Universal Homogeneous Subspaces, Affine Boundaries and Finite Counterexamples
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261101-r01-vector-subspace-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[1, -1]. Selected point=[-12, -3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-12, -3], [1, -1], -9, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-12, -3], [1, -1], -9, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
2. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[2, -2]. Selected point=[-11, -2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-11, -2], [2, -2], -18, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-11, -2], [2, -2], -18, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
3. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[3, -3]. Selected point=[-10, -1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-10, -1], [3, -3], -27, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-10, -1], [3, -3], -27, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
4. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[4, -4]. Selected point=[-9, 0]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-9, 0], [4, -4], -36, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-9, 0], [4, -4], -36, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
5. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[5, -5]. Selected point=[-8, 1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-8, 1], [5, -5], -45, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-8, 1], [5, -5], -45, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
6. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[6, -1]. Selected point=[-7, 2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-7, 2], [6, -1], -44, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-7, 2], [6, -1], -44, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
7. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[7, -2]. Selected point=[-6, 3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-6, 3], [7, -2], -48, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-6, 3], [7, -2], -48, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
8. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[8, -3]. Selected point=[-5, -3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-5, -3], [8, -3], -31, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-5, -3], [8, -3], -31, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
9. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[9, -4]. Selected point=[-4, -2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-4, -2], [9, -4], -28, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-4, -2], [9, -4], -28, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
10. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[10, -5]. Selected point=[-3, -1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-3, -1], [10, -5], -25, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-3, -1], [10, -5], -25, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
11. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[11, -1]. Selected point=[-2, 0]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-2, 0], [11, -1], -22, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-2, 0], [11, -1], -22, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
12. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[12, -2]. Selected point=[-1, 1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([-1, 1], [12, -2], -14, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([-1, 1], [12, -2], -14, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
13. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[13, -3]. Selected point=[0, 2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([0, 2], [13, -3], -6, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([0, 2], [13, -3], -6, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
14. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[14, -4]. Selected point=[1, 3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([1, 3], [14, -4], 2, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([1, 3], [14, -4], 2, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
15. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[15, -5]. Selected point=[2, -3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([2, -3], [15, -5], 45, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([2, -3], [15, -5], 45, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
16. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[16, -1]. Selected point=[3, -2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([3, -2], [16, -1], 50, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([3, -2], [16, -1], 50, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
17. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[17, -2]. Selected point=[4, -1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([4, -1], [17, -2], 70, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([4, -1], [17, -2], 70, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
18. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[18, -3]. Selected point=[5, 0]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([5, 0], [18, -3], 90, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([5, 0], [18, -3], 90, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
19. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[19, -4]. Selected point=[6, 1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([6, 1], [19, -4], 110, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([6, 1], [19, -4], 110, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
20. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[20, -5]. Selected point=[7, 2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([7, 2], [20, -5], 130, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([7, 2], [20, -5], 130, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
21. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[21, -1]. Selected point=[8, 3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([8, 3], [21, -1], 165, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([8, 3], [21, -1], 165, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
22. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[22, -2]. Selected point=[9, -3]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([9, -3], [22, -2], 204, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([9, -3], [22, -2], 204, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
23. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[23, -3]. Selected point=[10, -2]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([10, -2], [23, -3], 236, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([10, -2], [23, -3], 236, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
24. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[24, -4]. Selected point=[11, -1]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([11, -1], [24, -4], 268, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([11, -1], [24, -4], 268, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
25. Membership in a Homogeneous Real Line Is an Exact Equation Check
Use R² with its usual real vector operations and the homogeneous equation a*x+b*y=0, with nonzero normal [a,b]=[25, -5]. Selected point=[12, 0]. Give (point,normal,equation_value,in_line?).
View answer and explanation
([12, 0], [25, -5], 300, No)
Calculate a*x+b*y exactly. Its being zero decides membership of this point; the universal subspace certificate is a separate algebraic obligation, not a conclusion from one point. The exact result, retaining the requested order, is ([12, 0], [25, -5], 300, No).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-line-membership · candidate
26. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[1, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([1, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([1, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
27. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[2, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([2, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([2, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
28. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[3, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([3, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([3, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
29. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[4, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([4, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([4, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
30. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[5, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([5, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
31. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[6, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([6, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([6, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
32. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[7, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([7, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([7, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
33. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[8, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([8, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([8, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
34. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[9, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([9, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([9, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
35. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[10, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([10, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([10, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
36. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[11, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([11, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([11, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
37. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[12, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([12, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([12, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
38. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[13, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([13, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([13, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
39. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[14, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([14, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([14, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
40. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[15, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([15, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([15, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
41. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[16, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([16, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([16, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
42. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[17, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([17, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([17, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
43. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[18, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([18, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([18, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
44. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[19, -5] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([19, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([19, -5], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
45. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[20, -6] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([20, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([20, -6], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
46. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[21, -7] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([21, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([21, -7], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
47. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[22, -1] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([22, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([22, -1], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
48. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[23, -2] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([23, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([23, -2], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
49. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[24, -3] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([24, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([24, -3], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
50. A Homogeneous Kernel Has Universal Closure under Real Linear Combinations
Use the subset U={(x,y) in R²:a*x+b*y=0} with nonzero normal [a,b]=[25, -4] and usual real vector operations. Give (normal,zero_in?,addition_closed?,scalar_closed?,homogeneous_kernel_closed_for_all_real_scalars).
View answer and explanation
([25, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars)
The zero vector satisfies the equation. Linearity makes the equation hold for sums of solutions and for every real scalar multiple. This algebraic all-vector argument, not sampled points or a new Lean receipt, supplies the stated certificate. The exact result, retaining the requested order, is ([25, -4], Yes, Yes, Yes, homogeneous_kernel_closed_for_all_real_scalars).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-universal-line-certificate · candidate
51. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [1, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [1, 1]], [scalar_failure, 2, [1, 1], [2, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
52. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [2, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [2, 1]], [scalar_failure, 2, [2, 1], [4, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
53. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [3, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [3, 1]], [scalar_failure, 2, [3, 1], [6, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
54. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [4, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [4, 1]], [scalar_failure, 2, [4, 1], [8, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
55. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [5, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [5, 1]], [scalar_failure, 2, [5, 1], [10, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
56. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [6, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [6, 1]], [scalar_failure, 2, [6, 1], [12, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
57. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [7, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [7, 1]], [scalar_failure, 2, [7, 1], [14, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
58. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [8, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [8, 1]], [scalar_failure, 2, [8, 1], [16, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
59. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [9, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [9, 1]], [scalar_failure, 2, [9, 1], [18, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
60. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [10, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [10, 1]], [scalar_failure, 2, [10, 1], [20, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
61. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [11, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [11, 1]], [scalar_failure, 2, [11, 1], [22, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
62. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [12, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [12, 1]], [scalar_failure, 2, [12, 1], [24, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
63. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [13, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [13, 1]], [scalar_failure, 2, [13, 1], [26, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
64. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [14, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [14, 1]], [scalar_failure, 2, [14, 1], [28, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
65. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [15, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [15, 1]], [scalar_failure, 2, [15, 1], [30, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
66. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [16, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [16, 1]], [scalar_failure, 2, [16, 1], [32, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
67. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [17, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [17, 1]], [scalar_failure, 2, [17, 1], [34, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
68. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [18, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [18, 1]], [scalar_failure, 2, [18, 1], [36, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
69. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [19, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [19, 1]], [scalar_failure, 2, [19, 1], [38, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
70. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [20, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [20, 1]], [scalar_failure, 2, [20, 1], [40, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
71. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [21, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [21, 1]], [scalar_failure, 2, [21, 1], [42, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
72. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [22, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [22, 1]], [scalar_failure, 2, [22, 1], [44, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
73. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [23, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [23, 1]], [scalar_failure, 2, [23, 1], [46, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
74. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [24, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [24, 1]], [scalar_failure, 2, [24, 1], [48, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
75. A Finite Nonzero Real Vector Set Cannot Pass Universal Scalar Closure
Use the complete nonempty duplicate-free finite point set=[[0, 0], [25, 1]] as a candidate subspace of R² with usual operations. Give (points,witness,subspace?,classification). First prefer [zero_missing,[0,0]]; otherwise use [scalar_failure,scalar,point,image] from the first nonzero point in input order and the first missing integer multiple with scalar 2,3,... . A null witness denotes the singleton zero_subspace.
View answer and explanation
([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], No, not_subspace_exact_counterexample)
Missing zero immediately refutes a subspace. If zero is present but a nonzero point occurs, distinct nonzero integer multiples cannot all belong to this finite set, yielding an exact scalar-closure counterexample. Do not infer an arbitrary subspace from finitely many positive samples. The exact result, retaining the requested order, is ([[0, 0], [25, 1]], [scalar_failure, 2, [25, 1], [50, 2]], No, not_subspace_exact_counterexample).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-finite-set-counterexample · candidate
76. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 1, -12]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([0, 1, -12], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 1, -12], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
77. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 2, -11]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([1, 2, -11], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 2, -11], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
78. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 3, -10]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([2, 3, -10], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 3, -10], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
79. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 4, -9]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([3, 4, -9], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 4, -9], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
80. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 5, -8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([4, 5, -8], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 5, -8], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
81. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 6, -7]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([0, 6, -7], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 6, -7], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
82. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 0, -6]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([1, 0, -6], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 0, -6], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
83. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 1, -5]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([2, 1, -5], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 1, -5], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
84. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 2, -4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([3, 2, -4], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 2, -4], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
85. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 3, -3]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([4, 3, -3], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 3, -3], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
86. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 4, -2]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([0, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 4, -2], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
87. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 5, -1]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([1, 5, -1], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 5, -1], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
88. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 6, 0]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([2, 6, 0], homogeneous_line, Yes, universal_homogeneous_equation_certificate)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 6, 0], homogeneous_line, Yes, universal_homogeneous_equation_certificate).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
89. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 0, 1]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([3, 0, 1], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 0, 1], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
90. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 1, 2]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([4, 1, 2], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 1, 2], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
91. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 2, 3]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([0, 2, 3], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 2, 3], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
92. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 3, 4]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([1, 3, 4], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 3, 4], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
93. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 4, 5]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([2, 4, 5], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 4, 5], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
94. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 5, 6]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([3, 5, 6], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 5, 6], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
95. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 6, 7]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([4, 6, 7], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 6, 7], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
96. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[0, 0, 8]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([0, 0, 8], empty_set, No, empty_set)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([0, 0, 8], empty_set, No, empty_set).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
97. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[1, 1, 9]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([1, 1, 9], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([1, 1, 9], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
98. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[2, 2, 10]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([2, 2, 10], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([2, 2, 10], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
99. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[3, 3, 11]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([3, 3, 11], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([3, 3, 11], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
100. Separate the Whole Space, Homogeneous Line, Affine Line and Empty Set
In R² with usual operations consider a*x+b*y=c with [a,b,c]=[4, 4, 12]. Give (equation,classification,subspace?,certificate). Classification is whole_R2, empty_set, homogeneous_line or affine_line_not_through_zero.
View answer and explanation
([4, 4, 12], affine_line_not_through_zero, No, affine_line_not_through_zero)
If both coefficients vanish, c=0 gives the whole space and c≠0 gives the empty set. Otherwise c=0 gives a homogeneous line with a universal linear-equation certificate, and nonzero c gives an affine line missing zero. The exact result, retaining the requested order, is ([4, 4, 12], affine_line_not_through_zero, No, affine_line_not_through_zero).
Reviewed step-by-step derivation
Real vector subspace · qf-mod32-subspace-classification · candidate
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