{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:39428ebb-bc71-4f72-9ebe-c940f88d9879", "labels": { "zh-CN": "线性算符", "en": "linear operator" }, "alternative_labels": [ { "lang": "en", "value": "linear operators" } ], "scope_note": "", "identity_status": "active", "identity_revision": "bb8089e84bd858b27043049370d0ce4be829486de026d788c6f7e7bd640c1aae", "occurrences": [ { "concept_id": "urn:uuid:39428ebb-bc71-4f72-9ebe-c940f88d9879", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-03/03-vector-space-homomorphisms", "anchor": "concept-3-linear-operator", "chapter": "3", "section": "3.3", "offset": 5214, "range": [ 9635, 9871 ], "source_revision": "72ae9374dfcfec9db3a41e363d652333b272a45d6cc7ee9f35002d24091fd8bd", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "线性算符(linear operator)", "context": { "range": [ 9607, 10510 ], "quote": "线性映射 $S : V \\to V$ 称为 $V$ 上的线性算符(linear operator)。它们构成向量空间 $L ( V , V )$。若 $S$ 是 $V$ 上的可逆线性算符，则称其为 $V$ 上的线性变换(linear transformation)。它可以视为 $V$ 到自身的向量空间同构，或 $V$ 的自同构。$V$ 的线性变换关于复合乘法构成一个群，称为 $V$ 上的一般线性群(general linear group)，记作 $G L ( V )$。群的性质容易证明：", "quote_sha256": "0975b14945ff1d09f2aef0354e185d1a62d66e0e0068b1ba1d9978105ef8e9da", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-03/03-vector-space-homomorphisms", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e04f0bd803a527b60d9b5de6" }, { "concept_id": "urn:uuid:39428ebb-bc71-4f72-9ebe-c940f88d9879", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-04/01-vector-spaces-and-linear-maps", "anchor": "concept-4-linear-map", "chapter": "4", "section": "4.1", "offset": 2185, "range": [ 8225, 8489 ], "source_revision": "7b67eea3f4266b108e1e788c2e21bac9a046d13f101f5330ec8134a1a88d496c", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "linear operator", "context": { "range": [ 8131, 8500 ], "quote": "The choice of a basis $\\{ \\mathbf { e } _ { j } \\}$ also allows us to express the action of a linear operator $L$ on $V$", "quote_sha256": "bae4297988f7366499731806953b9a0b4ec4a50b3c061b62de73572de1a3aacd", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/01-vector-spaces-and-linear-maps", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; 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not-concept-equivalence" } ], "id": "occ-7e8d50c92836446cc3f84b74" }, { "concept_id": "urn:uuid:39428ebb-bc71-4f72-9ebe-c940f88d9879", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/01-vector-spaces-and-linear-maps", "anchor": "first-ca0cf9fe5048403896dc-zh", "chapter": "4", "section": "4.1", "offset": 3601, "range": [ 10410, 10711 ], "source_revision": "c0dc763683b75205317bd9c7651ff56544eb9e566deb8f6f7be4d14f18c70600", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "线性算符(linear operators)", "context": { "range": [ 10308, 11264 ], "quote": "读者应注意，本书常把线性算符 $L$ 与它在某个未指定基下的矩阵（矩阵元记为 $L_{jk}$）使用同一个符号。这是因为我们通常关心算符及其矩阵表示中与基无关的性质，例如它们是否可逆。$V$ 上所有可逆[线性算符(linear operators)](../../../../../concepts/linear-algebra/linear-map)在复合下构成一个群，记作 $GL(V)$；其中“GL”取自英文“General Linear”（一般线性）；这个群称为[一般线性群(general linear group)](../../../../../concepts/linear-algebra/general-linear-group)。选定一组基后，该群即可与可逆矩阵群对应，群运算就是矩阵乘法。若 $\\dim V=n$，则实数域与复数域上的群分别记作 $GL(n,\\mathbf R)$ 与 $GL(n,\\mathbf C)$。", "quote_sha256": "92c491b6d7b480d57ac6a690d69a3c0ef5ee15fbc9cfdaf06ff4cf6bdd3ac0ff", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-04/01-vector-spaces-and-linear-maps", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; 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