 0]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
37. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-1, -1, -8]. Give (scalar,vector,scaled vector).
View answer and explanation
(1, [-1, -1, -8], [-1, -1, -8])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-1, -1, -8], [-1, -1, -8]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
38. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[0, 0, -9]. Give (scalar,vector,scaled vector).
View answer and explanation
(2, [0, 0, -9], [0, 0, -18])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [0, 0, -9], [0, 0, -18]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
39. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[1, 1, -10]. Give (scalar,vector,scaled vector).
View answer and explanation
(3, [1, 1, -10], [3, 3, -30])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [1, 1, -10], [3, 3, -30]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
40. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[2, 2, -11]. Give (scalar,vector,scaled vector).
View answer and explanation
(-3, [2, 2, -11], [-6, -6, 33])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [2, 2, -11], [-6, -6, 33]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
41. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[3, -2, -12]. Give (scalar,vector,scaled vector).
View answer and explanation
(-2, [3, -2, -12], [-6, 4, 24])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [3, -2, -12], [-6, 4, 24]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
42. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[4, -1, -13]. Give (scalar,vector,scaled vector).
View answer and explanation
(-1, [4, -1, -13], [-4, 1, 13])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [4, -1, -13], [-4, 1, 13]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
43. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[5, 0, -14]. Give (scalar,vector,scaled vector).
View answer and explanation
(0, [5, 0, -14], [0, 0, 0])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [5, 0, -14], [0, 0, 0]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
44. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[6, 1, -15]. Give (scalar,vector,scaled vector).
View answer and explanation
(1, [6, 1, -15], [6, 1, -15])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [6, 1, -15], [6, 1, -15]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
45. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[7, 2, -16]. Give (scalar,vector,scaled vector).
View answer and explanation
(2, [7, 2, -16], [14, 4, -32])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [7, 2, -16], [14, 4, -32]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
46. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[8, -2, -17]. Give (scalar,vector,scaled vector).
View answer and explanation
(3, [8, -2, -17], [24, -6, -51])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [8, -2, -17], [24, -6, -51]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
47. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[9, -1, -18]. Give (scalar,vector,scaled vector).
View answer and explanation
(-3, [9, -1, -18], [-27, 3, 54])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [9, -1, -18], [-27, 3, 54]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
48. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[10, 0, -19]. Give (scalar,vector,scaled vector).
View answer and explanation
(-2, [10, 0, -19], [-20, 0, 38])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [10, 0, -19], [-20, 0, 38]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
49. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[11, 1, -20]. Give (scalar,vector,scaled vector).
View answer and explanation
(-1, [11, 1, -20], [-11, -1, 20])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [11, 1, -20], [-11, -1, 20]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
50. A Declared Real-Scalar Instance on Integer Coordinates
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[12, 2, -21]. Give (scalar,vector,scaled vector).
View answer and explanation
(0, [12, 2, -21], [0, 0, 0])
Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [12, 2, -21], [0, 0, 0]).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate
51. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-12, -2, 3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
52. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-11, -1, 2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
53. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-10, 0, 1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
54. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-9, 1, 0]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
55. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-8, 2, -1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
56. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-7, -2, -2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
57. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-6, -1, -3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
58. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-5, 0, -4]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
59. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-4, 1, -5]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
60. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-3, 2, -6]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
61. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-2, -2, -7]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
62. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-1, -1, -8]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
63. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[0, 0, -9]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
64. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[1, 1, -10]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
65. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[2, 2, -11]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
66. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[3, -2, -12]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
67. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[4, -1, -13]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
68. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[5, 0, -14]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
69. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[6, 1, -15]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
70. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[7, 2, -16]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
71. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[8, -2, -17]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
72. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[9, -1, -18]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
73. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[10, 0, -19]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
74. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[11, 1, -20]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
75. The Coordinate Additive Inverse and Its Zero Sum
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[12, 2, -21]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).
View answer and explanation
([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes)
Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-additive-inverse · candidate
76. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-2, vector=[-12, -1, 2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
77. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-1, vector=[-11, 0, 1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
78. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=0, vector=[-10, 1, 0]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
79. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=1, vector=[-9, 2, -1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
80. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=2, vector=[-8, -1, -2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
81. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=-2, vector=[-7, 0, -3]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
82. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-1, vector=[-6, 1, -4]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
83. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=0, vector=[-5, 2, -5]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
84. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=1, vector=[-4, -1, -6]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
85. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=2, vector=[-3, 0, -7]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
86. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=-2, vector=[-2, 1, -8]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
87. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=-1, vector=[-1, 2, -9]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
88. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=0, vector=[0, -1, -10]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
89. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=1, vector=[1, 0, -11]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
90. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=2, vector=[2, 1, -12]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
91. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-2, vector=[3, 2, -13]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
92. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=-1, vector=[4, -1, -14]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
93. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=0, vector=[5, 0, -15]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
94. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=1, vector=[6, 1, -16]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
95. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=2, vector=[7, 2, -17]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
96. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-2, vector=[8, -1, -18]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
97. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-1, vector=[9, 0, -19]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
98. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=0, vector=[10, 1, -20]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
99. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=1, vector=[11, 2, -21]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
100. One Exact Instance of Distributivity over Scalar Addition
Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=2, vector=[12, -1, -22]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).
View answer and explanation
(0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes)
Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes).
Reviewed step-by-step derivation
Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate
Canonical question JSONL · English JSONL · Validation report