{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:84fdb7c6-ad08-4e44-851b-c0fbfd67193b", "labels": { "zh-CN": "定向向量空间", "en": "oriented vector space" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "8ba314f39fb5d1053a8e2d4c8217155098270fb414c872fdf3abfc19e892e30c", "occurrences": [ { "concept_id": "urn:uuid:84fdb7c6-ad08-4e44-851b-c0fbfd67193b", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-08/05-oriented-vector-spaces", "anchor": "first-e02cdd7886d0c6f30a01-zh", "chapter": "8", "section": "8.5", "offset": 9509, "range": [ 15766, 16014 ], "source_revision": "2527bf3e5cc47b1791642cfc4fb125098de6990765a21be26887e873669071e2", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "定向向量空间(oriented vector space)", "context": { "range": [ 14936, 16050 ], "quote": "若两个基 $\\{ e _ { i } \\}$ 与 $\\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$ 的变换矩阵 $\\mathsf { A } = [ A _ { j } ^ { i ^ { \\prime } } ]$ 在式 (7.26) 中具有正的行列式，即 $det$ $\\mathsf { A } > 0 \\mathsf { j }$，则称它们具有相同的定向；否则称它们具有相反的定向。当且仅当 $\\{ e _ { i } \\}$ 与 $\\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$ 具有相同定向时，记作 $\\{ e _ { i } \\} o \\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$，则容易证明 $o$ 是一个等价关系，并将 $V$ 上所有基的集合划分为两个等价类，称为 定向(orientation)。向量空间 $V$ 连同其定向的一个选择称为一个 定向向量空间(oriented vector space)。属于所选定向的任意基都称为正定向的，而具有相反定向的基则称为负定向的。", "quote_sha256": "dce097ee3b68fc9ba77619d2c6bc469b040ba2aa9fae29b1e77b9378c5324f3e", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-08/05-oriented-vector-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-6de75e5202c0ef2284314efc" }, { "concept_id": "urn:uuid:84fdb7c6-ad08-4e44-851b-c0fbfd67193b", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-08/05-oriented-vector-spaces", "anchor": "first-e02cdd7886d0c6f30a01-en", "chapter": "8", "section": "8.5", "offset": 11360, "range": [ 17145, 17385 ], "source_revision": "3cb063d69fe7f90481323cd6e3876a8ef23120a60d5822c2638ad62a1fbd914a", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "oriented vector space", "context": { "range": [ 16035, 17542 ], "quote": "Two bases $\\{ e _ { i } \\}$ and $\\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$ are said to have the same orientation if the transformation matrix $\\mathsf { A } = [ A _ { j } ^ { i ^ { \\prime } } ]$ in Eq. (7.26) has positive determinant, $det$ $\\mathsf { A } > 0 \\mathsf { j }$ otherwise they are said to be oppositely oriented. Writing $\\{ e _ { i } \\} o \\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$ iff $\\{ e _ { i } \\}$ and $\\{ e _ { i ^ { \\prime } } ^ { \\prime } \\}$ have the same orientation, it is straightforward to show that $o$ is an equivalence relation and divides the set of all bases on $V$ into two equivalence classes, called orientations. A vector space $V$ together with a choice oforientation is called an oriented vector space. 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