{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r02-linear-transformation-0001","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d18ca12a10e76d3b204b207e8ce1963c038c3d5ac25c02cb47b87833b4cf0e0","family_id":"qf-linear-map-image","seed":202610072001,"semantic_template":"qf-linear-map-image/{\"inputs\":[\"zero_input\"],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 0], find T(v).","answer":{"kind":"expression","value":"[0, 0]","explanation":"Take the sum of row-by-column products for each component, giving [0, 0]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0002","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddddf289854dcb87a48b8949a064c6f92f862df0674e88fa2d04dc0c430fb97e","family_id":"qf-linear-map-image","seed":202610072002,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"1\",null],[\"2\",null]],[\"1\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[1, 0], find T(v).","answer":{"kind":"expression","value":"[1, 2]","explanation":"Take the sum of row-by-column products for each component, giving [1, 2]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0003","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"187f571298b1b4b640e7af319a50f2ceb1def684e135580bbf240d5ad5c23747","family_id":"qf-linear-map-image","seed":202610072003,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[\"-2\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-2, 1], find T(v).","answer":{"kind":"expression","value":"[1, -2]","explanation":"Take the sum of row-by-column products for each component, giving [1, -2]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0004","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5443211f3a21595a18f7098b7f20b8815e062a1157dc0e440882fb9b387d06f","family_id":"qf-linear-map-image","seed":202610072004,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"4\"],[null,\"4\"]],[\"0\",\"3\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 3], find T(v).","answer":{"kind":"expression","value":"[12, 12]","explanation":"Take the sum of row-by-column products for each component, giving [12, 12]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0005","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d0ad2bce3e476c5003555cc7fc7a30b14122ad66a56afcf4ad2b8d4b9196794","family_id":"qf-linear-map-image","seed":202610072005,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-8\",null],[\"-2\",null]],[\"4\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[4, 0], find T(v).","answer":{"kind":"expression","value":"[-32, -8]","explanation":"Take the sum of row-by-column products for each component, giving [-32, -8]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0006","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc53e63e04b55ec32caff0de652dba0711a2094fd94ee808be38f40cf6c09f18","family_id":"qf-linear-map-image","seed":202610072006,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[\"-5\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-5, 1], find T(v).","answer":{"kind":"expression","value":"[37, 16]","explanation":"Take the sum of row-by-column products for each component, giving [37, 16]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0007","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"672e1f3e943995f5a3aebe5e968b54f4492c65577ee98e3ae590e0cc66ce4049","family_id":"qf-linear-map-image","seed":202610072007,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"3\"],[null,\"7\"]],[\"0\",\"6\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 6], find T(v).","answer":{"kind":"expression","value":"[18, 42]","explanation":"Take the sum of row-by-column products for each component, giving [18, 42]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0008","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4746639a5deafa5df953d2006de20e59a9e6d697289be78d9f2d4e57a6a57d5","family_id":"qf-linear-map-image","seed":202610072008,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-5\",null],[\"-2\",null]],[\"7\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[7, 0], find T(v).","answer":{"kind":"expression","value":"[-35, -14]","explanation":"Take the sum of row-by-column products for each component, giving [-35, -14]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0009","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905c5d42705230141ad5c3e56f27c1706d6c29bf556bc14d6c1bb7e159afc682","family_id":"qf-linear-map-image","seed":202610072009,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[\"-8\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-8, 1], find T(v).","answer":{"kind":"expression","value":"[33, 25]","explanation":"Take the sum of row-by-column products for each component, giving [33, 25]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0010","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b07efd3191e2a704b8591693e3e75829e66340587a7b6860bb7f038b3a9fa15c","family_id":"qf-linear-map-image","seed":202610072010,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"2\"],[null,\"10\"]],[\"0\",\"9\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 9], find T(v).","answer":{"kind":"expression","value":"[18, 90]","explanation":"Take the sum of row-by-column products for each component, giving [18, 90]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0011","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cf4aaf70aee2731908ec802a6545f8c0e59bdb7ed63fb26f2d2605c72d49651","family_id":"qf-linear-map-image","seed":202610072011,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-2\",null],[\"-2\",null]],[\"10\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[10, 0], find T(v).","answer":{"kind":"expression","value":"[-20, -20]","explanation":"Take the sum of row-by-column products for each component, giving [-20, -20]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0012","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"698513e0f463c7c730cb062a45a399710d1c34a116de71fc7672d2e3ec86724c","family_id":"qf-linear-map-image","seed":202610072012,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[\"-11\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-11, 1], find T(v).","answer":{"kind":"expression","value":"[15, 34]","explanation":"Take the sum of row-by-column products for each component, giving [15, 34]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0013","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2ed34bed9b2c6aafc0b6827719174886be79fd04be2d766ed6676d642c8b43a","family_id":"qf-linear-map-image","seed":202610072013,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"1\"],[null,\"13\"]],[\"0\",\"12\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 12], find T(v).","answer":{"kind":"expression","value":"[12, 156]","explanation":"Take the sum of row-by-column products for each component, giving [12, 156]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0014","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d84137bc08f8fc82b32b98b073da1e1e6cf786cc6397eac25113a4aa6f19be1","family_id":"qf-linear-map-image","seed":202610072014,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"1\",null],[\"-2\",null]],[\"13\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[13, 0], find T(v).","answer":{"kind":"expression","value":"[13, -26]","explanation":"Take the sum of row-by-column products for each component, giving [13, -26]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0015","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4fe0629b99591dbe1df407246aac1cc16ee4135bf57502ee26da594c4aac1de","family_id":"qf-linear-map-image","seed":202610072015,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[\"-14\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-14, 1], find T(v).","answer":{"kind":"expression","value":"[-25, 43]","explanation":"Take the sum of row-by-column products for each component, giving [-25, 43]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0016","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b02f6784558b8539fa682db639d8a5705bc3ca874217ff01ce27339c207a6c5b","family_id":"qf-linear-map-image","seed":202610072016,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"4\"],[null,\"16\"]],[\"0\",\"15\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 15], find T(v).","answer":{"kind":"expression","value":"[60, 240]","explanation":"Take the sum of row-by-column products for each component, giving [60, 240]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0017","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c207b69d16f41bec1a3f8ddf7306aa7de6d5af9a1e59aece2f46192a88f9f76d","family_id":"qf-linear-map-image","seed":202610072017,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"4\",null],[\"-2\",null]],[\"16\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[16, 0], find T(v).","answer":{"kind":"expression","value":"[64, -32]","explanation":"Take the sum of row-by-column products for each component, giving [64, -32]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0018","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1636ed6b7a29add5c5763a2e0518ab52289de23edfb34e4c8c729d970768bdd8","family_id":"qf-linear-map-image","seed":202610072018,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[\"-17\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-17, 1], find T(v).","answer":{"kind":"expression","value":"[-83, 52]","explanation":"Take the sum of row-by-column products for each component, giving [-83, 52]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0019","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df18db0124a28e906da859c91aaefffcabe89c4c434df6c9f354138b9f05164","family_id":"qf-linear-map-image","seed":202610072019,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"3\"],[null,\"19\"]],[\"0\",\"18\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 18], find T(v).","answer":{"kind":"expression","value":"[54, 342]","explanation":"Take the sum of row-by-column products for each component, giving [54, 342]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0020","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3123eda4d903160f8e3516ce70a31c47c14270fa12bc23fbca7a4db3c7551f","family_id":"qf-linear-map-image","seed":202610072020,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"7\",null],[\"-2\",null]],[\"19\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[19, 0], find T(v).","answer":{"kind":"expression","value":"[133, -38]","explanation":"Take the sum of row-by-column products for each component, giving [133, -38]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0021","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27f43a55e18541d6830350705d0f805057bd9aeca81c350900510d074f4e6cd3","family_id":"qf-linear-map-image","seed":202610072021,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[\"-20\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-20, 1], find T(v).","answer":{"kind":"expression","value":"[-159, 61]","explanation":"Take the sum of row-by-column products for each component, giving [-159, 61]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0022","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9ef4d406e7f39d2c3a17fbed9c6e198ec336bf44caf5a1d4a500c8998bcdac4","family_id":"qf-linear-map-image","seed":202610072022,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"2\"],[null,\"22\"]],[\"0\",\"21\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 21], find T(v).","answer":{"kind":"expression","value":"[42, 462]","explanation":"Take the sum of row-by-column products for each component, giving [42, 462]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0023","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cbe64c0c9312378e8c8d94da758cd9581eb3832c950de9e54dcd2db6aef24e3","family_id":"qf-linear-map-image","seed":202610072023,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"10\",null],[\"-2\",null]],[\"22\",\"0\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[22, 0], find T(v).","answer":{"kind":"expression","value":"[220, -44]","explanation":"Take the sum of row-by-column products for each component, giving [220, -44]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0024","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11021ea19edd9ab619c67d9c462159769205084e2f09aad204a06bd78ec8be3b","family_id":"qf-linear-map-image","seed":202610072024,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[\"-23\",\"1\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-23, 1], find T(v).","answer":{"kind":"expression","value":"[-249, 70]","explanation":"Take the sum of row-by-column products for each component, giving [-249, 70]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0025","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7143f2d030e57f7eab5debb2b22a1fdf85d114e3c75f4d38972a4530f76ccc20","family_id":"qf-linear-map-image","seed":202610072025,"semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"1\"],[null,\"25\"]],[\"0\",\"24\"]],\"operation\":\"linear_image\"}"},"fields":{"title":"Vector Image under a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 24], find T(v).","answer":{"kind":"expression","value":"[24, 600]","explanation":"Take the sum of row-by-column products for each component, giving [24, 600]. A zero input has zero image; the output is a vector, not a matrix or a guessed point."},"skills":["Vector Image under a 2×2 Matrix"],"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-20261006-advance-20261007-r02-linear-transformation-0026","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06d45430a5913829000116a027359a0802c4d7f2729d87ba1726c5802414f148","family_id":"qf-linear-map-affine-test","seed":202610072026,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"0\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[0, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"Yes","explanation":"Here F(0)=b=[0, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving Yes. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0027","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44bcec3840e5654d11e224b4f2560ba04acc1aff1190137409e9e706c8afbcdf","family_id":"qf-linear-map-affine-test","seed":202610072027,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-1\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-1, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-1, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0028","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e336a4b748d8f9e621c607e88af92e26dc105379c1f4a2ba9110dedaf19a6fde","family_id":"qf-linear-map-affine-test","seed":202610072028,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-2\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-2, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-2, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0029","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cb4b8271de0d7540704edb841d0f86fc38fdae9313b0c14d08d7cec5e1a7a7a","family_id":"qf-linear-map-affine-test","seed":202610072029,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-3\",\"5\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-3, 5]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-3, 5]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0030","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc6d016c338a083990339f8bd8c1842be8e540d55be7dc2401fd796d97ab8dad","family_id":"qf-linear-map-affine-test","seed":202610072030,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-4\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-4, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-4, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0031","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc17085a0aa5285a1785b2e72b542c5dcc7517753acdd10e0b3c24e489e8ec05","family_id":"qf-linear-map-affine-test","seed":202610072031,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-5\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-5, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-5, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0032","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ba1da3f5b98cdfdb87f7b17f8cd250ceae0700aa811b0aa4f425588b4f2ef6","family_id":"qf-linear-map-affine-test","seed":202610072032,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-6\",\"8\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-6, 8]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-6, 8]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0033","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02411a483696a63cf05e14b277f3e3d28af4f6b0a77a692e64c99fdcb82fe24a","family_id":"qf-linear-map-affine-test","seed":202610072033,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-7\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-7, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-7, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0034","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"290260e8baf5729c85245258d18d21bf3be4e59083993184441c60623269deb7","family_id":"qf-linear-map-affine-test","seed":202610072034,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-8\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-8, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-8, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0035","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1296d852645683876d50c96ceb435c809ad2c45272497d68d845f126095008ab","family_id":"qf-linear-map-affine-test","seed":202610072035,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-9\",\"11\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-9, 11]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-9, 11]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0036","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7757f54ad4d144fa7e39932f877c3e3e7480fa21302b5a3597b68635f08ea9b","family_id":"qf-linear-map-affine-test","seed":202610072036,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-10\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-10, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-10, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0037","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"991df045a6d92c34bce0b5ac84e85c5625cf8c1062c13ba99ef53dfb432a3524","family_id":"qf-linear-map-affine-test","seed":202610072037,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-11\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-11, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-11, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0038","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15d168501de74078d38eb4a81f85ab45d2b0049e8b3eed6fd2222e720ec06975","family_id":"qf-linear-map-affine-test","seed":202610072038,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-12\",\"14\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-12, 14]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-12, 14]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0039","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"896e331c316e6e9c508edae73e58705aee3142110cc48efa7d64b3993a27a1b6","family_id":"qf-linear-map-affine-test","seed":202610072039,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-13\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-13, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-13, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0040","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d37078fa09cc6aa703092b7935b990b9bd95c317a272d883693b70a0248f2017","family_id":"qf-linear-map-affine-test","seed":202610072040,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-14\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-14, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-14, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0041","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f24aa1b3e443b904d4272d2ddfe8be76439a2c6b49d8f8cb22cc6c778055f7a","family_id":"qf-linear-map-affine-test","seed":202610072041,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-15\",\"17\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-15, 17]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-15, 17]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0042","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973d0259b9908535a82509d9f555ac0d10efeb38e960bc5167e880276d5e217a","family_id":"qf-linear-map-affine-test","seed":202610072042,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-16\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-16, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-16, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0043","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0027be22c98442acfb1124eff813a24c35daa808fb95869fa2eb63c4d50d2b9f","family_id":"qf-linear-map-affine-test","seed":202610072043,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-17\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-17, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-17, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0044","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22138d5ea168df918b6a5e0403398d01ef54dbf5a8e1a2d51f50cbccd7e2a058","family_id":"qf-linear-map-affine-test","seed":202610072044,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-18\",\"20\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-18, 20]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-18, 20]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0045","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a5ae6ed65e4ab29c7777b7e45b25edbd8ed383183a2bbb8147c1a9742be7901","family_id":"qf-linear-map-affine-test","seed":202610072045,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-19\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-19, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-19, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0046","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"625d81dd21cac9778cf11a6242c8ef1b8a0517bcf3fe6114db972e9ac661c919","family_id":"qf-linear-map-affine-test","seed":202610072046,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-20\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-20, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-20, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0047","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee1ff3091ded39c65c62b47a369b105e7a4f22b1f413866524e1fb8189849ef","family_id":"qf-linear-map-affine-test","seed":202610072047,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-21\",\"23\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-21, 23]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-21, 23]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0048","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"798230e4411f92d3d429fa70d0c16f8c2a93d20e7516ece4e2aeaf649b86c406","family_id":"qf-linear-map-affine-test","seed":202610072048,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-22\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-22, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-22, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0049","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1c9e9b06278602dda74351596c7eaaa110d6f245046f0350e83bc72c5259472","family_id":"qf-linear-map-affine-test","seed":202610072049,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-23\",\"0\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-23, 0]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-23, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0050","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e60f306cb23c76a810bb7e783c381713f4f5cf1878159b2a69fac2c3cc7f3ec9","family_id":"qf-linear-map-affine-test","seed":202610072050,"semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-24\",\"26\"],\"operation\":\"affine_linearity\"}"},"fields":{"title":"Linearity of an Explicit Affine Map","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-24, 26]. Is F linear on the whole R²?","answer":{"kind":"short_text","value":"No","explanation":"Here F(0)=b=[-24, 26]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs."},"skills":["Linearity of an Explicit Affine Map"],"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-20261006-advance-20261007-r02-linear-transformation-0051","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec7c21fca6f51d35deace55383c5a5e696d36f1e46fd3fc944ce5ff2998077b","family_id":"qf-linear-map-additivity-values","seed":202610072051,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"0\"]],[[\"0\",\"1\"],[\"0\",\"2\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 1] and v=[0, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0052","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bdc0f41fac7a6af2faa5b76451bf76b3b45f063d95ff7afd3109dc2a40e3a35","family_id":"qf-linear-map-additivity-values","seed":202610072052,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],[[\"-1\",\"1\"],[\"1\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[1, 1] and v=[-1, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 8] = [4, 8]","explanation":"Evaluate the two expressions exactly, obtaining [4, 8] = [4, 8]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0053","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d51b6f245f1fd4f1b0f3e7a3b4f3019712dabcd09377ba3c6f44d49571c106de","family_id":"qf-linear-map-additivity-values","seed":202610072053,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[[\"-2\",\"0\"],[\"2\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[2, 1] and v=[-2, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[1, 0] = [1, 0]","explanation":"Evaluate the two expressions exactly, obtaining [1, 0] = [1, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0054","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"378bb5f0e28b4a9bd41d53f77a4a31124cb074026be041427a7dd9fdba2191f0","family_id":"qf-linear-map-additivity-values","seed":202610072054,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[[\"-3\",\"-1\"],[\"3\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 1] and v=[-3, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0055","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"694a9b16eaf44dc1eeb25009e9adc720c3742edbfce1517a628d9edd5237337b","family_id":"qf-linear-map-additivity-values","seed":202610072055,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[[\"-4\",\"2\"],[\"4\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 1] and v=[-4, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 15] = [3, 15]","explanation":"Evaluate the two expressions exactly, obtaining [3, 15] = [3, 15]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0056","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b52f7b3ab6ab91f38b6b63f82a99cb9cf0ddcea0e08e63c1b7240588b14bc34","family_id":"qf-linear-map-additivity-values","seed":202610072056,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[[\"-5\",\"1\"],[\"5\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 1] and v=[-5, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 12] = [4, 12]","explanation":"Evaluate the two expressions exactly, obtaining [4, 12] = [4, 12]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0057","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d55f551ebe09b761c3f625cb8bb8665359692133b497c13343f2aadc3d8cec8","family_id":"qf-linear-map-additivity-values","seed":202610072057,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[[\"-6\",\"0\"],[\"6\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 1] and v=[-6, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 7] = [3, 7]","explanation":"Evaluate the two expressions exactly, obtaining [3, 7] = [3, 7]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0058","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f6e64f0bd8d270e0b3c03871b7b47ae67a9a68f2d72187809b169a3e357735c","family_id":"qf-linear-map-additivity-values","seed":202610072058,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[[\"-7\",\"-1\"],[\"7\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 1] and v=[-7, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0059","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0fdd2528e57740674d6e62e51179d6ca31ef212a0ab1089c896575c90474bc9","family_id":"qf-linear-map-additivity-values","seed":202610072059,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[[\"-8\",\"2\"],[\"8\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[8, 1] and v=[-8, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 27] = [3, 27]","explanation":"Evaluate the two expressions exactly, obtaining [3, 27] = [3, 27]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0060","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02c7ad849eaaee957bb7aa2cd21429f9cb70b768cad039cdb775920aa82887d4","family_id":"qf-linear-map-additivity-values","seed":202610072060,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[[\"-9\",\"1\"],[\"9\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 1] and v=[-9, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 20] = [4, 20]","explanation":"Evaluate the two expressions exactly, obtaining [4, 20] = [4, 20]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0061","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b020f928ebba162ebe48803d5ad8315fdf3563359bbdac77289968a2ceecf1b2","family_id":"qf-linear-map-additivity-values","seed":202610072061,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[[\"-10\",\"0\"],[\"10\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 1] and v=[-10, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 11] = [3, 11]","explanation":"Evaluate the two expressions exactly, obtaining [3, 11] = [3, 11]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0062","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fc7600b784503aacc6fe11ccaf27d8c27c26091d557a8b876ab01ccc1fcd96f","family_id":"qf-linear-map-additivity-values","seed":202610072062,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[[\"-11\",\"-1\"],[\"11\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 1] and v=[-11, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0063","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a0424bcd8de4101098db3e2912fd8cf4dffafcda62775949229432ef5ee42f4","family_id":"qf-linear-map-additivity-values","seed":202610072063,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[[\"-12\",\"2\"],[\"12\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[12, 1] and v=[-12, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 39] = [3, 39]","explanation":"Evaluate the two expressions exactly, obtaining [3, 39] = [3, 39]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0064","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15745c7dce51bfba6022a6f889ecfac35ae59d19f70d96f73e93d5092b99342a","family_id":"qf-linear-map-additivity-values","seed":202610072064,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[[\"-13\",\"1\"],[\"13\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[13, 1] and v=[-13, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 28] = [4, 28]","explanation":"Evaluate the two expressions exactly, obtaining [4, 28] = [4, 28]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0065","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6535ff9d5b7d71199853b8c860463ebf7d3303faf1ef7292dca8d6ec6b2347d2","family_id":"qf-linear-map-additivity-values","seed":202610072065,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[[\"-14\",\"0\"],[\"14\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[14, 1] and v=[-14, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 15] = [3, 15]","explanation":"Evaluate the two expressions exactly, obtaining [3, 15] = [3, 15]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0066","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25b94c688c37c3cd11886acd308e63948452779b830b2aadb5042c927018327a","family_id":"qf-linear-map-additivity-values","seed":202610072066,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],[[\"-15\",\"-1\"],[\"15\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[15, 1] and v=[-15, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0067","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a8a83b1d831e70d81085b1096a02dcbf87e5890dbf706a29dad45a8fe68cc7c","family_id":"qf-linear-map-additivity-values","seed":202610072067,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],[[\"-16\",\"2\"],[\"16\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[16, 1] and v=[-16, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 51] = [3, 51]","explanation":"Evaluate the two expressions exactly, obtaining [3, 51] = [3, 51]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0068","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04456cbd535d8aab1905b8b3336442c882805130720197e649b4aedc8659d524","family_id":"qf-linear-map-additivity-values","seed":202610072068,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[[\"-17\",\"1\"],[\"17\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[17, 1] and v=[-17, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 36] = [4, 36]","explanation":"Evaluate the two expressions exactly, obtaining [4, 36] = [4, 36]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0069","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b38ab9f1ce6e1a30a6a98cc4ffe6fcbb12aa56977fd8d330b58968508c8900b7","family_id":"qf-linear-map-additivity-values","seed":202610072069,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],[[\"-18\",\"0\"],[\"18\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[18, 1] and v=[-18, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 19] = [3, 19]","explanation":"Evaluate the two expressions exactly, obtaining [3, 19] = [3, 19]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0070","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65e00c7555173b1096d1cf3fc90aaa1a1c96bbcc299e0830dfbb17dcd8bd9a6a","family_id":"qf-linear-map-additivity-values","seed":202610072070,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],[[\"-19\",\"-1\"],[\"19\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[19, 1] and v=[-19, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0071","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32fc4bed1734b5b189e0ca5062aef80be709b5695c0f0801412362afc5b06c88","family_id":"qf-linear-map-additivity-values","seed":202610072071,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[[\"-20\",\"2\"],[\"20\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[20, 1] and v=[-20, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 63] = [3, 63]","explanation":"Evaluate the two expressions exactly, obtaining [3, 63] = [3, 63]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0072","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71431b269f47191c674ec5bdc9716fe315cb0d9f2da08cb1f86b76f0f44fcc02","family_id":"qf-linear-map-additivity-values","seed":202610072072,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],[[\"-21\",\"1\"],[\"21\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[21, 1] and v=[-21, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[4, 44] = [4, 44]","explanation":"Evaluate the two expressions exactly, obtaining [4, 44] = [4, 44]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0073","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6aba98486a89f84d5047c6e40260b8c5cdca65406d18d834f4450e9322ba16b","family_id":"qf-linear-map-additivity-values","seed":202610072073,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[[\"-22\",\"0\"],[\"22\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[22, 1] and v=[-22, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 23] = [3, 23]","explanation":"Evaluate the two expressions exactly, obtaining [3, 23] = [3, 23]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0074","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36cdfac74f12b8df9ae0804e1ee61120b6afb0cdd3f5f7c9ede1310c902d7ab1","family_id":"qf-linear-map-additivity-values","seed":202610072074,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[[\"-23\",\"-1\"],[\"23\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[23, 1] and v=[-23, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0075","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"432df475408d6ddaf995834bc56ff66c873b4ac89c224f420320bd7442490883","family_id":"qf-linear-map-additivity-values","seed":202610072075,"semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[[\"-24\",\"2\"],[\"24\",\"1\"]]],\"operation\":\"additivity_values\"}"},"fields":{"title":"Both Values in a Matrix Additivity Identity","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[24, 1] and v=[-24, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[3, 75] = [3, 75]","explanation":"Evaluate the two expressions exactly, obtaining [3, 75] = [3, 75]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Additivity Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0076","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"373d0d1ad6433d95f352247508e684be4e8ad30fbe894c8473aba8ca24080043","family_id":"qf-linear-map-homogeneity-values","seed":202610072076,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[\"zero_scalar\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-12, 1] and c=0, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate both sides in the declared model to obtain [0, 0] = [0, 0], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0077","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"038b689b7b958b003f229cb33fa874fe27fefa65b35dc93140571c28c1895e55","family_id":"qf-linear-map-homogeneity-values","seed":202610072077,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[\"zero_input\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 0] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation":"Evaluate both sides in the declared model to obtain [0, 0] = [0, 0], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0078","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"430ebefe11506266a13f97811e068ef277ef1401c16370d79e8f185d190d3fe2","family_id":"qf-linear-map-homogeneity-values","seed":202610072078,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[\"-10\",\"3\"],\"3\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-10, 3] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[9, -30] = [9, -30]","explanation":"Evaluate both sides in the declared model to obtain [9, -30] = [9, -30], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0079","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af146f20ad89348c4861e42476a82916e938ce8e2e763c58249c8bd0ae8a897e","family_id":"qf-linear-map-homogeneity-values","seed":202610072079,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[\"-9\",\"4\"],\"-4\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-9, 4] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-388, -136] = [-388, -136]","explanation":"Evaluate both sides in the declared model to obtain [-388, -136] = [-388, -136], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0080","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a19999e40985f8a084187f600af3fee96a5fcf1d99640dead537e8fc5547bf88","family_id":"qf-linear-map-homogeneity-values","seed":202610072080,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[\"-8\",\"5\"],\"5\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-8, 5] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[345, 205] = [345, 205]","explanation":"Evaluate both sides in the declared model to obtain [345, 205] = [345, 205], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0081","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b612cd3198860ec77e4a9d74bd8625407a86db2fe87de38face22a9647213f","family_id":"qf-linear-map-homogeneity-values","seed":202610072081,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[\"-7\",\"6\"],\"-1\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-7, 6] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-61, -50] = [-61, -50]","explanation":"Evaluate both sides in the declared model to obtain [-61, -50] = [-61, -50], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0082","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d3ddc81907045dd9132e8d001459f3132d3bc330c99ad46713189366f827bd3","family_id":"qf-linear-map-homogeneity-values","seed":202610072082,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[\"-6\",\"7\"],\"2\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-6, 7] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[114, 122] = [114, 122]","explanation":"Evaluate both sides in the declared model to obtain [114, 122] = [114, 122], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0083","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d88fc7cb16421d34b255c7acf3473035521aa941525b07f4de6182df2f923bbe","family_id":"qf-linear-map-homogeneity-values","seed":202610072083,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[\"-5\",\"8\"],\"-3\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-5, 8] and c=-3, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-171, -222] = [-171, -222]","explanation":"Evaluate both sides in the declared model to obtain [-171, -222] = [-171, -222], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0084","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59896d34e85d3749666661ca923da05a1c20ca836b9510c949a864a33c456d11","family_id":"qf-linear-map-homogeneity-values","seed":202610072084,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[\"-4\",\"9\"],\"4\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-4, 9] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[100, 356] = [100, 356]","explanation":"Evaluate both sides in the declared model to obtain [100, 356] = [100, 356], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0085","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47d6f10b886725a4cb709cf8c6b2d49fcce01171e59019bf83616957b5f27cd6","family_id":"qf-linear-map-homogeneity-values","seed":202610072085,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[\"-3\",\"10\"],\"-5\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-3, 10] and c=-5, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-145, -530] = [-145, -530]","explanation":"Evaluate both sides in the declared model to obtain [-145, -530] = [-145, -530], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0086","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff815199d46d5a1e1b706737534d79b1ebddf94384a3e42e5e9cb479b4e9dfcc","family_id":"qf-linear-map-homogeneity-values","seed":202610072086,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[\"-2\",\"11\"],\"1\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-2, 11] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[37, 125] = [37, 125]","explanation":"Evaluate both sides in the declared model to obtain [37, 125] = [37, 125], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0087","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28e547352c9ff5607e9f3239c54e23d768f62c827faae5854e37a02b6755bf76","family_id":"qf-linear-map-homogeneity-values","seed":202610072087,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[\"-1\",\"12\"],\"-2\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-1, 12] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-98, -292] = [-98, -292]","explanation":"Evaluate both sides in the declared model to obtain [-98, -292] = [-98, -292], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0088","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea10d5441fe9d7d37082e368fb3eef0a57a9c037f5fefca716269f535fa9f5c9","family_id":"qf-linear-map-homogeneity-values","seed":202610072088,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[\"0\",\"13\"],\"3\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 13] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[39, 507] = [39, 507]","explanation":"Evaluate both sides in the declared model to obtain [39, 507] = [39, 507], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0089","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a437b337839cf0a88e61698f3933dc0627699a50d134d915d693102831a1ed4d","family_id":"qf-linear-map-homogeneity-values","seed":202610072089,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[\"1\",\"14\"],\"-4\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[1, 14] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-116, -776] = [-116, -776]","explanation":"Evaluate both sides in the declared model to obtain [-116, -776] = [-116, -776], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0090","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f03d8b2b1b981cec53ddaa97cf3d90c79e3e072467f68807f61f083d30af757e","family_id":"qf-linear-map-homogeneity-values","seed":202610072090,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[\"2\",\"15\"],\"5\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[2, 15] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[245, 1105] = [245, 1105]","explanation":"Evaluate both sides in the declared model to obtain [245, 1105] = [245, 1105], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0091","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba1c238245689d94c4fcfb51df2a177c6962c034adcd4a35aa7f1de0750e30fb","family_id":"qf-linear-map-homogeneity-values","seed":202610072091,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],[\"3\",\"16\"],\"-1\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 16] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-73, -250] = [-73, -250]","explanation":"Evaluate both sides in the declared model to obtain [-73, -250] = [-73, -250], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0092","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"609d28251404251f28a52bf986d0eaa24fa6ecb8b567ae027b83764d8fa04e61","family_id":"qf-linear-map-homogeneity-values","seed":202610072092,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],[\"4\",\"17\"],\"2\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 17] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[66, 562] = [66, 562]","explanation":"Evaluate both sides in the declared model to obtain [66, 562] = [66, 562], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0093","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee7b3d22dec4fb55fc585f5974f28adba14020fb196b747ae44b1cf9b4130822","family_id":"qf-linear-map-homogeneity-values","seed":202610072093,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[\"5\",\"18\"],\"-3\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 18] and c=-3, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-183, -942] = [-183, -942]","explanation":"Evaluate both sides in the declared model to obtain [-183, -942] = [-183, -942], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0094","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"124541b72714a5bdd3cb8371970321a191ac6da8e9a3d036a68a16224f6e9b61","family_id":"qf-linear-map-homogeneity-values","seed":202610072094,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],[\"6\",\"19\"],\"4\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 19] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[372, 1396] = [372, 1396]","explanation":"Evaluate both sides in the declared model to obtain [372, 1396] = [372, 1396], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0095","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3fa1642e770ffda177ced4277ddbf193bc71d2aeb45cde8ddd348b7690c1648","family_id":"qf-linear-map-homogeneity-values","seed":202610072095,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],[\"7\",\"20\"],\"-5\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 20] and c=-5, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-645, -1930] = [-645, -1930]","explanation":"Evaluate both sides in the declared model to obtain [-645, -1930] = [-645, -1930], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0096","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e0ebaa2ee0230bc299c23157bde357f2eb48990e9b6d968903a7aa94a8338a","family_id":"qf-linear-map-homogeneity-values","seed":202610072096,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[\"8\",\"21\"],\"1\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[8, 21] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[85, 425] = [85, 425]","explanation":"Evaluate both sides in the declared model to obtain [85, 425] = [85, 425], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0097","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f2220b91c68f160ed3b8f0387b6f9ed231806ab442b48080d81bc11ec71819","family_id":"qf-linear-map-homogeneity-values","seed":202610072097,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],[\"9\",\"22\"],\"-2\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-250, -932] = [-250, -932]","explanation":"Evaluate both sides in the declared model to obtain [-250, -932] = [-250, -932], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0098","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95fd96c9dcce62beddf3f1a924c368c504299ca7d73f06a11151e700e3d14f76","family_id":"qf-linear-map-homogeneity-values","seed":202610072098,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[\"10\",\"23\"],\"3\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 23] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[507, 1527] = [507, 1527]","explanation":"Evaluate both sides in the declared model to obtain [507, 1527] = [507, 1527], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0099","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3210efd1a7c0cef8cfcdb880be00a203ce239828be589523b8253bab481144ca","family_id":"qf-linear-map-homogeneity-values","seed":202610072099,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[\"11\",\"24\"],\"-4\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 24] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[-868, -2216] = [-868, -2216]","explanation":"Evaluate both sides in the declared model to obtain [-868, -2216] = [-868, -2216], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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-20261006-advance-20261007-r02-linear-transformation-0100","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2467cfa79b16e0c1ff49c09a84d5353039a49dc1d9a74873d6be30916823ba9a","family_id":"qf-linear-map-homogeneity-values","seed":202610072100,"semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[\"12\",\"25\"],\"5\"],\"operation\":\"homogeneity_values\"}"},"fields":{"title":"Both Values in a Matrix Scalar Identity","stem":"Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[12, 25] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector.","answer":{"kind":"expression","value":"[845, 3005] = [845, 3005]","explanation":"Evaluate both sides in the declared model to obtain [845, 3005] = [845, 3005], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function."},"skills":["Both Values in a Matrix Scalar Identity"],"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."}}