{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-dual-vector-space-01", "mko_id": "mko-dual-vector-space", "title": { "zh-Hant": "雙基底與covector換基", "en": "Dual basis and covector change" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-dual-vector-space-01", "source": { "mko_id": "mko-dual-vector-space", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "english_sha256": "1e23ee2fa2f740284ff7f2e81139957b6196a81326cb161bae3a56d54fe07153" }, "mode": "worked_example", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "對b₁=(1,1)、b₂=(1,−1)，φ₁=(x+y)/2、φ₂=(x−y)/2為雙基底。", "en": "For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "V=R²，B=((1,1),(1,−1))，P=[[1,1],[1,−1]]為基底column矩陣。", "en": "V=R2, B=((1,1),(1,-1)), P=[[1,1],[1,-1]] is the basis-column matrix." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "\\alpha b_1+\\beta b_2=(\\alpha+\\beta,\\alpha-\\beta)", "after_tex": "(x,y)=\\frac{x+y}{2}b_1+\\frac{x-y}{2}b_2\\text{ uniquely}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "每個座標有唯一這種表示，先認證B為基底。", "en": "Every vector has this unique representation, qualifying B as a basis first." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "除2合法，未用尚待證的雙基底來證B。", "en": "Division by 2 is valid; the dual basis being constructed is not used to qualify B." } }, "leaf_basis": "calc" }, { "id": "S02", "uses": [ "G1", "S01" ], "before_tex": "\\phi_1(x,y)=\\frac{x+y}{2},\\quad\\phi_2(x,y)=\\frac{x-y}{2}", "after_tex": "\\phi_i(au+bv)=a\\phi_i(u)+b\\phi_i(v)\\quad(i=1,2)", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "固定row covector乘座標，分配律對任意實a、b證線性。", "en": "Each fixed row covector is linear by distributivity for all real a,b." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01" ], "explanation": { "zh-Hant": "φᵢ輸出實數；泛函不等同primal向量。", "en": "phi_i returns a real number; a functional is not a primal vector." } }, "leaf_basis": "calc" }, { "id": "S03", "uses": [ "G1", "S02" ], "before_tex": "b_1=(1,1)", "after_tex": "\\phi_1(b_1)=1,\\quad\\phi_2(b_1)=0", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "代入第一基底向量，核第一column的delta值。", "en": "Evaluating the first basis vector checks the first delta column." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02" ], "explanation": { "zh-Hant": "(1−1)/2為0，不能兩格皆1。", "en": "(1-1)/2 is zero; both entries cannot be one." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S02", "S03" ], "before_tex": "b_2=(1,-1)", "after_tex": "\\phi_1(b_2)=0,\\quad\\phi_2(b_2)=1", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "代入第二基底向量，四pairings矩陣完整為I₂。", "en": "Evaluating the second vector completes the four-pairing matrix I2." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S03" ], "explanation": { "zh-Hant": "φ₂(b₂)=(1−(−1))/2=1，需保留負號。", "en": "phi2(b2)=(1-(-1))/2=1, with the minus sign retained." } }, "leaf_basis": "calc" }, { "id": "S05", "uses": [ "G1", "S01", "S02", "S03", "S04" ], "before_tex": "\\ell\\in V^*,\\quad v=\\alpha b_1+\\beta b_2", "after_tex": "\\ell(v)=\\alpha\\ell(b_1)+\\beta\\ell(b_2)=\\ell(b_1)\\phi_1(v)+\\ell(b_2)\\phi_2(v)", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "任意線性泛函由基底值決定，因此兩φ生成全部V*。", "en": "An arbitrary linear functional is determined by its basis values, so the two phi functionals span V*." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02", "S03", "S04" ], "explanation": { "zh-Hant": "ℓ是任意的，不是只檢查特定幾個函數。", "en": "Ell is arbitrary, not a few sampled functions." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S03", "S04", "S05" ], "before_tex": "a\\phi_1+b\\phi_2=0\\text{ as functions}", "after_tex": "\\text{evaluate at }b_1,b_2:\\quad a=b=0", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "評值直接證獨立，與生成合成真正雙基底。", "en": "Evaluation directly proves independence, completing the dual basis with spanning." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "S04", "S05" ], "explanation": { "zh-Hant": "零函數意味所有輸入值皆零，所以可在兩個基底向量核。", "en": "The zero functional vanishes on all inputs, so in particular on both basis vectors." } }, "leaf_basis": "mko_definition" }, { "id": "S07", "uses": [ "G1", "S01" ], "before_tex": "[v]_{\\rm std}=P[v]_B", "after_tex": "[v]_B=P^{-1}[v]_{\\rm std},\\quad P^{-1}=\\frac12\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "primal座標乘逆矩陣；P²=2I精確核逆矩陣。", "en": "Primal coordinates use the inverse; P^2=2I exactly verifies it." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01" ], "explanation": { "zh-Hant": "P與P^{-1}不可混用，即使本例只差純量。", "en": "P and P inverse cannot be interchanged even though they differ by a scalar here." } }, "leaf_basis": "calc" }, { "id": "S08", "uses": [ "G1", "S01", "S07" ], "before_tex": "\\ell(x,y)=3x+y,\\quad[\\ell]_{\\rm std}=(3,1)", "after_tex": "[\\ell]_B=(3,1)P=(4,2),\\quad\\ell(v)=4\\alpha+2\\beta", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "covector row乘P而非P^{-1}，與primal換基方向相反。", "en": "The covector row multiplies P rather than P inverse, opposite to primal coordinate change." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S07" ], "explanation": { "zh-Hant": "代入x=α+β,y=α−β得4α+2β，核同一泛函。", "en": "Substitution x=alpha+beta,y=alpha-beta gives 4alpha+2beta, checking the same functional." } }, "leaf_basis": "calc" } ], "result": { "zh-Hant": "對b₁=(1,1)、b₂=(1,−1)，φ₁=(x+y)/2、φ₂=(x−y)/2為雙基底。", "en": "For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis." }, "negative_controls": [ { "claim": { "zh-Hant": "b₁與φ₁是同一典範物件。", "en": "b1 and phi1 are canonically the same object." }, "expected": "reject", "reason": { "zh-Hant": "一個是向量，一個是V→R函數。", "en": "One is a vector, the other a function V to R." } }, { "claim": { "zh-Hant": "φ₁(b₂)=1。", "en": "phi1(b2)=1." }, "expected": "reject", "reason": { "zh-Hant": "(1+(−1))/2=0。", "en": "(1+(-1))/2=0." } }, { "claim": { "zh-Hant": "covector換基也乘P^{-1}。", "en": "Covectors also multiply P inverse." }, "expected": "reject", "reason": { "zh-Hant": "本例ℓ的B值為(4,2)，不是(2,1)。", "en": "Here ell's B-values are (4,2), not (2,1)." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "calc", "kind": "prerequisite", "statement": { "zh-Hant": "假設通常實數域與座標向量運算、矩陣乘法、線性獨立及生成的定義；所用消去均為可逆初等列運算。", "en": "Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations." } }, { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "本輪限有限維V=R^n，以row covector給出線性泛函的symbolic全稱證書，並以phi_i(v_j)=delta_ij核dual basis。primal vector與dual functional不混同；沒有額外內積時不宣稱V與V*典範識別。", "en": "This slice uses finite-dimensional V=R^n. A row covector gives a symbolic linearity certificate, and a dual basis is checked by phi_i(v_j)=delta_ij. A primal vector and a dual functional are distinct; without extra structure no canonical identification V→V* is claimed." }, "source": { "mko_id": "mko-dual-vector-space", "sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } } ], "revision_log": [], "numeric_checks": [ { "id": "pair", "lhs": "(1-(-1))/2", "rhs": "1", "step": "S04" }, { "id": "covector", "lhs": "3+1", "rhs": "4", "step": "S08" } ] }, "presentation": { "md_scope": { "zh-Hant": "R²明示泛函、全稱線性及四個pairings；不作無額外結構的典範識別。", "en": "Explicit R2 functionals, universal linearity and four pairings; no canonical identification without extra structure." }, "md_symbols": [ [ "V^*", "實線性對偶空間", "real linear dual" ], [ "\\phi_i", "雙基底泛函", "dual-basis functional" ], [ "P", "primal基底columns", "primal basis columns" ] ], "md_error_controls": [ { "title": "四pairings / Four pairings", "zh": "逐對核全部δᵢⱼ，不只對角格。", "en": "Check all delta_ij pairings, not only diagonal entries." }, { "title": "型別 / Types", "zh": "V*是函數空間，row座標需指定基底。", "en": "V* is a function space, and row coordinates require a chosen basis." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "284271845add69df42d2cff4d06bc73a7247e856826b15e73926becaccedbd91", "original_source_sha256": "284271845add69df42d2cff4d06bc73a7247e856826b15e73926becaccedbd91", "owner_adopted": true, "bilingual_sync": "complete_source_bound", "formal_Lean": "NOT_RUN", "reader_review": "142_of_142_source_bound", "wording_review": "original_Lead_decisions_resolved", "glossary_reader_paraphrases": [], "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW", "issue_ids": [], "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a", "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite." } }