{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:9d1874b8-6c8c-465e-849a-f2ddc348d3d0", "labels": { "zh-CN": "线性变换", "en": "linear transformation" }, "alternative_labels": [ { "lang": "en", "value": "linear transformations" } ], "scope_note": "§4.2 permits linear maps without a universal invertibility requirement.", "identity_status": "active", "identity_revision": "20b07fd6634690d02a1e63b7c7fe7cc8e7eb7e2aef3132ace565040fc365b046", "occurrences": [ { "concept_id": "urn:uuid:9d1874b8-6c8c-465e-849a-f2ddc348d3d0", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations", "anchor": "first-5e64326d1ddbf841b8da-zh", "chapter": "1", "section": "1.3", "offset": 5553, "range": [ 17097, 17403 ], "source_revision": "4ee587fc4d72d2916ca8f6bce8080ecd42e2ae3d45d92d067027a95982f00008", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "线性变换(linear transformation)", "context": { "range": [ 16633, 17461 ], "quote": "群常以变换群(transformation group)的形式出现，也就是说，群元素作为某个特定几何对象上的变换而作用。以上述三维旋转群为例，这些旋转就是 $\\mathbf R^3$ 上的[线性变换(linear transformation)](../../../../../concepts/linear-algebra/linear-map)。一般而言：", "quote_sha256": "6be6e24167ae26e9f9a65dbeb9fa322ef3b4635a2085b5c6891182e6e21ab316", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-3900ad58db3727883392e4c9" }, { "concept_id": "urn:uuid:9d1874b8-6c8c-465e-849a-f2ddc348d3d0", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces", "anchor": "source-block-027", "chapter": "4", "section": "4.2", "offset": 2516, "range": [ 10538, 10809 ], "source_revision": "938d19d061a8b69c758beeb064276f3de2f0639a77ac963f124b6481149d29e7", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "线性变换(linear transformation)", "context": { "range": [ 10525, 10941 ], "quote": "给定作用在 $V$ 上的[线性变换(linear transformation)](../../../../../concepts/linear-algebra/linear-map) $L$，我们定义：\n", "quote_sha256": "157dbb3d3fe79646fb71f2511da7c7b6f2605261dba8386fd4fa966a17d779a3", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; 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not-concept-equivalence" } ], "id": "occ-e3051ee43cd3f7f6e8b35259" }, { "concept_id": "urn:uuid:9d1874b8-6c8c-465e-849a-f2ddc348d3d0", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group", "anchor": "first-8e4012da08917067c3b4-en", "chapter": "16", "section": "16.1", "offset": 443, "range": [ 7376, 7677 ], "source_revision": "ed2414a74ed2fd73363cf684a60f2b8e6245a4dc84ed67db852efd43fec8ffec", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "linear transformations", "context": { "range": [ 7017, 7992 ], "quote": "Recall that the [orthogonal group](../../../../../concepts/linear-algebra/orthogonal-group) consists of the [linear transformations](../../../../../concepts/linear-algebra/linear-map) that preserve an inner product, a symmetric bilinear form. We now consider the analogous group obtained by replacing the inner product with the antisymmetric bilinear form $\\Omega$ that determines the symplectic geometry of phase space. 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