{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0001","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19a8e85da0f8b618b51994416178fa9cd68d49b348dec4e429e49ab2a615a858","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163001,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[0,0],[0,0]],\"v\":[0,0]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0002","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"412a06aa9b169c5192ea9c45a8af8cc781e119d3a9173dc1aa349cec673dd6ea","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163002,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[1,2],[3,4]],\"v\":[0,0]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([1, 3], [2, 4], [0, 0], [0, 0])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0003","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"456ead4d50097ef61f3a6c4a399cb46653095aecc2e8a66dacb003f81ddc84ce","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163003,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-10,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-10, 2], [1, -1], [-11, 3], [-11, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0004","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97bdb70b2d4a214884103b04c692f290a41a7f0dcf7acb422214fb7d99e04e3","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163004,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-9,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-9, 2], [1, -1], [-10, 3], [-10, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0005","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34693287fa5eb42e94c65469cee3f49e13be594bdd2ab61dd6119dd94b474207","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163005,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-8,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-8, 2], [1, -1], [-9, 3], [-9, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0006","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab06bb3113cf97718f0023593a244499b71117b4c2a1ecf579e36fb04c9d42f","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163006,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-7,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-7, 2], [1, -1], [-8, 3], [-8, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0007","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd8f7f17c95c7fe817a201243d0ae1ecc46af5299e11a8b8b9bb4778ea568cd3","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163007,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-6,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-6, 2], [1, -1], [-7, 3], [-7, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0008","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71764cdf0c2b7f7a503aa49018f058809141f1e366329ee50c0bce38b1bbd5b","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163008,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-5,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-5, 2], [1, -1], [-6, 3], [-6, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0009","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9b3c61340438d8de81b1e4c5b1a39d4a7c786a5bbb4689ff23c689d5570bf24","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163009,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-4,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-4, 2], [1, -1], [-5, 3], [-5, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0010","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc5352ae90426049f413865d7ea0b51764ddcce9fe264e59b46eb70489976e6d","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163010,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-3,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-3, 2], [1, -1], [-4, 3], [-4, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0011","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0675500b946858e63f7be8ce75396b1a32040af6ec16c1a6316d47254db79e63","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163011,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-2,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-2, 2], [1, -1], [-3, 3], [-3, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0012","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4effd94c401c615d55a43042f97b59281b58f5ca0f7f5458d13cf8b4d5e64173","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163012,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-1,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([-1, 2], [1, -1], [-2, 3], [-2, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0013","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815c2aab8d70da98b896bb75264e69e03b0630ceb8b42f1724a014776a07ad17","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163013,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[0,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([0, 2], [1, -1], [-1, 3], [-1, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0014","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d02a61ef150ac2775d1d47b93a72db9f99d6b98b9993eb7e8820ddda2c87f66a","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163014,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[1,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([1, 2], [1, -1], [0, 3], [0, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0015","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4a9e883220d76ef5daa9d481fb2c7483353fb3e4223a5df5819c2cbc4d2bdd","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163015,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[2,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([2, 2], [1, -1], [1, 3], [1, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0016","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a9b5efe93cbfbfa2260f98ec7d827d1d1a610cbe4deb8ab4cfaae871d01654d","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163016,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[3,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([3, 2], [1, -1], [2, 3], [2, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0017","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6264a47b26cdc5dac950aab99fffc837cb8251ba0efe1d8961eabd6e403a3d","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163017,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[4,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([4, 2], [1, -1], [3, 3], [3, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0018","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"978230b55df576ab3b63804dd75036343319c61687e08836e2ae4c9508b740ae","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163018,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[5,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([5, 2], [1, -1], [4, 3], [4, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0019","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f969f3c51cd43ccb501c9c28c13df32d66a30d675627d211f9d09125ee2bccf5","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163019,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[6,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([6, 2], [1, -1], [5, 3], [5, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0020","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e47152f015de4616015b8a10580cefd571f2731b622eb07f1b9aa2a54d288a24","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163020,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[7,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([7, 2], [1, -1], [6, 3], [6, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0021","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a606cd4eedfeb12a3efbf54db5fff13eb5479228345372747744aec13962f049","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163021,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[8,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([8, 2], [1, -1], [7, 3], [7, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0022","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"612523110323d4a99cb5ea3f0fa776c900a28d071fa4cdf812f8d7a0a65f9b84","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163022,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[9,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([9, 2], [1, -1], [8, 3], [8, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0023","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e82453bf170e65e26f8a3caa4ef413646605c58b02af7f7e68a3d552550432f","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163023,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[10,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([10, 2], [1, -1], [9, 3], [9, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0024","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"241f4432743afdd98002d53e5d97e7e0c2fb16f337d8bb33d2da37c0e1ac5906","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163024,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[11,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([11, 2], [1, -1], [10, 3], [10, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0025","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c7e83e49a449ff08fd8597f77c63e89b8d07210184722b21a414f1fa59a6853","family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163025,"semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[12,1],[2,-1]],\"v\":[1,-1]}"},"fields":{"title":"Basis-Image Columns and Two Matrix-Vector Routes","stem":"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).","answer":{"kind":"expression","value":"([12, 2], [1, -1], [11, 3], [11, 3])","explanation":"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])."},"skills":["Basis-Image Columns and Two Matrix-Vector Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0026","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eca1d4bccaf145f7a85f2a9aa2306d2c53185dcd320d9dcbeb7a78749b3a756","family_id":"qf-density16-linear-additivity-values","seed":202610163026,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[0,0],[0,0]],\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0027","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eb2f5962fa439a65c61d8978a4e75d0758aac3417fbe0f271e4f4479c37b607","family_id":"qf-density16-linear-additivity-values","seed":202610163027,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[1,0],[0,1]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([1, 0], [0, 1], [1, 1], [1, 1], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0028","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"617cc1d140cfea8b55217751f9030580ed75dd8ada7a4f2d78de6141791038d9","family_id":"qf-density16-linear-additivity-values","seed":202610163028,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-10,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-10, 0], [1, 2], [-9, 2], [-9, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0029","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2a09f71901a853f217ef95095eccafec2e25be509bd2967033c9be64706d14b","family_id":"qf-density16-linear-additivity-values","seed":202610163029,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-9,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-9, 0], [1, 2], [-8, 2], [-8, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0030","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2d8f1b251f3f484500589f209e1067effb55e81a89208dacdc363137097900","family_id":"qf-density16-linear-additivity-values","seed":202610163030,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-8,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-8, 0], [1, 2], [-7, 2], [-7, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0031","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5184ca87ce143be72969cbea68d50149b86cf37863dab2543a65989e340f09e","family_id":"qf-density16-linear-additivity-values","seed":202610163031,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-7,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-7, 0], [1, 2], [-6, 2], [-6, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0032","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eb21c60bb89636803593d606507a21e972698b19347c0c8d1e843fbffe6ce6f","family_id":"qf-density16-linear-additivity-values","seed":202610163032,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-6,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-6, 0], [1, 2], [-5, 2], [-5, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0033","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9270176d8678bfd05378324c24446e593a60fc55f70718daa0ec64f08293bde","family_id":"qf-density16-linear-additivity-values","seed":202610163033,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-5,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-5, 0], [1, 2], [-4, 2], [-4, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0034","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7512d15dc98dbe3347bff94ae1bbce7b1c5c3e27ac26c0396b54b53d8872720f","family_id":"qf-density16-linear-additivity-values","seed":202610163034,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-4,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-4, 0], [1, 2], [-3, 2], [-3, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0035","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"127d62d2577bef27d85e8f40e048804144cafb631e781595fb864e79f7a47aad","family_id":"qf-density16-linear-additivity-values","seed":202610163035,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-3,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-3, 0], [1, 2], [-2, 2], [-2, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0036","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564ad5c7b0f3db57e0fe49cd9e7df4f2c44b638ccc7417fa194adcb752fd0fe2","family_id":"qf-density16-linear-additivity-values","seed":202610163036,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-2,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-2, 0], [1, 2], [-1, 2], [-1, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0037","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e61b6970f2fcfacd8cfae940f1ce5f98e7dea15abaa0b6f0c6b4e70c75f1ea49","family_id":"qf-density16-linear-additivity-values","seed":202610163037,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-1,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([-1, 0], [1, 2], [0, 2], [0, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0038","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25dfda4a8a959cf79b7e1e136c7534548c0108b09848c767f45896200fbc226c","family_id":"qf-density16-linear-additivity-values","seed":202610163038,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[0,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([0, 0], [1, 2], [1, 2], [1, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0039","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d72c7e066e6e61c198c3b463711a4556d475087a83f2c768c8a87ab582f440","family_id":"qf-density16-linear-additivity-values","seed":202610163039,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[1,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([1, 0], [1, 2], [2, 2], [2, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0040","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c079367dafbf45395d75890a15d9673e205ac36f02714b49fcf93e51a1960fb8","family_id":"qf-density16-linear-additivity-values","seed":202610163040,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[2,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([2, 0], [1, 2], [3, 2], [3, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0041","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43171436f13c2b624ef98d36b24790382c09bd6244a2d474dcfd7e0ddaafb11c","family_id":"qf-density16-linear-additivity-values","seed":202610163041,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[3,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([3, 0], [1, 2], [4, 2], [4, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0042","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2705983e7a9f905b301a1c7f1cda84402030a5ed2e5caded895b805069550ba","family_id":"qf-density16-linear-additivity-values","seed":202610163042,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[4,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([4, 0], [1, 2], [5, 2], [5, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0043","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61f44dbfc4db9022978eb1f13a46273a29375df0e890aaf4e37f779821719a2f","family_id":"qf-density16-linear-additivity-values","seed":202610163043,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[5,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([5, 0], [1, 2], [6, 2], [6, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0044","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea24e7296f988f03deeda713defc3341e54f0cf8373fd858f3f47ac2be23fd86","family_id":"qf-density16-linear-additivity-values","seed":202610163044,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[6,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([6, 0], [1, 2], [7, 2], [7, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0045","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f64db268eb002061d5ba28b8b8eb4976065e498bbc36c8eef0022b42e228b86","family_id":"qf-density16-linear-additivity-values","seed":202610163045,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[7,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([7, 0], [1, 2], [8, 2], [8, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0046","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd085b8f2c280c48c6f2b4e51450325a363405fc7f6c8ccb265bef339b62e390","family_id":"qf-density16-linear-additivity-values","seed":202610163046,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[8,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([8, 0], [1, 2], [9, 2], [9, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0047","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"945e05a1fab9f9479741a04292daab2fa29e916cb01e0fd22176777f8b3e7f0f","family_id":"qf-density16-linear-additivity-values","seed":202610163047,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[9,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([9, 0], [1, 2], [10, 2], [10, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0048","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc9a5f9dc95ebdc77f5dc3ca17472fdf0071c50474841b3bd6d00b109935e85e","family_id":"qf-density16-linear-additivity-values","seed":202610163048,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[10,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([10, 0], [1, 2], [11, 2], [11, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0049","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e002a98c37577997be30677367aaa633dccc6b6a56eebff19ea4608daf1d13","family_id":"qf-density16-linear-additivity-values","seed":202610163049,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[11,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([11, 0], [1, 2], [12, 2], [12, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0050","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6710dab98c796f59287ac59c46ebe842cee49461db7ee6bde718b5422a8455d5","family_id":"qf-density16-linear-additivity-values","seed":202610163050,"semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[12,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Additivity Values and Defect","stem":"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).","answer":{"kind":"expression","value":"([12, 0], [1, 2], [13, 2], [13, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Additivity Values and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0051","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92cebe7d256ddcb730e0fe54256e2ee98e8998d03d8423fea864120b3deeccf1","family_id":"qf-density16-linear-scalar-values","seed":202610163051,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[0,0],[0,0]],\"c\":0,\"v\":[0,0]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0052","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2ea76f319aa261c49af4a6c06e659eaf7a11c6873c5e51ed3bfa0fe7ec98f5a","family_id":"qf-density16-linear-scalar-values","seed":202610163052,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,1]],\"c\":0,\"v\":[1,-1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([1, -1], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0053","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d336432bb5ddf82c6e4fdd6d525c89fea25f8b020dbf84613e370c2086b85822","family_id":"qf-density16-linear-scalar-values","seed":202610163053,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-10,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-10, 2], [-10, 2], [-10, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0054","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"debeda983eb49d7fcc19fbd3e519419ad0c2521658ff032a46c0c04c7f72a859","family_id":"qf-density16-linear-scalar-values","seed":202610163054,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-9,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-9, 2], [9, -2], [9, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0055","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc280294b4bb1c614703caba38ad51a2df3cee406011e5ce86eb2ef6bf8fdaeb","family_id":"qf-density16-linear-scalar-values","seed":202610163055,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-8,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-8, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0056","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b077532e0af7be306c0431dcc7a50ce8a32e8adb39f4882a5ab30a849a14a79","family_id":"qf-density16-linear-scalar-values","seed":202610163056,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-7,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-7, 2], [-7, 2], [-7, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0057","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec50531030f32985177e6978ea88a888a642bd08a33b9b14a88c203011919ac","family_id":"qf-density16-linear-scalar-values","seed":202610163057,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-6,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-6, 2], [6, -2], [6, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0058","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbdf6421e9b54e4bca838d69b7b6099df179d5e63a3a37d151d0f70d3d74fdc4","family_id":"qf-density16-linear-scalar-values","seed":202610163058,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-5,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-5, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0059","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"581884920eef9b042a25acb76d2e873fb68b867930459d5d6b008c397a663430","family_id":"qf-density16-linear-scalar-values","seed":202610163059,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-4,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-4, 2], [-4, 2], [-4, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0060","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67b393d80a2a1f6795db587a0dd6acac1c1da06557c5148443888469465303dc","family_id":"qf-density16-linear-scalar-values","seed":202610163060,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-3,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-3, 2], [3, -2], [3, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0061","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b88d43b82336df8563019a5af9062447456dd1382274cb6240173c8b44fbd62","family_id":"qf-density16-linear-scalar-values","seed":202610163061,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-2,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-2, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0062","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e734e4956dd5c6961af24a6bca23a75b59adc68573b1acec651620b2f1452895","family_id":"qf-density16-linear-scalar-values","seed":202610163062,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-1,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([-1, 2], [-1, 2], [-1, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0063","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ffaead9dd8a3748658bcd992745bb24b98bd5c32f6e444387ab08f0565210a4","family_id":"qf-density16-linear-scalar-values","seed":202610163063,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[0,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([0, 2], [0, -2], [0, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0064","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acdf06a0a164af0e1468e4fea43ac7650e2ee191d089eaedab4a23ab85f2f2de","family_id":"qf-density16-linear-scalar-values","seed":202610163064,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[1,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([1, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0065","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bf120f2e8654a95da5d9dee25a8f2eb5d6b894cad358961f672eeaf338d80fa","family_id":"qf-density16-linear-scalar-values","seed":202610163065,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[2,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([2, 2], [2, 2], [2, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0066","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c62bf838f923b9ec8a8e4458923b226a8fb4f758b2f398881d91f49c4512e9d","family_id":"qf-density16-linear-scalar-values","seed":202610163066,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[3,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([3, 2], [-3, -2], [-3, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0067","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bcbf2d4ad113736493ad48987272b91063d4dea2278791bfcf1c719d963b106","family_id":"qf-density16-linear-scalar-values","seed":202610163067,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[4,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([4, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0068","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81aa964c29cafbd67fde0a3330afd48aaee05b0f805529ef746bfb90640df56","family_id":"qf-density16-linear-scalar-values","seed":202610163068,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[5,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([5, 2], [5, 2], [5, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0069","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c46d23111e11bd7f1bd6a72785743a77e4b667c59a8d96fbf69f7c45dc319772","family_id":"qf-density16-linear-scalar-values","seed":202610163069,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[6,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([6, 2], [-6, -2], [-6, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0070","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02d3b0bf25f403b1398d432f5ad2772c2fc5b15d69979b1df0ac1c6f58958063","family_id":"qf-density16-linear-scalar-values","seed":202610163070,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[7,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([7, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0071","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"650a70027e52e2b1daccf28c90f273ef31ec48a9a996cc3d9457ff7e8272a2ff","family_id":"qf-density16-linear-scalar-values","seed":202610163071,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[8,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([8, 2], [8, 2], [8, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0072","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"466449207569f7d76bbd7ed2036a1ac1c7e92b803dee52463d01b1540f64f469","family_id":"qf-density16-linear-scalar-values","seed":202610163072,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[9,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([9, 2], [-9, -2], [-9, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0073","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b243211356b641b7842d7d23424059aeeca95540cf558bf29424d18f80b77cc","family_id":"qf-density16-linear-scalar-values","seed":202610163073,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[10,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([10, 2], [0, 0], [0, 0], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0074","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55a4f202d72a50596dbdc2f2b8884e9817452446dbdc9110f2bbfcbde71a7c07","family_id":"qf-density16-linear-scalar-values","seed":202610163074,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[11,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([11, 2], [11, 2], [11, 2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0075","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e68ba9d27a090dcea5216b708a3787c25771b53917d43ba606b2d68d8fb17fe","family_id":"qf-density16-linear-scalar-values","seed":202610163075,"semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[12,1]}"},"fields":{"title":"Matrix-Model Scalar Routes and Defect","stem":"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).","answer":{"kind":"expression","value":"([12, 2], [-12, -2], [-12, -2], [0, 0])","explanation":"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])."},"skills":["Matrix-Model Scalar Routes and Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0076","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3544bcd77ca06be2a96df4bc851196aea7ce2d8c007c18842e34b8b44fa5d67","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163076,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[0,0],[0,0]],\"b\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0077","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"628b823473ca50b286a94a3aaa0404b90a33fb5caaf2c8eafd9ea514a3e6ab1b","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163077,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[0,0],[0,0]],\"b\":[1,-1],\"v\":[99,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([0, 0], [1, -1], [1, -1], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0078","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4f1a4c64742c976ad4db70aeb0974cbaf575c80864fe34c396b2ae56f467a16","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163078,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-10,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-10, 2], [-10, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0079","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b6cd785c6a4c014105c225fb9e8abc6446966c3540e919e231105555fc331f","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163079,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-9,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-9, 2], [-8, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0080","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dcba2dd2db6c5669e931dbd8a09d725ea11faeaf344ece5af3b3afde9ab8e81","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163080,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-8,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-8, 2], [-8, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0081","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e273412d179b4eca8d60de937bb61aa62a80abf6c2195c0a6be7dc218971bd6","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163081,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-7,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-7, 2], [-6, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0082","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25cb50c926bc922f67a892560078dd93939f50bdf8f1409dc420f7ea0d69e57a","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163082,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-6,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-6, 2], [-6, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0083","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a29b114ea7ba2cd3a1a49a95113f8caff42ef6320b447b3292b4bfab9adc04a0","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163083,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-5,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-5, 2], [-4, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0084","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffb95cc1fcc0e6e58f2bee1e1fddbe69df7aa2a0bab3f0715be94c8264616243","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163084,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-4,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-4, 2], [-4, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0085","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe92f202833223112a5e3ac9465c677ac43906725373a39bd6c48b96d0398d15","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163085,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-3,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-3, 2], [-2, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0086","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8476149af841d36b5035ff72a5f4d27c9859e2e3313af2b3a2eb692917373a3","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163086,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-2,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-2, 2], [-2, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0087","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e1906e0cc4b6ac302876c330c1c2cfdb143a735065cb3632549b2b89ab514c","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163087,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-1,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([-1, 2], [0, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0088","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"481542dbb7b89e929ab46bbcdb6e0e1148df1e32cfd5606b446bdea358624faf","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163088,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[0,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([0, 2], [0, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0089","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36d65a981d561164ce1da096ae522d8cacd8f1cefa396712a1bc6249d1bea9fd","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163089,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[1,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([1, 2], [2, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0090","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e3e7e6021e439db4c7c91ae9b81b399c09365ea08e0103f3ce2d5f3ed733b07","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163090,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[2,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([2, 2], [2, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0091","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf8a7cfc8d2701df3986f77d01482388fe09b6d33b171ccec0597fe678eaaa5","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163091,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[3,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([3, 2], [4, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0092","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e71dcc4644bc8490543b3acd010475a45e8837ef86d351fb1ba33c7a1dfa42d","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163092,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[4,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([4, 2], [4, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0093","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06ec1954bb84c240790d272d70e370b8c52fb2f5e175808872410840cdf0f83f","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163093,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[5,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([5, 2], [6, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0094","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7715199cca12c0c00b1640174ecd926275e79b03a773e200e9ec3c9409e9a89","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163094,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[6,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([6, 2], [6, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0095","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15dd9a15b2d592ef0bdc255b52fe8c3ee0d7a18818df23f5c517e5d70becbf5c","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163095,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[7,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([7, 2], [8, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0096","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684303284355b7187d7f4db656b8fe778d86aac18aca1e4a32ba6c293d2989b6","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163096,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[8,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([8, 2], [8, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0097","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5646ee173ca8528b95b902cfe9ecaf88420d464f18798cef43ad63bf5e2e174d","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163097,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[9,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([9, 2], [10, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0098","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c36d098ba6b3a6a560dcbca9ed89f044d7bbdf203d60d7bf8b74fc93837c8fb","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163098,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[10,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([10, 2], [10, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0099","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e28c2d43e0fe1a1eb06dc3bf039b0c68d7be98bedfdfb7898b2ff49b0523e2","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163099,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[11,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([11, 2], [12, 2], [1, 0], No)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r03-linear-transformation-0100","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"465a25fff6c7a805e3b956175a57303b2ead222f59c4380083b80a516d3cbef2","family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163100,"semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[12,1]}"},"fields":{"title":"Declared Affine Model and Zero-Point Linearity Guard","stem":"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?).","answer":{"kind":"expression","value":"([12, 2], [12, 2], [0, 0], Yes)","explanation":"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)."},"skills":["Declared Affine Model and Zero-Point Linearity Guard"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}