{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r01-linear-independence-basis-0001","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca48d42b4f6fae1b2642f07a3c467fd32c42808003d4c8faace0f46e749f60a1","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241001,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-12,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-12, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-12, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-12, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0002","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee1efd6d8b303f818ba1846bb0bef1cace982996d847ef5a0140ec4087d9ff7","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241002,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-11,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-11, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-11, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-11, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0003","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6ee61b7f40ac984aa31ec5a5ea85ec387dcadb18585de28bc4f388a1940a590","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241003,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-10,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-10, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-10, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-10, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0004","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2b9513e20e490936ff6b5200a937cad5c69e7945f6bc89675d99a432ecbeaf","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241004,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-9,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-9, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-9, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-9, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0005","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0601b9595af4a186b68669435c2b4ab9254e512742e12d279964a4a73e723c4","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241005,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-8,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-8, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-8, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-8, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0006","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"258a4db2b9f07cc98da8f3115df5e43800f064e079163819e356ad5922ef0d16","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241006,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-7,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-7, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-7, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-7, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0007","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21a90d5d7175e881f06184961020e5a70d09785c5115cbe355489968d4b12cf8","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241007,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-6,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-6, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-6, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-6, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0008","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4143fb8f1042d0099c03fcba3c4c822726bd45725fb3f4a7eabda8ca014c4c65","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241008,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-5,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-5, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-5, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-5, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0009","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"484bf951d0e4e4974798e67cebc35a42e5c6ee7850e2b96eb8ddbd3276de0b01","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241009,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-4,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-4, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-4, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-4, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0010","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9385bdb999434164f10522cba7976fbefee614fa7865aee9eefbb47add8a4474","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241010,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-3,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-3, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-3, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-3, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0011","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4145f7fd5fa2751a62ca450629e06f770ae704f31c36ef2d772d150852e1c1b","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241011,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-2,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-2, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-2, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-2, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0012","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e0997b64223c141e9fc1d9bc8ad07bdc61e4a432db851940e7018f9ba4ccad","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241012,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-1,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-1, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [-1, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-1, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0013","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8c9ed4625655631ce3f78d6c6a4e0d892578f1d5e1d14459573efd5e637038e","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241013,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0014","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54fae7558d6243b5d01e3d842ea777e2224873d5b7c8816b3fcc5dd76504f4f0","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241014,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[1,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [1, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [1, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [1, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0015","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3bc373b32373c89795acd294ada5342bf18db6d09ec91fd23d13fa3f20e03fb","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241015,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[2,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [2, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [2, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [2, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0016","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a25fbfdd5ab82a1fc5490c0d8742eb0323254f4b90f10463f2a5e0dd5b8d67","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241016,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[3,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [3, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [3, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0017","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8b7cf2ad0dbc6a2a7c03ea9c438891002eefd5c6197d472add275459397a264","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241017,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[4,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [4, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [4, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [4, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0018","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee7b993ee920df2fab40e757ed0048f57a65113197efea41ae044cf9d0b86437","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241018,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[5,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [5, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [5, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [5, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0019","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e3f793f685c54d22316fbc439ecc5dbcf2e749cec78e392b27ab23991ce3437","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241019,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[6,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [6, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [6, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [6, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0020","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a2ce4f58b788c83aae7d1880f01f84f4eea5d06eb1a18e65a70bfe8baf29608","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241020,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[7,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [7, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [7, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [7, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0021","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a527dc8af55559547e2f2900bf4406ff51b822418e526de94fa330ef1d59b96","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241021,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[8,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [8, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [8, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [8, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0022","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12678b69a52c6c7697b807aea8e0a8002e2508e2b7b586a435a628795d4fb1b8","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241022,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[9,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [9, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [9, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [9, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0023","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5c5274453dc6e1e7a9fce3f51ad9f348a98b795616cee3e5bf05c441cad4ae2","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241023,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[10,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [10, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [10, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [10, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0024","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85700ac41d9d4f64d45116e363edd3c25d4c2a09d726476edd0ac85d6eb5337f","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241024,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[11,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [11, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [11, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [11, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0025","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"273691ba12a94c155705dc3a8b80513307596c30f1f9526db19d769d7339d095","family_id":"qf-mod24-linbasis-independence-audit","seed":202610241025,"semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[12,1]]}"},"fields":{"title":"Exact Rank versus Independence of a Finite Vector Family","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [12, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).","answer":{"kind":"expression","value":"([[1, 0], [12, 1]], 2, Yes)","explanation":"Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [12, 1]], 2, Yes)."},"skills":["Exact Rank versus Independence of a Finite Vector Family"],"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-20261013-advance-20261024-r01-linear-independence-basis-0026","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a591afb038d98ad9a831ab4695b327938b9a0e67cc8420dcb16900bfb180504e","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241026,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-12],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -12], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -12], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -12], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0027","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf03bddc78706c7ca19584947bf4b40ef915814a4edcb00ec7cd6619aeee716d","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241027,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-11],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -11], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -11], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -11], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0028","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8980a80b1bdfcb7079997a6575b5feff87d3afd6d691b011d4e85ffa18009ed4","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241028,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-10],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -10], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -10], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -10], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0029","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64bb330a622d57c5134cb919ccd1fb62a989a495464033ac7080227d8685fe0","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241029,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-9],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -9], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -9], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -9], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0030","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66c98a1975750a468297d4d4c21394caae012e50abc43d1c3c40ed178909ab78","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241030,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-8],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -8], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -8], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -8], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0031","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a2d93a2ff7884ebc50d2ffb1f829f9e90ef9def77223813342540c54eda2f01","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241031,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-7],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -7], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -7], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -7], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0032","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b41ab75f9bd1e80d49938653d3bbcf14e13091f1af85b16f37acdf1b30753898","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241032,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-6],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -6], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -6], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -6], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0033","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f55759a78bfd43e785f6bd6202d997cabce1cc80c0fd6c1367092225a29c575","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241033,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-5],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -5], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -5], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0034","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf367c0558a9c63199fd77e05e62558df73c388ce82e9df38edd90244e382c3e","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241034,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-4],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -4], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -4], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -4], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0035","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7165650df3da41350640dbe5bf5a62f6dcab389f574ee4516e27f3954068b2","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241035,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-3],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -3], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -3], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -3], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0036","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de4a7c5fde6ce6fb0acb2f2f02b65abca3f8a916eb9b90d9e7410fe61b9d43a9","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241036,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-2],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -2], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0037","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85a081500183948105fdcba5b06ab1e26ac3dbaa7af9d267b427b458c034dd5e","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241037,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-1],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -1], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0038","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc189fa6b9a8a24ca29f7633d64a6308bf09e2f0c19fea13e9d971334e84b70b","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241038,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0039","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1324274c06116593212d4f97d1a008fd5ba7d2ffedad13cf79e45291ec184691","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241039,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,1],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 1], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0040","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25f93d2ebc00ae1adbe1a816fde012792a1c40d1c5d041e3f5427fbfc1e61aeb","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241040,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,2],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 2], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0041","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e742d5fc9f874e5356455f9b712e6878adcdb7fdd7e0e28a9ba52735cf296526","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241041,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,3],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 3], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 3], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 3], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0042","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"124df9cce011e6125f4bf16671f9eeb42d88c4539e19617b6d0ea85c26bba857","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241042,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,4],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 4], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 4], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 4], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0043","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1dced685aafbc110deae05c00bed237417d02476198a547cd3891ba0d67de0e","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241043,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,5],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 5], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 5], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0044","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"221087e3ba35507d0ea882cf291631abe38f3e36c162e82d9aeb0d93dbbffd59","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241044,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,6],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 6], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 6], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 6], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0045","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd1ba8fd5c08cd8577e81d4cd8561a798d9a8f1ddd033e938ac9330be8dfeec8","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241045,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,7],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 7], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 7], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 7], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0046","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"397e03d32806f98a2c36e76e67446d163eee25248bbfcf35a7ef7b9e3258d3f5","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241046,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,8],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 8], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 8], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 8], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0047","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a672326775a3a34a1210d677b47f373445aeb88f9d1aa3ec85d70977bc5af4a","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241047,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,9],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 9], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 9], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 9], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0048","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31e6251cb379474afffa685bf136a17f2249e81afa87690d35a9f3310a1b7a31","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241048,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,10],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 10], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 10], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 10], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0049","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be7793055f9b81c3b62904b3c4d0080dbb9d155736f066ac96d1d4b6fff621d8","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241049,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,11],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 11], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 11], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 11], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0050","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a9afec83efae4a143c24d54802bc5eda6254bdfa084f635e56a0f4e5fc4d57a","family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241050,"semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,12],[0,1]]}"},"fields":{"title":"Independence and Spanning Are Separate Plane-Basis Obligations","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 12], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.","answer":{"kind":"expression","value":"([[1, 12], [0, 1]], 2, Yes, Yes, Yes)","explanation":"An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 12], [0, 1]], 2, Yes, Yes, Yes)."},"skills":["Independence and Spanning Are Separate Plane-Basis Obligations"],"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-20261013-advance-20261024-r01-linear-independence-basis-0051","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"331801ce73622c451da72d3d5b4668cbaeb133757db9ea4d0a19c9e5abdb1e12","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241051,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-12,-3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-12, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0052","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fd4fcdfff5d237c7b97ade567eadf8d4e27b7c6a185ecfef730a2103dd286bf","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241052,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-11,-2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-11, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0053","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5b3a33934a5d4115d98a395ab05f5e2adbcf2f98c538f406e7c0c14318ecac7","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241053,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[-10,-1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[-10, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0054","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e7e3acc60003d34220e789ca0e73aac76bf28660bfb55b8baae4d8a22df87b5","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241054,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[-9,0]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[-9, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0055","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91595126174d7b274d27fe3f85344389fd94ccd2c6808545f4d68a1b065c5cf6","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241055,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-8,1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-8, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0056","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c71b369bba534e38ec7b3332d3f8fdfbf4195396843a808f3d44fdf91a773603","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241056,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-7,2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-7, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0057","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e9b36794d31c4c99e1dbd53fefa930138c28a942cef15502818c2905c934377","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241057,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-6,3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-6, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0058","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0dc56acd9fa2eb31e1a59fcb523bc8906ff1c877a8a4624376de7acf5619505","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241058,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[-5,-3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[-5, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0059","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9914129bde92459d6c7f6473cf47deb02935e1c92641c3e0e48e6510749ca93f","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241059,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[-4,-2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[-4, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0060","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcfcd8df5847c2846e1eaa6ebd1c75802da4e5a5286d194f5bcca98e43dbec4b","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241060,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-3, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0061","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0896fcc5c31fd4ee1de56ecaa4cb2bf244acdc763841f3be9fa97842a4d8bf43","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241061,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-2,0]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-2, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0062","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8b2b337dbd5a5a6b700e279843236d36024c787256c7e705d1cfccfedbd9aa","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241062,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-1,1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-1, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0063","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f99b957ce76a3c799b625783b5809b8c7d1fbe5ddc3bf86d7a54bec1dce77b7c","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241063,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[0,2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[0, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0064","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07802691ec92347a8f980b100a806ce16ab340706b9d8033a3bd8d67124f012a","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241064,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[1,3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[1, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0065","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93510428f511310e4a92114232c2fd047cbbb8b995a09ee44cac4fc0db162b85","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241065,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[2,-3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[2, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0066","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"697326a01dcf1f4d0a03294310f7b03848328172b818c5fefb0bc2641b597402","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241066,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[3,-2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[3, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0067","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"819a56e305590d23ceebc70e21e038009e4da5d40b36cf5b94ae92d57832fcaf","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241067,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[4,-1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[4, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0068","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb04b1a212501dff216f02900762b6b2149b4f62ab6a36f8594ea68eb4c48152","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241068,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[5,0]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[5, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0069","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"243f987ea087b930aba881fb727722495613dd4e137bb430320d59088dc08565","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241069,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[6,1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[6, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0070","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d48d2dbb1e89d65b6bd43d265963524d760764077cc947781aba6c551fb540b","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241070,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[7,2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[7, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0071","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abcd7ec2d3b7d85fafd93371833ffe4fbd24ae563db13af94db968bf11807380","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241071,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[8,3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[8, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0072","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb9251015f11ec14a9f384f3cc999c2dccc228e14556cc34017fba26a24926a6","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241072,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[9,-3]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[9, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0073","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5ebef0a8f7da1642a1144fd3a93e7957722bd303158ea4367cf3a7a1e6cfcb1","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241073,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[10,-2]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[10, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0074","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0c8fafa3407b04e167224ac1d6242c0b16cdfce7b0a79adf5f030099f5ebcf1","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241074,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[11,-1]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[11, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0075","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a53699a6243e5fd77ec02ad36564c2cc1422c6f2b139ffdf82e5ea91074140f","family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241075,"semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[12,0]}"},"fields":{"title":"Unique Coordinates Only after Qualifying an Ordered Plane Basis","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[12, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0])","explanation":"Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0])."},"skills":["Unique Coordinates Only after Qualifying an Ordered Plane Basis"],"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-20261013-advance-20261024-r01-linear-independence-basis-0076","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1da28db6a21d49d6d8ad639ae46cce332207eaedc309c9ad06d59005b95bae","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241076,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,12,1],\"vectors\":[[1,0],[0,1],[1,-12]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -12]], coefficients=[-1, 12, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0077","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4737f6ce4b6b172447fcfe8f6c87aa4767c08f0f777c4e545b7f82dafabc31c","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241077,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,11,1],\"vectors\":[[1,0],[0,1],[1,-11]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -11]], coefficients=[-1, 11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0078","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e48c05292b1780919109448747d890ae7f1b79140380557283a46a4f20b8eff","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241078,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,10,1],\"vectors\":[[1,0],[0,1],[1,-10]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -10]], coefficients=[-1, 10, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0079","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe4b3d2a78fd24c2c95432c02efe38c8bc05b1f2b57dabc67bd711a9bc9730f","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241079,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,9,1],\"vectors\":[[1,0],[0,1],[1,-9]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -9]], coefficients=[-1, 9, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0080","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"943882556557fd2d154e0fa8055b5a4e50debfbaef365b8434b2559ea774a11d","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241080,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,8,1],\"vectors\":[[1,0],[0,1],[1,-8]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -8]], coefficients=[-1, 8, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0081","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"354533464bb0d19277133552c7f9478782e3f83c19cc077d27e8aef852488665","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241081,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,7,1],\"vectors\":[[1,0],[0,1],[1,-7]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -7]], coefficients=[-1, 7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0082","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c48b8ac4094dd1ef5c6c19a2e391dc43446117e6cf319eadf06ca07a2f79c8","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241082,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,6,1],\"vectors\":[[1,0],[0,1],[1,-6]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -6]], coefficients=[-1, 6, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0083","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea366b7cd389f9e254bfcf7f298f1239345b79ddfef6a065209fc3d067e4815","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241083,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,5,1],\"vectors\":[[1,0],[0,1],[1,-5]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -5]], coefficients=[-1, 5, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0084","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26f22122c7078b30bac6e0062096f6ca8beab2c6a7010d36bfb5d61cc2e38648","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241084,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,4,1],\"vectors\":[[1,0],[0,1],[1,-4]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -4]], coefficients=[-1, 4, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0085","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cbcf37b86e010b6b3cd5833a746c52e998fd67b576ba286ca2070d734e8618a","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241085,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,3,1],\"vectors\":[[1,0],[0,1],[1,-3]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -3]], coefficients=[-1, 3, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0086","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb4a1958d899a45c214e0a0d62b9abb652a598af5121af4cd34fce10e8b9a9f","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241086,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,2,1],\"vectors\":[[1,0],[0,1],[1,-2]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -2]], coefficients=[-1, 2, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0087","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47d5dfbc9b64ab949fda4412e7815fb1df14f0e9e9fd6bfae749770c50120782","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241087,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,1,1],\"vectors\":[[1,0],[0,1],[1,-1]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -1]], coefficients=[-1, 1, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0088","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dbf7c7c86254bd1813e90fcd18131d1c53a3604989f45211fe882dcf1dcc561","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241088,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,0,1],\"vectors\":[[1,0],[0,1],[1,0]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 0]], coefficients=[-1, 0, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0089","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4d5e1599bcd6046aae6bf158f8a31364aa6df45d4afbbc18544e06bda4f9da","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241089,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-1,1],\"vectors\":[[1,0],[0,1],[1,1]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, -1, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0090","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8048797236d8dd4efec7fedd4e35468d81b16b8dc77578077957ee64709cd3","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241090,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-2,1],\"vectors\":[[1,0],[0,1],[1,2]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 2]], coefficients=[-1, -2, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0091","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c29114e4436afa178cfcea6876a2ba5d08a0beaf12ff47270a7a1e57ddcd66d","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241091,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-3,1],\"vectors\":[[1,0],[0,1],[1,3]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 3]], coefficients=[-1, -3, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0092","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63f3c219cf6bd86c4ddf2f6c6361418727aa38892cdcf40a1526048edb577265","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241092,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-4,1],\"vectors\":[[1,0],[0,1],[1,4]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 4]], coefficients=[-1, -4, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0093","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f91fb9ee84fc8d4ac09fa139999c359725b6678ae0b5fed3304825eaa79d008","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241093,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-5,1],\"vectors\":[[1,0],[0,1],[1,5]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 5]], coefficients=[-1, -5, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0094","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40a69928c98e702fa21b6a3e199f9511f3a54f19744ac229726b1ec671ab80d0","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241094,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-6,1],\"vectors\":[[1,0],[0,1],[1,6]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 6]], coefficients=[-1, -6, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0095","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad5fe5610a6b85ed1281e9fccb4846de7a786ad3ca2aa37b86c07f2de10e6106","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241095,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-7,1],\"vectors\":[[1,0],[0,1],[1,7]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 7]], coefficients=[-1, -7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0096","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42e68eb72afa96c647d5adec25efae0138e124bdd060c01bf6fc55b32af82756","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241096,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-8,1],\"vectors\":[[1,0],[0,1],[1,8]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 8]], coefficients=[-1, -8, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0097","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96552502e6ca844d197de4d1162990444dbbc17021d03c43b66ed65edc6ab5a3","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241097,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-9,1],\"vectors\":[[1,0],[0,1],[1,9]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 9]], coefficients=[-1, -9, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0098","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7782f8adc8f86b18b0e624fe88728017f6dd16ab2e1c388a3247b069f5d14b98","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241098,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-10,1],\"vectors\":[[1,0],[0,1],[1,10]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 10]], coefficients=[-1, -10, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0099","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"615519781cb5066fdf229d1550df02f2e9e74323a22c55bde5f1b76d38e6cd48","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241099,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-11,1],\"vectors\":[[1,0],[0,1],[1,11]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 11]], coefficients=[-1, -11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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-20261013-advance-20261024-r01-linear-independence-basis-0100","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe65d34ba05c95894b99ffc97ab86c62faa4c762fe3e6c805bd15139c6f35df8","family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241100,"semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-12,1],\"vectors\":[[1,0],[0,1],[1,12]]}"},"fields":{"title":"A Zero Combination Must Also Be Nontrivial to Witness Dependence","stem":"Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 12]], coefficients=[-1, -12, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], Yes, Yes, Yes)","explanation":"Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], Yes, Yes, Yes)."},"skills":["A Zero Combination Must Also Be Nontrivial to Witness Dependence"],"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."}}