Declared Plane Matrix Maps and Affine Zero Obstructions
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261012-advance-20261016-r03-linear-transformation-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[0, 0], [0, 0]], v=[0, 0]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([0, 0], [0, 0], [0, 0], [0, 0])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
2. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 2], [3, 4]], v=[0, 0]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([1, 3], [2, 4], [0, 0], [0, 0])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [2, 4], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
3. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-10, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-10, 2], [1, -1], [-11, 3], [-11, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [1, -1], [-11, 3], [-11, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
4. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-9, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-9, 2], [1, -1], [-10, 3], [-10, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 2], [1, -1], [-10, 3], [-10, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
5. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-8, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-8, 2], [1, -1], [-9, 3], [-9, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [1, -1], [-9, 3], [-9, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
6. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-7, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-7, 2], [1, -1], [-8, 3], [-8, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 2], [1, -1], [-8, 3], [-8, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
7. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-6, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-6, 2], [1, -1], [-7, 3], [-7, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [1, -1], [-7, 3], [-7, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
8. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-5, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-5, 2], [1, -1], [-6, 3], [-6, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 2], [1, -1], [-6, 3], [-6, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
9. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-4, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-4, 2], [1, -1], [-5, 3], [-5, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [1, -1], [-5, 3], [-5, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
10. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-3, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-3, 2], [1, -1], [-4, 3], [-4, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 2], [1, -1], [-4, 3], [-4, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
11. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-2, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-2, 2], [1, -1], [-3, 3], [-3, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [1, -1], [-3, 3], [-3, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
12. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-1, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([-1, 2], [1, -1], [-2, 3], [-2, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 2], [1, -1], [-2, 3], [-2, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
13. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[0, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([0, 2], [1, -1], [-1, 3], [-1, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [1, -1], [-1, 3], [-1, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
14. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([1, 2], [1, -1], [0, 3], [0, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [1, -1], [0, 3], [0, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
15. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[2, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([2, 2], [1, -1], [1, 3], [1, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [1, -1], [1, 3], [1, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
16. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[3, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([3, 2], [1, -1], [2, 3], [2, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [1, -1], [2, 3], [2, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
17. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[4, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([4, 2], [1, -1], [3, 3], [3, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [1, -1], [3, 3], [3, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
18. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[5, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([5, 2], [1, -1], [4, 3], [4, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [1, -1], [4, 3], [4, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
19. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[6, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([6, 2], [1, -1], [5, 3], [5, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [1, -1], [5, 3], [5, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
20. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[7, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([7, 2], [1, -1], [6, 3], [6, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2], [1, -1], [6, 3], [6, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
21. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[8, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([8, 2], [1, -1], [7, 3], [7, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [1, -1], [7, 3], [7, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
22. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[9, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([9, 2], [1, -1], [8, 3], [8, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 2], [1, -1], [8, 3], [8, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
23. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[10, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([10, 2], [1, -1], [9, 3], [9, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [1, -1], [9, 3], [9, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
24. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[11, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([11, 2], [1, -1], [10, 3], [10, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 2], [1, -1], [10, 3], [10, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
25. Basis-Image Columns and Two Matrix-Vector Routes
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[12, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).
View answer and explanation
([12, 2], [1, -1], [11, 3], [11, 3])
Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [1, -1], [11, 3], [11, 3]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-basis-column-reconstruction · candidate
26. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[0, 0], [0, 0]], u=[0, 0], v=[0, 0]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
27. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 1]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([1, 0], [0, 1], [1, 1], [1, 1], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0], [0, 1], [1, 1], [1, 1], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
28. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-10, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-10, 0], [1, 2], [-9, 2], [-9, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 0], [1, 2], [-9, 2], [-9, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
29. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-9, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-9, 0], [1, 2], [-8, 2], [-8, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 0], [1, 2], [-8, 2], [-8, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
30. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-8, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-8, 0], [1, 2], [-7, 2], [-7, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 0], [1, 2], [-7, 2], [-7, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
31. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-7, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-7, 0], [1, 2], [-6, 2], [-6, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 0], [1, 2], [-6, 2], [-6, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
32. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-6, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-6, 0], [1, 2], [-5, 2], [-5, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 0], [1, 2], [-5, 2], [-5, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
33. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-5, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-5, 0], [1, 2], [-4, 2], [-4, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 0], [1, 2], [-4, 2], [-4, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
34. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-4, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-4, 0], [1, 2], [-3, 2], [-3, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 0], [1, 2], [-3, 2], [-3, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
35. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-3, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-3, 0], [1, 2], [-2, 2], [-2, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 0], [1, 2], [-2, 2], [-2, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
36. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-2, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-2, 0], [1, 2], [-1, 2], [-1, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 0], [1, 2], [-1, 2], [-1, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
37. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-1, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([-1, 0], [1, 2], [0, 2], [0, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 0], [1, 2], [0, 2], [0, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
38. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[0, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([0, 0], [1, 2], [1, 2], [1, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [1, 2], [1, 2], [1, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
39. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([1, 0], [1, 2], [2, 2], [2, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0], [1, 2], [2, 2], [2, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
40. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[2, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([2, 0], [1, 2], [3, 2], [3, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [1, 2], [3, 2], [3, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
41. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[3, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([3, 0], [1, 2], [4, 2], [4, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0], [1, 2], [4, 2], [4, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
42. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[4, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([4, 0], [1, 2], [5, 2], [5, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 0], [1, 2], [5, 2], [5, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
43. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[5, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([5, 0], [1, 2], [6, 2], [6, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 0], [1, 2], [6, 2], [6, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
44. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[6, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([6, 0], [1, 2], [7, 2], [7, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 0], [1, 2], [7, 2], [7, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
45. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[7, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([7, 0], [1, 2], [8, 2], [8, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 0], [1, 2], [8, 2], [8, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
46. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[8, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([8, 0], [1, 2], [9, 2], [9, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 0], [1, 2], [9, 2], [9, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
47. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[9, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([9, 0], [1, 2], [10, 2], [10, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 0], [1, 2], [10, 2], [10, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
48. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[10, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([10, 0], [1, 2], [11, 2], [11, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 0], [1, 2], [11, 2], [11, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
49. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[11, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([11, 0], [1, 2], [12, 2], [12, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 0], [1, 2], [12, 2], [12, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
50. Matrix-Model Additivity Values and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[12, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, difference of the last two sides).
View answer and explanation
([12, 0], [1, 2], [13, 2], [13, 2], [0, 0])
Evaluate both routes componentwise for the declared matrix model and retain the actual signed residual. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 0], [1, 2], [13, 2], [13, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-additivity-values · candidate
51. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[0, 0], [0, 0]], v=[0, 0], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([0, 0], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
52. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 1]], v=[1, -1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([1, -1], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, -1], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
53. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-10, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-10, 2], [-10, 2], [-10, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [-10, 2], [-10, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
54. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-9, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-9, 2], [9, -2], [9, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 2], [9, -2], [9, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
55. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-8, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-8, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
56. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-7, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-7, 2], [-7, 2], [-7, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 2], [-7, 2], [-7, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
57. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-6, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-6, 2], [6, -2], [6, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [6, -2], [6, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
58. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-5, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-5, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
59. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-4, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-4, 2], [-4, 2], [-4, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-4, 2], [-4, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
60. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-3, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-3, 2], [3, -2], [3, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 2], [3, -2], [3, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
61. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-2, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-2, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
62. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-1, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([-1, 2], [-1, 2], [-1, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 2], [-1, 2], [-1, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
63. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[0, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([0, 2], [0, -2], [0, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0, -2], [0, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
64. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[1, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([1, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
65. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[2, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([2, 2], [2, 2], [2, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [2, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
66. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[3, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([3, 2], [-3, -2], [-3, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [-3, -2], [-3, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
67. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[4, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([4, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
68. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[5, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([5, 2], [5, 2], [5, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [5, 2], [5, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
69. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[6, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([6, 2], [-6, -2], [-6, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [-6, -2], [-6, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
70. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[7, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([7, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
71. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[8, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([8, 2], [8, 2], [8, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [8, 2], [8, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
72. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[9, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([9, 2], [-9, -2], [-9, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 2], [-9, -2], [-9, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
73. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[10, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([10, 2], [0, 0], [0, 0], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [0, 0], [0, 0], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
74. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[11, 1], c=1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([11, 2], [11, 2], [11, 2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 2], [11, 2], [11, 2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
75. Matrix-Model Scalar Routes and Defect
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[12, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).
View answer and explanation
([12, 2], [-12, -2], [-12, -2], [0, 0])
One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [-12, -2], [-12, -2], [0, 0]).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-scalar-values · candidate
76. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[0, 0], [0, 0]], v=[0, 0], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([0, 0], [0, 0], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
77. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[0, 0], [0, 0]], v=[99, 1], b=[1, -1]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([0, 0], [1, -1], [1, -1], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [1, -1], [1, -1], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
78. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-10, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-10, 2], [-10, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [-10, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
79. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-9, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-9, 2], [-8, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 2], [-8, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
80. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-8, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-8, 2], [-8, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [-8, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
81. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-7, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-7, 2], [-6, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 2], [-6, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
82. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-6, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-6, 2], [-6, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [-6, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
83. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-5, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-5, 2], [-4, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 2], [-4, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
84. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-4, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-4, 2], [-4, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-4, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
85. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-3, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-3, 2], [-2, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 2], [-2, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
86. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-2, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-2, 2], [-2, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-2, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
87. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[-1, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([-1, 2], [0, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 2], [0, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
88. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[0, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([0, 2], [0, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
89. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[1, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([1, 2], [2, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [2, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
90. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[2, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([2, 2], [2, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
91. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[3, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([3, 2], [4, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [4, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
92. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[4, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([4, 2], [4, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [4, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
93. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[5, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([5, 2], [6, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [6, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
94. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[6, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([6, 2], [6, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [6, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
95. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[7, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([7, 2], [8, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2], [8, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
96. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[8, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([8, 2], [8, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [8, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
97. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[9, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([9, 2], [10, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 2], [10, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
98. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[10, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([10, 2], [10, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [10, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
99. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[11, 1], b=[1, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([11, 2], [12, 2], [1, 0], No)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 2], [12, 2], [1, 0], No).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
100. Declared Affine Model and Zero-Point Linearity Guard
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[12, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
View answer and explanation
([12, 2], [12, 2], [0, 0], Yes)
The image of zero is b; in this explicitly given matrix-affine model both components of b must vanish. This is not a sample-based test for arbitrary functions. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [12, 2], [0, 0], Yes).
Reviewed step-by-step derivation
Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate
Canonical question JSONL · English JSONL · Validation report