{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:9ac8f727-4eb8-406f-8ac4-006f5b7d7a7f", "labels": { "zh-CN": "泛性质", "en": "universal property" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "dcc0cd27bcf3a257b95b6fe262f6c33ead231a523e5e9b4b69a85b8eec4d3474", "occurrences": [ { "concept_id": "urn:uuid:9ac8f727-4eb8-406f-8ac4-006f5b7d7a7f", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "anchor": "first-315dbfc6265865430b46-zh", "chapter": "7", "section": "7.1", "offset": 17655, "range": [ 23456, 23698 ], "source_revision": "724009ba3be50eb7848a22be25a944d8dcc36fc25a0a8a39f383d373f899b86b", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "泛性质(universal property)", "context": { "range": [ 23367, 24298 ], "quote": "> [!exercise] 习题 7.2 \n>\n> 证明自由向量空间的所谓泛性质(universal property)。设$\\varphi:S\\to F(S)$为将任意元素$s \\in S$映到其特征函数$\\chi _ { s } \\in F ( S )$的映射。若$V$为任意向量空间，且$\\alpha:S\\to V$为从$S$到$V$的任意映射，则存在唯一的线性映射$T : F ( S ) \\to V$使得$\\alpha = T \\circ \\varphi$，如交换图所示\n>\n> \n>\n> ![自由向量空间泛性质的交换图](../../assets/chapter-07/part-001/2129a035115fbe81cc748c70582220262a2580a4fab593944fddcfd583c96316.jpg)\\\n> *自由向量空间泛性质的交换图（习题 7.2）。*\n>\n> 证明该过程是可逆的，并可用于将 $S$ 上的自由向量空间定义为使得上述交换图成立的唯一向量空间 $F ( S )$。", "quote_sha256": "d87aba03726e87741124d2174b5060e4abd1236dc877ee292a6aae5fcd8c1a44", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5b070ed2f7a85f8eb9cfad7c" }, { "concept_id": "urn:uuid:9ac8f727-4eb8-406f-8ac4-006f5b7d7a7f", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "anchor": "first-315dbfc6265865430b46-en", "chapter": "7", "section": "7.1", "offset": 23447, "range": [ 28771, 29008 ], "source_revision": "5975ff488ab3c30cf5496df825f7f213a867478d71c61476831ea0a9e7b76c64", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "universal property", "context": { "range": [ 28668, 30003 ], "quote": "> [!exercise] Problem 7.2 \n>\n> Prove the so-called universal property of free vector spaces. Let $\\varphi:S\\to F(S)$ be the map that assigns to any element $s \\in S$ its characteristic function $\\chi _ { s } \\in F ( S )$ . If $V$ is any vector space and $\\alpha:S\\to V$ any map from $S$ to $V$ , then there exists a unique linear map $T : F ( S ) \\to V$ such that $\\alpha = T \\circ \\varphi$ , as depicted by the commutative diagram\n>\n> \n>\n> ![Universal property of a free vector space](../../assets/chapter-07/part-001/2129a035115fbe81cc748c70582220262a2580a4fab593944fddcfd583c96316.jpg)\\\n> *Commutative diagram for the universal property of a free vector space (Problem 7.2).*\n>\n> Show that this process is reversible and may be used to define the free vector space on $S$ as being the unique vector space $F ( S )$ for which the above commutative diagram holds.", "quote_sha256": "27290a33db96ab087f92c55d70821fd9564cb7bb565a14e41942bc20aead3f79", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d13e6bcd8358b8cd81fbca8d" } ], "registered_positions": [ { "route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "lang": "en", "section": "7.1", "signal": "term-mark", "label": "universal property", "english": "universal property", "offset": 22145 }, { "route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "lang": "zh-CN", "section": "7.1", "signal": "term-mark", "label": "泛性质", "english": "universal property", "offset": 16318 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#problem-7-2\"", "href": "#problem-7-2", "range": [ 23409, 23428 ], "relation": "references", "target_route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "target_anchor": "problem-7-2", "resolved": true, "reason": null, "source_revision": "724009ba3be50eb7848a22be25a944d8dcc36fc25a0a8a39f383d373f899b86b", "source_route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "occurrence_id": "occ-5b070ed2f7a85f8eb9cfad7c" }, { "raw": "href=\"#problem-7-2\"", "href": "#problem-7-2", "range": [ 24096, 24115 ], "relation": "references", "target_route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "target_anchor": "problem-7-2", "resolved": true, "reason": null, "source_revision": "724009ba3be50eb7848a22be25a944d8dcc36fc25a0a8a39f383d373f899b86b", "source_route": "zh/chapter-07/01-free-vector-spaces-and-tensor-spaces", "occurrence_id": "occ-5b070ed2f7a85f8eb9cfad7c" }, { "raw": "href=\"#problem-7-2\"", "href": "#problem-7-2", "range": [ 28710, 28729 ], "relation": "references", "target_route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "target_anchor": "problem-7-2", "resolved": true, "reason": null, "source_revision": "5975ff488ab3c30cf5496df825f7f213a867478d71c61476831ea0a9e7b76c64", "source_route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "occurrence_id": "occ-d13e6bcd8358b8cd81fbca8d" }, { "raw": "href=\"#problem-7-2\"", "href": "#problem-7-2", "range": [ 29673, 29692 ], "relation": "references", "target_route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "target_anchor": "problem-7-2", "resolved": true, "reason": null, "source_revision": "5975ff488ab3c30cf5496df825f7f213a867478d71c61476831ea0a9e7b76c64", "source_route": "en/chapter-07/01-free-vector-spaces-and-tensor-spaces", "occurrence_id": "occ-d13e6bcd8358b8cd81fbca8d" } ], "reading_order": [ "occ-5b070ed2f7a85f8eb9cfad7c", "occ-d13e6bcd8358b8cd81fbca8d" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "d5e555b03c5e0a6bd7fb157d4574481c53b65045781dd9e66c1c2b74bf1958ac", "optional_enrichments": [] }