# Declared Plane Matrix Maps and Affine Zero Obstructions

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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-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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r03-linear-transformation-0100/)

Matrix-represented linear maps on the real plane · qf-density16-linear-affine-zero-obstruction · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r03-linear-transformation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r03-linear-transformation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r03-linear-transformation-foundations/validation-report.json)