{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:68e0c96d-dfe2-418d-b27a-df73bf8da23b", "labels": { "zh-CN": "转置", "en": "transpose" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "0b9c4849a7b53ebae6f30642406b9a5807f83eb1de692967c7a19cdd0dc56e67", "occurrences": [ { "concept_id": "urn:uuid:68e0c96d-dfe2-418d-b27a-df73bf8da23b", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-02/03-matrix-groups", "anchor": "concept-2-transpose", "chapter": "2", "section": "2.3", "offset": 5301, "range": [ 11622, 11844 ], "source_revision": "3dd2476e51de9c40a8d6ddb8f88871c179362c4e1209a1a91c0b6b3f4e535a63", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "转置(transpose)", "context": { "range": [ 11590, 11876 ], "quote": "以下例子都采用矩阵乘法，因此结合律自动成立。我们会经常使用矩阵的转置(transpose) $\\mathsf{A}^{T}$，它由沿主对角线翻转矩阵得到：", "quote_sha256": "a450007d16474475bce43776c39336d3fa6ab768626d26aba9e14f4c03202fa1", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-02/03-matrix-groups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-953ba7a9b48cfb935bea42a4" }, { "concept_id": "urn:uuid:68e0c96d-dfe2-418d-b27a-df73bf8da23b", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-03/07-dual-spaces", "anchor": "concept-3-transpose", "chapter": "3", "section": "3.7", "offset": 20976, "range": [ 28315, 28537 ], "source_revision": "938dfcbbffde5cbdbd7e55327dc24b68109b332b4f17d94eeedfe4be90a42e74", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "转置(transpose)", "context": { "range": [ 28209, 29530 ], "quote": "> [!exercise] 习题 3.19 \n>\n> 若 $A : V \\to V$ 是线性算符，则定义其转置(transpose)为线性映射 $A ^ { \\prime }$ : $V ^ { * } \\to V ^ { * }$，使得\n>\n> $$\n> A ^ {\\prime} \\omega (u) = \\omega (A u), \\quad \\forall u \\in V, \\omega \\in V ^ {*}.\n> $$\n>\n> 证明该关系唯一地定义了线性算符 $A ^ { \\prime }$，并且\n>\n> $$\n> O ^ {\\prime} = O, \\quad (\\mathrm{id} _ {V}) ^ {\\prime} = \\mathrm{id} _ {V ^ {*}}, \\quad (a B + b A) ^ {\\prime} = a B ^ {\\prime} + b A ^ {\\prime}, \\quad \\forall a, b \\in \\mathbb {K}.\n> $$\n>\n> (a) 证明 $( B A ) ^ { \\prime } = A ^ { \\prime } B ^ { \\prime }$\n>\n> (b) 若 $A$ 是可逆算子，则证明 $( A ^ { \\prime } ) ^ { - 1 } = ( A ^ { - 1 } ) ^ { \\prime }$\n>\n> (c) 若 $V$ 是有限维的，则证明 $A ^ { \\prime \\prime } = A .$，其中我们对 $V ^ { * * }$ 与 $V$ 作自然等同。\n>\n> (d) 证明转置映射 $A$ 在对偶基下的分量矩阵是 $A , \\mathsf { A } ^ { \\prime } = \\mathsf { A } ^ { T }$ 的矩阵的转置\n>\n> (e) 利用问题 3.16 证明 $\\ker A ^ { \\prime } = ( \\mathrm { i m } A ) ^ { \\perp }$\n>\n> (f) 利用式 (3.10) 证明 $A ^ { \\prime }$ 的秩等于 $A$ 的秩。", "quote_sha256": "8f44a2a0d8138f7efeb394171c86d4aba48433b3da1f308069fb222b0f98e811", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-03/07-dual-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; 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