{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0001","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3cad01a1b3d027ecc1eca3073eb23be185f96442c2396a3fb6244f3aa5afb6c","family_id":"qf-mod35-ranknull-qualification","seed":202611041001,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,0],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0002","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f1d248f5d35506250dacf3ae4f28088dd33a872a3a9e74e540f11b5f435836b","family_id":"qf-mod35-ranknull-qualification","seed":202611041002,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,1],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0003","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65bc32765a6e20a5b4c18b86e00d03f5f2b44a359447146ded130b8d9379867a","family_id":"qf-mod35-ranknull-qualification","seed":202611041003,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,2],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0004","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06cad57ac43990b6c40dc6974dfa85d375efadd8e8402e043e877f46c047d2c2","family_id":"qf-mod35-ranknull-qualification","seed":202611041004,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,3],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0005","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a59096582914f7f68168debe04ed3794e85902ec215d857a0720b4485949c1ee","family_id":"qf-mod35-ranknull-qualification","seed":202611041005,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,4],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0006","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4421da85078439a39878893e158154b7ef95e4d0747a4a1b6da365d8b78afde","family_id":"qf-mod35-ranknull-qualification","seed":202611041006,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,5],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0007","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb440b636b40910e351f7eaf66135867395ae7a6a2558ae885318ff2c4bf947","family_id":"qf-mod35-ranknull-qualification","seed":202611041007,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,6],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0008","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c16d9c8ce1b093b816f8544090e206c9829049098e745de92d371caf2446d94b","family_id":"qf-mod35-ranknull-qualification","seed":202611041008,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,7],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0009","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca6fb0cf5267edf5d28fe80799d992473e5f8ec82e1d1661132d626cc87eb008","family_id":"qf-mod35-ranknull-qualification","seed":202611041009,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,8],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0010","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"114b8f2ece80caa85736c314d2e5ca19fee460dc86009da784577a21a411d790","family_id":"qf-mod35-ranknull-qualification","seed":202611041010,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,9],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0011","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e346f79533baf98fa02837bcecdb8e6cb65d443c7150b3627d59632519f5f569","family_id":"qf-mod35-ranknull-qualification","seed":202611041011,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,10],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0012","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f809d999ab771293cb82b9359f5d638996051c5cac933583cc4ba793f126b8e","family_id":"qf-mod35-ranknull-qualification","seed":202611041012,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,11],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0013","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1af52da1ac882931ed63567da8d695558c7596934da2b61887d6e6fabe95164","family_id":"qf-mod35-ranknull-qualification","seed":202611041013,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,12],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0014","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f0a9afa0fab4f7d52fbd65cf3ba682bbfd964ba7904eaeee38eb1571f3884a5","family_id":"qf-mod35-ranknull-qualification","seed":202611041014,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,13],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0015","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed5175ee083104882e33e1b8a7f0bc3665688a4abd8d519cb690b634d8668400","family_id":"qf-mod35-ranknull-qualification","seed":202611041015,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,14],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0016","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9546c2bcba422c0de9bb0ba0b4a54cdb273ef519097dae0048f6d2e11a9ec0c2","family_id":"qf-mod35-ranknull-qualification","seed":202611041016,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,15],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0017","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdf8b9f214b7db37e6721799cba516059f18eb1993483212aea7547549c635b2","family_id":"qf-mod35-ranknull-qualification","seed":202611041017,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,16],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0018","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915dd596a3549aef51399412e11ef040bdcc816024c3e4ec555d77ddff0b48b1","family_id":"qf-mod35-ranknull-qualification","seed":202611041018,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,17],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0019","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ff8b9981c5f5bac03492c2b17655994d403b3e1a02432ca49be11452ea570ff","family_id":"qf-mod35-ranknull-qualification","seed":202611041019,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,18],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0020","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52d63fe86906d7a0c68abf1342074c881be4d4a93367070ab123f852244527b7","family_id":"qf-mod35-ranknull-qualification","seed":202611041020,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,19],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0021","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db46605c66653cf4c53a0034d49ba70bc51ab019a8f26be054fc47cc6843b727","family_id":"qf-mod35-ranknull-qualification","seed":202611041021,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,20],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0022","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34c7ffbeeec56dfe8fe8115d621636a32dfd63239e6b956ac5c3920cc3937f98","family_id":"qf-mod35-ranknull-qualification","seed":202611041022,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,21],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0023","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"497d35d4b712ea884d0daf8c88e80bca5f8e021aeb04a4ea3e4d8aefa33c9f18","family_id":"qf-mod35-ranknull-qualification","seed":202611041023,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,22],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0024","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ffe29c06667cab37235d77b7588a4525a6d16ceb251856d239d686caa39b091","family_id":"qf-mod35-ranknull-qualification","seed":202611041024,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,23],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0025","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"018a56d8dc996c3538ce0aca5cde75bfa98674c4c79f94cf0fd0b4124f4522cf","family_id":"qf-mod35-ranknull-qualification","seed":202611041025,"semantic_template":"qf-mod35-ranknull-qualification/{\"matrix\":[[1,0,24],[0,1,1]]}"},"fields":{"title":"Rectangular Rank-Nullity Starts with the Correct Map Direction","stem":"Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).","answer":{"kind":"expression","value":"(2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1)","explanation":"Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1)."},"skills":["Rectangular Rank-Nullity Starts with the Correct Map Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0026","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f60f5243c7faa92c2822438f33fde3ddcfc433c0e4cd51d4f4e25b2f8465f0","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041026,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,0],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0027","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf8d913a55f9a9703722ea5382a403d8b6c503836e96613d430ec71ce5a1897e","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041027,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,1],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0028","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4959c75ee581917af1fb90772f5ad673d4e18fa5f1ecb2fb12f6a6fbc5d0ff9f","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041028,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,2],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0029","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92f84730fe67adce06ac4bd97221d9b777f22e67d15e34ca861f87ed55f2cbf9","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041029,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,3],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0030","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da60ce1e47e87078117c4e745bd6bd1599839d02bee1b6aafab0c43b6e3f49e","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041030,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,4],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0031","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1728529a9a8034f3ca80f9d6bf75fe66e2de03469591b90edc65182864d77cd4","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041031,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,5],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0032","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"347e8a3e9c6738fd73abd6bc23a37c576f105a142d01cfe5c33c6db8cd7194ae","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041032,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,6],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0033","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a9fd619d31d23edea69c69c4cc9c12ddcaeb7e0a412e17272d5022dff9778af","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041033,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,7],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0034","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e84138a7aa987e18dde3e75f49b2c62a3e501db93b5e073135e168df88219a54","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041034,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,8],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0035","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"692131cdd4e26435c8d726994aae625b09fd6dbcd332a01e5259566f2c870c30","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041035,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,9],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0036","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2a1d66994290a11f17ff4a071fbe3413686dda1258aef11d12fedbe63fbd34f","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041036,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,10],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0037","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2078da65aa2c0211d64b94726f6355efff6034e4bfe3eb14b122a8d0ea4a7cb","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041037,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,11],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0038","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67be19b101f3db22e825eeaa83007990651e55548f5491809cd2df31585e1c84","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041038,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,12],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0039","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b736a8cf8128b4574f13883a56c0c8d6dc309a62638dc16c04103d612274122","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041039,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,13],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0040","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adb9d1180faaede855d9c93d99080456b9645dd21e3fa359465400b8bcff7992","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041040,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,14],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0041","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38c99abc2f238aed03b43fd3a97e1689c35226f789e554f2241d85fcda35dd7a","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041041,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,15],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0042","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"828b97c82810548821f82886faedfb03c0d77145566ab9fdfd6ec4bd40b45e4d","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041042,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,16],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0043","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e63bbdd1e071866bbc8cf20fa446b1b2fa0fd3694cb7c3bb0fb18ea33b303ac","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041043,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,17],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0044","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f90700b956cfe8bd4658ab3caee8d7190e9020d519664a17d24a6b7cf39f243c","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041044,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,18],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0045","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b2b85ec380a816b8c2e6a8c8769f1dcaae563a2af413cf23aca79262c30d689","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041045,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,19],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0046","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07a73230c8789dbc2e5789c3330ab9f862344687c245168efaef1e5cfa9ddb09","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041046,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,20],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0047","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f00e1e864f368009a8d7bf26b9e9b05faa85c123c3a8ac771cddb71b85efb9a8","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041047,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,21],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0048","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d64d76f8ce83b2e93bf5ce45625d70865a220c8d49525f3f0f7dc0b05a1c754c","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041048,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,22],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0049","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c8b662931d8380a37b85a433171b56cb1db3ccc6c24bba80dddb9291eebaccb","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041049,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,23],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0050","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2aa3666f92f3e1e50db3d05096b719790aee63a25daeee14d760c371ed5601c","family_id":"qf-mod35-ranknull-dimension-identity","seed":202611041050,"semantic_template":"qf-mod35-ranknull-dimension-identity/{\"matrix\":[[1,0,24],[0,1,1]]}"},"fields":{"title":"The Source Dimension Equals Nullity Plus Rank","stem":"Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).","answer":{"kind":"expression","value":"(3, 1, 2, 3, Yes)","explanation":"Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes)."},"skills":["The Source Dimension Equals Nullity Plus Rank"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0051","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"facae7f5222dc2a62645f9aeffde8ac9f88c37b241af501f8188c0e90e376687","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041051,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,0],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0052","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a64e42da4539137dfac20d40dcae88506a0406c7b969b91fbe76653ad453f51","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041052,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,1],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0053","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b37b9a0d42e23782c140226cb7b8cc6bfe123f288afa2b66f4e73310f3e78756","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041053,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,2],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0054","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea0f88672d7ee6456fa4e34235d80d4539be60e505fed6ffc313f9288cd6d099","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041054,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,3],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0055","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba4619f3deb7a6b69361c11cc9e9383b9ae3142bb7c65b5bd825f9140af8732d","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041055,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,4],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0056","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40e28d650a84ff97382541fd565ed07b2dbe6077de5d0a78cf723565ef9850cc","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041056,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,5],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0057","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7d0caa13a9d6970b32cb7c0e041612ed63e0d376c28126c7575f349caf3c442","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041057,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,6],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0058","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33b64f6f43eac5fa34bbfffb6ee87f5feb77157774b2706a444a8c6a5da5acd4","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041058,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,7],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0059","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e899b068a4b63161b10930e0596aedd2988de596c84a737de3c8be182c390a5","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041059,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,8],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0060","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e622d978c24b73954afb3e1c24076c4469d349555a4bcfd230895200fbdbc9b","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041060,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,9],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0061","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a031b8d586e964f2d4b25b61bc87b48384d544d025c7518ef95885b985d2189","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041061,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,10],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0062","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d79a10210239b98eda04e553d6edcc96d09aff5e940a8cc07e0f73689bbd3c73","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041062,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,11],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0063","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37013e9c0442ea99070b9915645044ea9c9529ff0d9fb8970c0944c421a2643","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041063,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,12],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0064","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d15f84e16d8b3c8ceb20010828fa48969bee6663969100de91522e5cfa25f57","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041064,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,13],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0065","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feaaefc149b8b310bd017be620c7230db756b0455318b9055dbcb856be2b5e4b","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041065,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,14],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0066","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"500bc79ba06383a8d7ad0fff8a764cecd1cf605c17894428f973641a4520d701","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041066,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,15],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0067","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c6303c4fcafb945b5b8a2549918f491372b49aa4ae4341db273005c62e88b74","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041067,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,16],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0068","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9edc7576dc1e2ff5e182193d48927befbd7c8defbd3a3a2c716aea2ed452d0fd","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041068,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,17],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0069","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc432cc71825ead35b753d44ca2ae684eeef8e9eb754b9ce4890435d5b7e741d","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041069,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,18],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0070","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7e5a3b584f9d4483d41afaae342ab009e0261f53a56a6b8190dbc0c0882cec","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041070,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,19],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0071","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93d93f6097079c474343c4d039130ab1256d633515b6783a29d30395d02266fa","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041071,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,20],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0072","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad11424fe89d876bcd21db4011cff99b2ee3c61e93f89e2bbbaf50ad8895472b","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041072,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,21],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0073","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6256d3f49602343cff9a4b3160bdc25c2635a63abc7d00a28ea46bab9e708281","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041073,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,22],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0074","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22e83255c3ebfb65861dcac2ae5a068f219ee06322109cbdeb9da82c634515e5","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041074,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,23],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0075","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4e7b05fb1a2dec34035e34f2132940f5f89ba8b8d03520c37af5b067ed75f50","family_id":"qf-mod35-ranknull-kernel-image-certificate","seed":202611041075,"semantic_template":"qf-mod35-ranknull-kernel-image-certificate/{\"matrix\":[[1,0,24],[0,1,1]]}"},"fields":{"title":"Use the Full Kernel and Original Pivot Columns","stem":"Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).","answer":{"kind":"expression","value":"([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)","explanation":"Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)."},"skills":["Use the Full Kernel and Original Pivot Columns"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0076","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a47eeac17fcaa899b98b2a02dc4612755961731ae8e70405089ae32a72721ce","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041076,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,0],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0077","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811cf3ecedc207e1e5bec9951ab4b23df9a4d5b01fa2a5931db37c037537177c","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041077,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,1],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0078","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"352329891de61b6f14fb8a8490e3ffad0ef880e482003cbd5858b50a4e397a48","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041078,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0079","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"183aeb98d3fc271c18ed1e2db18ad108cf4d639ca977f084aae7f2d6f45e1337","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041079,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,3],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0080","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2558249a20ecb943f0708efd979a694315a2d78c9d3d261177edb04934d106dc","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041080,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,4],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0081","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8b353b385243a20fd29ffac80ae99981967bc07061116664e425f39dd44d6ec","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041081,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,5],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0082","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701f819ac393f0b300843f353c7cf1cb8b3ffadad14cc4a349a82aa97280fec8","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041082,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,6],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0083","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1181f0e18ea80e3b61800b7598fa848dcc96f247b708f3af6b2f89b4f8edfb","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041083,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,7],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0084","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"955fb1ad7601215caa98baf2e6b3190b62ee769228020e4b7d89dce76d3a2cfe","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041084,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,8],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0085","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05d8b63a77b633245843bcd6022fe39ef158562eb423d48f92a2a31f963b943a","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041085,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,9],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0086","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"316f2892640ccaad2cf0a8db654826a1ce8a4f3eefaa04e64d3639174e7c64e8","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041086,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,10],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0087","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5de0fb6c0632211082d240b29c8d141ae5685a4c9811f29ab7f3f516e9a878a9","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041087,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,11],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0088","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96fd65626719739e53a5facf76b1c7c916422423f3046b07bdb67eed5a0237a1","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041088,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,12],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0089","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7893cbaa3965c042c0ca4d86e36c1325e3cd1e224a0d4e2ca7b1ca8384a55f8e","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041089,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,13],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0090","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e8df7f2ea66602924adfa30f30e592c10d412da92b6be1e844a276a689ce1ab","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041090,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,14],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0091","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a611ccba9ed6692909c6b4cab879b325184e30bf2ff27d24871900d182f8c8","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041091,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,15],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0092","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d37658392af68b84448975a2b980dddc438d65b68b2376f8f1e2b5ff0c3b714e","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041092,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,16],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0093","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f1500df4b30536b04148dfd86220d78fccb3c313f49b2f3cb79177586ae6be","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041093,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,17],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0094","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f71ba0575283db90e86e16f5d40e0ec95700218996deccf0674926307c665aaf","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041094,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,18],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0095","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57182a98c4586ed9132b20094242cb7953e0cdb8251705e49f778aa8ed611bab","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041095,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,19],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0096","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68db4ba198a84c76ca14b6c98b8733f9791c41f7f3bba5b5387c43541e359f5f","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041096,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,20],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0097","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"317ab0441cb69f1ae167cf90dc35f78acc567c04ea2abee4b618a08df97568c8","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041097,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,21],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0098","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce7f68a46181c20760a0945f254b5c9d73799dadf4529ce815e9e9166005dfee","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041098,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,22],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0099","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e4a257b7fef8a16cc205ebe2c9592d13ea9ad3599a60807cdad9d30ae6fe30","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041099,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,23],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r01-rank-nullity-theorem-0100","batch_id":"batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d145a91d2074eb9a2f9e82091ec6793a27c4978dfb2e0c316b04552cb0d9973c","family_id":"qf-mod35-ranknull-sampled-boundary","seed":202611041100,"semantic_template":"qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,24],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"},"fields":{"title":"Sampled Vectors Cannot Replace the Full Kernel/Image Certificates","stem":"Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).","answer":{"kind":"expression","value":"([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)","explanation":"The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."},"skills":["Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}