{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r03-linear-transformation-0003",
  "pilot_number": "R0014-07553",
  "number": 7553,
  "family_id": "qf-density16-linear-basis-column-reconstruction",
  "title": {
    "zh-Hant": "第 7553 題：基底像與列向量重建",
    "en": "Question 7553: Basis-Image Columns and Two Matrix-Vector Routes"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r03-linear-transformation-0003",
    "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-transformation"
    ],
    "title_zh": "基底像與列向量重建",
    "stem_zh": "R²到R²列向量，明示2×2矩陣模型。M=[[-10, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。",
    "answer": {
      "kind": "expression",
      "value": "([-10, 2], [1, -1], [-11, 3], [-11, 3])",
      "explanation_zh": "有限結構核對=([-10, 2], [1, -1], [-11, 3], [-11, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "基底像與列向量重建"
      ]
    },
    "variant": {
      "family_id": "qf-density16-linear-basis-column-reconstruction",
      "seed": 202610163003,
      "parameters": {
        "M": [
          [
            -10,
            1
          ],
          [
            2,
            -1
          ]
        ],
        "v": [
          1,
          -1
        ],
        "domain": "R2_column_integer_slice",
        "operation": "linear-basis-column-reconstruction",
        "semantic_template": "qf-density16-linear-basis-column-reconstruction/{\"M\":[[-10,1],[2,-1]],\"v\":[1,-1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r03-linear-transformation-0003",
    "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "456ead4d50097ef61f3a6c4a399cb46653095aecc2e8a66dacb003f81ddc84ce",
      "family_id": "qf-density16-linear-basis-column-reconstruction",
      "seed": 202610163003,
      "semantic_template": "qf-density16-linear-basis-column-reconstruction/{\"M\":[[-10,1],[2,-1]],\"v\":[1,-1]}"
    },
    "fields": {
      "title": "Basis-Image Columns and Two Matrix-Vector Routes",
      "stem": "Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-10, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).",
      "answer": {
        "kind": "expression",
        "value": "([-10, 2], [1, -1], [-11, 3], [-11, 3])",
        "explanation": "Read the two basis-image columns and combine their given coefficients; independently evaluate Mv by row-by-column multiplication. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [1, -1], [-11, 3], [-11, 3])."
      },
      "skills": [
        "Basis-Image Columns and Two Matrix-Vector Routes"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "R²到R²列向量，明示2×2矩陣模型。M=[[-10, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。",
        "en": "Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[-10, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "M=[[-10, 1], [2, -1]];v=[1, -1]",
      "after": "declared R2 matrix model;finite values are not a test of an arbitrary function",
      "relation": "implies",
      "rule_id": "declared_linear_model_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明示矩陣模型Mv在全R²上由分配律給出線性映射；本題列出的有限代值不是對任意函數的線性證明。零映射也線性，不要求非零或可逆。",
        "en": "The declared matrix model Mv gives a linear map on all R² by distributivity.The listed finite evaluations do not prove an arbitrary function linear.The zero map is linear;nonzero values or invertibility are not required."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Mv[0] term0 := mul(-10, 1)",
      "after": "Mv[0] term0 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] term0，不以小數近似替代。",
          "en": "Recompute Mv[0] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-10",
          "1"
        ],
        "output": "-10",
        "label": "Mv[0] term0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Mv[0] term1 := mul(1, -1)",
      "after": "Mv[0] term1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] term1，不以小數近似替代。",
          "en": "Recompute Mv[0] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Mv[0] term1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Mv[0] row sum := sum(-10, -1)",
      "after": "Mv[0] row sum = -11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] row sum，不以小數近似替代。",
          "en": "Recompute Mv[0] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "-1"
        ],
        "output": "-11",
        "label": "Mv[0] row sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Mv[1] term0 := mul(2, 1)",
      "after": "Mv[1] term0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] term0，不以小數近似替代。",
          "en": "Recompute Mv[1] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Mv[1] term0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Mv[1] term1 := mul(-1, -1)",
      "after": "Mv[1] term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] term1，不以小數近似替代。",
          "en": "Recompute Mv[1] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Mv[1] term1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Mv[1] row sum := sum(2, 1)",
      "after": "Mv[1] row sum = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] row sum，不以小數近似替代。",
          "en": "Recompute Mv[1] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "Mv[1] row sum"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "M=[[-10, 1], [2, -1]];e1=[1, 0];e2=[0, 1]",
      "after": "M e1=[-10, 2];M e2=[1, -1]",
      "relation": "implies",
      "rule_id": "basis_images_are_columns",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一個標準基底向量選第一個直欄，第二個選第二個直欄；不要把橫列誤作基底像。",
        "en": "The first standard basis vector selects the first column,and the second selects the second column.Do not substitute rows for basis images."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "v0 column1[0] := mul(1, -10)",
      "after": "v0 column1[0] = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 v0 column1[0]，不以小數近似替代。",
          "en": "Recompute v0 column1[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-10"
        ],
        "output": "-10",
        "label": "v0 column1[0]"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "v0 column1[1] := mul(1, 2)",
      "after": "v0 column1[1] = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 v0 column1[1]，不以小數近似替代。",
          "en": "Recompute v0 column1[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "v0 column1[1]"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "v1 column2[0] := mul(-1, 1)",
      "after": "v1 column2[0] = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 v1 column2[0]，不以小數近似替代。",
          "en": "Recompute v1 column2[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "v1 column2[0]"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "v1 column2[1] := mul(-1, -1)",
      "after": "v1 column2[1] = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 v1 column2[1]，不以小數近似替代。",
          "en": "Recompute v1 column2[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "v1 column2[1]"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "basis reconstruction[0] := sum(-10, -1)",
      "after": "basis reconstruction[0] = -11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis reconstruction[0]，不以小數近似替代。",
          "en": "Recompute basis reconstruction[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "-1"
        ],
        "output": "-11",
        "label": "basis reconstruction[0]"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "basis reconstruction[1] := sum(2, 1)",
      "after": "basis reconstruction[1] = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis reconstruction[1]，不以小數近似替代。",
          "en": "Recompute basis reconstruction[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "basis reconstruction[1]"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "v=[1, -1];Mv=[-11, 3]",
      "after": "column reconstruction=[-11, 3]",
      "relation": "implies",
      "rule_id": "basis_reconstruction_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先由橫列點乘列向量算Mv，再用基底像的線性組合重建，兩條路不共用候選答案。",
        "en": "First compute Mv by row times column vector,then reconstruct it from the basis images.The two routes do not reuse a proposed answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        -10,
        2
      ],
      [
        1,
        -1
      ],
      [
        "-11",
        "3"
      ],
      [
        "-11",
        "3"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-10, 2], [1, -1], [-11, 3], [-11, 3])",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r03-linear-transformation-foundations/questions.jsonl",
      "batch_sha256": "915d95dff60999b46e55ccc20b8ae66dc4b4ee601782d8d932e1810cf76686f1",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r03-linear-transformation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r03-linear-transformation-foundations.en.jsonl"
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    "mko_bindings": {
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        "english_path": "public/data/i18n/editions/mko-linear-transformation.en.json"
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    "original_candidate_sha256": "3e95f4da510521ae26a813b3d8cbc7c46eb46f68c8178ea44afae12d7f21c46d",
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      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p15",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    "bilingual_semantics": true,
    "steps_reviewed": 15,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}