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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"Rzut (algebra liniowa)"@pl .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uC0AC\uC601 \uC791\uC6A9\uC18C(\u5C04\u5F71\u4F5C\u7528\u7D20, \uC601\uC5B4: projection operator)\uB294 \uBA71\uB4F1 \uC120\uD615 \uBCC0\uD658\uC774\uB2E4."@ko .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"En matem\u00E0tiques, un operador de projecci\u00F3 P en un espai vectorial \u00E9s una transformaci\u00F3 lineal , \u00E9s a dir, que satisf\u00E0 la igualtat P 2 = P ."@ca .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"In der Mathematik ist eine Projektion oder ein Projektor eine spezielle lineare Abbildung (Endomorphismus) \u00FCber einem Vektorraum , die alle Vektoren in ihrem Bild (ein Unterraum von ) unver\u00E4ndert l\u00E4sst. Bei geeigneter Wahl einer Basis von setzt die Projektion einige Komponenten eines Vektors auf null und beh\u00E4lt die \u00FCbrigen bei. Damit ist auch anschaulich die Bezeichnung Projektion gerechtfertigt, wie etwa bei der Abbildung eines Hauses in einem zweidimensionalen Grundriss."@de .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"\u0412 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0435 \u0438 \u0444\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C \u0430\u043D\u0430\u043B\u0438\u0437\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 , \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435, \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u0440\u043E\u0435\u0301\u043A\u0442\u043E\u0440\u043E\u043C (\u0430 \u0442\u0430\u043A\u0436\u0435 \u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440\u043E\u043C \u043F\u0440\u043E\u0435\u0446\u0438\u0301\u0440\u043E\u0432\u0430\u043D\u0438\u044F \u0438 \u043F\u0440\u043E\u0435\u043A\u0446\u0438\u043E\u0301\u043D\u043D\u044B\u043C \u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440\u043E\u043C) \u0435\u0441\u043B\u0438 . \u0422\u0430\u043A\u043E\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442 \u0438\u0434\u0435\u043C\u043F\u043E\u0442\u0435\u043D\u0442\u043D\u044B\u043C. \u041D\u0435\u0441\u043C\u043E\u0442\u0440\u044F \u043D\u0430 \u0441\u0432\u043E\u044E \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u043E\u0441\u0442\u044C, \u044D\u0442\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u0431\u043E\u0431\u0449\u0430\u0435\u0442 \u0438\u0434\u0435\u044E \u043F\u043E\u0441\u0442\u0440\u043E\u0435\u043D\u0438\u044F \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043F\u0440\u043E\u0435\u043A\u0446\u0438\u0438. \u0412 \u043E\u0431\u0449\u0435\u043C \u0441\u043B\u0443\u0447\u0430\u0435, \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0432 \u043F\u0440\u044F\u043C\u0443\u044E \u0441\u0443\u043C\u043C\u0443 \u043D\u0435 \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u043E. \u041F\u043E\u044D\u0442\u043E\u043C\u0443, \u0434\u043B\u044F \u043F\u043E\u0434\u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 , \u0432\u043E\u043E\u0431\u0449\u0435 \u0433\u043E\u0432\u043E\u0440\u044F, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043C\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0435\u043A\u0442\u043E\u0440\u043E\u0432, \u043E\u0431\u0440\u0430\u0437 \u0438\u043B\u0438 \u044F\u0434\u0440\u043E \u043A\u043E\u0442\u043E\u0440\u044B\u0445 \u0441\u043E\u0432\u043F\u0430\u0434\u0430\u0435\u0442 \u0441 ."@ru .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"\u041A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u044F \u0437 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u043C\u0438 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0430\u043C\u0438 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u0440\u043E\u0454\u043A\u0446\u0456\u0439\u043D\u043E\u044E, \u044F\u043A\u0449\u043E \u0432\u0438\u043A\u043E\u043D\u0443\u0454\u0442\u044C\u0441\u044F \u042F\u043A\u0449\u043E \u0432\u0438\u043A\u043E\u043D\u0443\u0454\u0442\u044C\u0441\u044F \u0442\u043E \u043C\u0430\u0442\u0440\u0438\u0446\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u043E-\u043F\u0440\u043E\u0454\u043A\u0446\u0456\u0439\u043D\u043E\u044E. \n* \u041F\u0440\u043E\u0454\u043A\u0446\u0456\u0439\u043D\u0456 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u043D\u0430\u0437\u0438\u0432\u0430\u044E\u0442\u044C\u0441\u044F \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u0438\u043C\u0438, \u044F\u043A\u0449\u043E \u0417 \u0442\u043E\u0447\u043A\u0438 \u0437\u043E\u0440\u0443 \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u043E\u0457 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 \u043F\u0440\u043E\u0454\u043A\u0446\u0456\u0439\u043D\u0456 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u2014 \u0446\u0435 \u0456\u0434\u0435\u043C\u043F\u043E\u0442\u0435\u043D\u0442\u043D\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u043A\u0456\u043B\u044C\u0446\u044F \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u0438\u0445 \u043C\u0430\u0442\u0440\u0438\u0446\u044C."@uk .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"\u0641\u064A \u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062E\u0637\u064A \u0648\u0627\u0644\u062A\u062D\u0644\u064A\u0644 \u0627\u0644\u062F\u0627\u0644\u064A\u060C \u0627\u0644\u0625\u0633\u0642\u0627\u0637 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Projection)\u200F \u0647\u0648 \u0643\u0644 \u062A\u062D\u0648\u064A\u0644 \u062E\u0637\u064A \u0645\u0646 \u0627\u0644\u0641\u0636\u0627\u0621 \u0627\u0644\u0645\u062A\u062C\u0647\u064A \u0646\u062D\u0648 \u0646\u0641\u0633\u0647 \u062D\u064A\u062B . \u0628\u062A\u0639\u0628\u064A\u0631 \u0622\u062E\u0631\u060C P \u0647\u0648 \u062D\u064A\u062B \u0625\u0630\u0627 \u0637\u064F\u0628\u0642 \u0645\u0631\u062A\u064A\u0646 \u0639\u0644\u0649 \u0642\u064A\u0645\u0629 \u0645\u0639\u064A\u0646\u0629\u060C \u0641\u0643\u0623\u0646\u0645\u0627 \u0637\u064F\u0628\u0642 \u0645\u0631\u0629 \u0648\u0627\u062D\u062F\u0629 (\u062A\u0633\u0645\u0649 \u0647\u0630\u0647 \u0627\u0644\u062E\u0627\u0635\u064A\u0629 \u0628\u062A\u0633\u0627\u0648\u064A \u0627\u0644\u0642\u0648\u0649). \u0627\u0644\u0639\u0646\u0627\u0635\u0631 \u0627\u0644\u0623\u0633\u0627\u0633\u064A\u0629 \u0641\u064A \u062C\u0645\u064A\u0639 \u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u0627\u0633\u0642\u0627\u0637 \u0647\u064A \u0645\u0631\u0643\u0632 \u0648\u0645\u0633\u062A\u0648\u0649 \u0627\u0644\u0627\u0633\u0642\u0627\u0637. \u0648\u0641\u0642\u0627 \u0644\u0637\u0628\u064A\u0639\u0629 \u0645\u0631\u0643\u0632 \u0627\u0644\u0627\u0633\u0642\u0627\u0637: \u0646\u0642\u0637\u0629 \u0646\u0647\u0627\u0626\u064A\u0629 \u0623\u0648 \u0644\u0627\u0646\u0647\u0627\u0626\u064A\u0629\u060C \u0627\u0644\u0627\u0633\u0642\u0627\u0637 \u064A\u0646\u0642\u0633\u0645 \u0625\u0644\u0649 \u0646\u0648\u0639\u064A\u0646 \u0627\u0644\u0625\u0633\u0642\u0627\u0637 \u0627\u0644\u0645\u062A\u0648\u0627\u0632\u064A \u0648\u0627\u0644\u0625\u0633\u0642\u0627\u0637 \u0627\u0644\u0645\u0631\u0643\u0632\u064A (\u0623\u0648 \u0627\u0644\u0645\u0646\u0638\u0648\u0631). \u0627\u0646\u0638\u0631 \u0625\u0644\u0649 \u0625\u0633\u0642\u0627\u0637 \u062A\u0645\u062B\u064A\u0644\u064A \u062B\u0644\u0627\u062B\u064A \u0627\u0644\u0623\u0628\u0639\u0627\u062F."@ar .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"Projekce (line\u00E1rn\u00ED algebra)"@cs .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"\u0625\u0633\u0642\u0627\u0637 (\u062C\u0628\u0631 \u062E\u0637\u064A)"@ar .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"\u5728\u7EBF\u6027\u4EE3\u6570\u548C\u6CDB\u51FD\u5206\u6790\u4E2D\uFF0C\u6295\u5F71\u662F\u4ECE\u5411\u91CF\u7A7A\u95F4\u6620\u5C04\u5230\u81EA\u8EAB\u7684\u4E00\u79CD\u7EBF\u6027\u53D8\u6362\uFF0C\u6EE1\u8DB3,\u4E5F\u5C31\u662F\u8BF4\uFF0C\u5F53\u4E24\u6B21\u4F5C\u7528\u4E8E\u67D0\u4E2A\u503C\uFF0C\u4E0E\u4F5C\u7528\u4E00\u6B21\u5F97\u5230\u7684\u7ED3\u679C\u76F8\u540C\uFF08\u5E42\u7B49\uFF09\u3002\u662F\u65E5\u5E38\u751F\u6D3B\u4E2D\u201C\u5E73\u884C\u6295\u5F71\u201D\u6982\u5FF5\u7684\u5F62\u5F0F\u5316\u548C\u4E00\u822C\u5316\u3002\u540C\u73B0\u5B9E\u4E2D\u9633\u5149\u5C06\u4E8B\u7269\u6295\u5F71\u5230\u5730\u9762\u4E0A\u4E00\u6837\uFF0C\u6295\u5F71\u53D8\u6362\u5C06\u6574\u4E2A\u5411\u91CF\u7A7A\u95F4\u6620\u5C04\u5230\u5B83\u7684\u5176\u4E2D\u4E00\u4E2A\u5B50\u7A7A\u95F4\uFF0C\u5E76\u4E14\u5728\u8FD9\u4E2A\u5B50\u7A7A\u95F4\u4E2D\u662F\u6052\u7B49\u53D8\u6362\u3002"@zh .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"V line\u00E1rn\u00ED algeb\u0159e a funkcion\u00E1ln\u00ED anal\u00FDze je projekce line\u00E1rn\u00ED transformace n\u011Bjak\u00E9ho vektorov\u00E9ho prostoru na sebe takov\u00E1, \u017Ee . To znamen\u00E1, \u017Ee pokud aplikujeme na jakoukoli hodnotu opakovan\u011B, v\u00FDsledek je stejn\u00FD, jako kdybychom ji pou\u017Eili jen jednou (je to idempotentn\u00ED zobrazen\u00ED, kter\u00E9 nem\u011Bn\u00ED prostor sv\u00FDch obraz\u016F). Tato definice formalizuje a zobec\u0148uje my\u0161lenku geometrick\u00E9 projekce."@cs .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/QR_decomposition> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Riemannian_submersion> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Expectation_value_(quantum_mechanics)> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Schauder_basis> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"En alg\u00E8bre lin\u00E9aire, un projecteur (ou une projection) est une application lin\u00E9aire qu'on peut pr\u00E9senter de deux fa\u00E7ons \u00E9quivalentes : \n* une projection lin\u00E9aire associ\u00E9e \u00E0 une d\u00E9composition de E comme somme de deux sous-espaces suppl\u00E9mentaires, c'est-\u00E0-dire qu'elle permet d'obtenir un des termes de la d\u00E9composition correspondante ; \n* une application lin\u00E9aire idempotente : elle v\u00E9rifie p2 = p. Dans un espace hilbertien ou m\u00EAme seulement pr\u00E9hilbertien, une projection pour laquelle les deux suppl\u00E9mentaires sont orthogonaux est appel\u00E9e projection orthogonale."@fr .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Unit_vector> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Diagonalizable_matrix> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Cauchy\u2013Schwarz_inequality> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"Em \u00E1lgebra linear e an\u00E1lise funcional, uma proje\u00E7\u00E3o \u00E9 uma transforma\u00E7\u00E3o linear de um espa\u00E7o vetorial em si mesmo, de modo que , ou seja, sempre que \u00E9 aplicado duas vezes a algum vetor, o resultado \u00E9 o mesmo que se tivesse sido aplicado uma \u00FAnica vez (uma propriedade conhecida como idempot\u00EAncia). Embora abstrata, esta defini\u00E7\u00E3o de \"proje\u00E7\u00E3o\" formaliza e generaliza adequadamente a ideia de proje\u00E7\u00E3o gr\u00E1fica. Tamb\u00E9m se pode considerar o efeito de uma proje\u00E7\u00E3o em um objeto geom\u00E9trico, examinando o efeito que a proje\u00E7\u00E3o tem nos pontos do objeto."@pt .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Endomorphism> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uC0AC\uC601 \uC791\uC6A9\uC18C(\u5C04\u5F71\u4F5C\u7528\u7D20, \uC601\uC5B4: projection operator)\uB294 \uBA71\uB4F1 \uC120\uD615 \uBCC0\uD658\uC774\uB2E4."@ko .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Moore\u2013Penrose_inverse> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Self-adjoint_operator> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Quaternionic_analysis> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Gram\u2013Schmidt_decomposition> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Diagonalizable> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"En matem\u00E1ticas, un operador de proyecci\u00F3n P en un espacio vectorial es una transformaci\u00F3n lineal idempotente, es decir, satisface la igualdad P2 = P.\u200B"@es .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"\u041F\u0440\u043E\u0435\u043A\u0442\u043E\u0440 (\u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430)"@ru .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"In der Mathematik ist eine Projektion oder ein Projektor eine spezielle lineare Abbildung (Endomorphismus) \u00FCber einem Vektorraum , die alle Vektoren in ihrem Bild (ein Unterraum von ) unver\u00E4ndert l\u00E4sst. Bei geeigneter Wahl einer Basis von setzt die Projektion einige Komponenten eines Vektors auf null und beh\u00E4lt die \u00FCbrigen bei. Damit ist auch anschaulich die Bezeichnung Projektion gerechtfertigt, wie etwa bei der Abbildung eines Hauses in einem zweidimensionalen Grundriss."@de .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"\u6295\u5F71"@zh .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"Projecteur (math\u00E9matiques)"@fr .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Linear_algebra> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Banach_space> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Identity_matrix> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://xmlns.com/foaf/0.1/topic>	<http://dbpedia.org/resource/Fr\u00F6licher\u2013Nijenhuis_bracket> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#label>	"\uC0AC\uC601\uC791\uC6A9\uC18C"@ko .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"\u7DDA\u578B\u4EE3\u6570\u5B66\u304A\u3088\u3073\u51FD\u6570\u89E3\u6790\u5B66\u306B\u304A\u3051\u308B\u5C04\u5F71\u4F5C\u7528\u7D20\u3042\u308B\u3044\u306F\u5358\u306B\u5C04\u5F71\uFF08\u3057\u3083\u3048\u3044\u3001\u82F1: projection\uFF09\u3068\u306F\u3001\u3044\u308F\u3086\u308B\u5C04\u5F71\uFF08\u6295\u5F71\uFF09\u3092\u4E00\u822C\u5316\u3057\u305F\u6982\u5FF5\u3067\u3042\u308B\u3002\u6709\u9650\u6B21\u5143\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593 V \u306E\u5834\u5408\u306F\u3001V \u4E0A\u306E\u7DDA\u578B\u5909\u63DB P: V \u2192 V \u3067\u3042\u3063\u3066\u3001\u51AA\u7B49\u5F8B P2 = P \u3092\u6E80\u305F\u3059\u3082\u306E\u3092\u8A00\u3046\u3002\u30D9\u30AF\u30C8\u30EB v \u306E\u50CF Pv \u3092 v \u306E\u5C04\u5F71\u3068\u3044\u3046\u3002\u5C04\u5F71\u4F5C\u7528\u7D20\u306F\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593 V \u3092 U\u2295W \u3068\u76F4\u548C\u5206\u89E3\u3057\u305F\u3068\u304D\u306B\u3001V \u306E\u5143 v = u + w (u \u2208 U, w \u2208 W) \u3092 u \u306B\u5199\u3059\u3088\u3046\u306A\u5909\u63DB\u3067\u3042\u308B\u3002\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306E\u6B21\u5143\u304C\u7121\u9650\u6B21\u5143\u306E\u5834\u5408\u306B\u306F\u3001\u9023\u7D9A\u6027\u3092\u8003\u616E\u3057\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u3002\u4F8B\u3048\u3070\u30D2\u30EB\u30D9\u30EB\u30C8\u7A7A\u9593 \u306B\u304A\u3051\u308B\u5C04\u5F71\u4F5C\u7528\u7D20\u3068\u306F\u3001 \u4E0A\u306E\u6709\u754C\u7DDA\u578B\u4F5C\u7528\u7D20 \u3067\u3042\u3063\u3066\u3001\u51AA\u7B49\u5F8B P2 = P \u3092\u6E80\u305F\u3059\u3082\u306E\u3092\u8A00\u3046\u3002\u3053\u306E\u3068\u304D\u3055\u3089\u306B\u81EA\u5DF1\u5171\u5F79\u6027 P\u2217 = P \u3092\u6301\u3064\u3068\u304D\u306B\u306F\u76F4\u4EA4\u5C04\u5F71\uFF08\u3061\u3087\u3063\u3053\u3046\u3057\u3083\u3048\u3044\u3001\u82F1: orthogonal projection\uFF09\u3068\u3044\u3046\u3002\u76F4\u4EA4\u5C04\u5F71\u306E\u3053\u3068\u3092\u5358\u306B\u5C04\u5F71\u3068\u547C\u3076\u3053\u3068\u3082\u3042\u308B\u3002 \u3053\u306E\u5B9A\u7FA9\u306F\u62BD\u8C61\u7684\u3067\u306F\u3042\u308B\u304C\u3001\u6295\u5F71\u56F3\u6CD5\u306E\u8003\u3048\u65B9\u3092\u4E00\u822C\u5316\u3057\u3001\u5B9A\u5F0F\u5316\u3057\u305F\u3082\u306E\u306B\u306A\u3063\u3066\u3044\u308B\u3002 \u4E0A\u306E\u5C04\u5F71\u306E\u5F71\u97FF\u306F\u3001\u305D\u306E\u5BFE\u8C61\u306E\u5404\u70B9\u306B\u304A\u3051\u308B\u5C04\u5F71\u306E\u5F71\u97FF\u3092\u8ABF\u3079\u308B\u3053\u3068\u3067\u308F\u304B\u308B\u3002"@ja .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its image unchanged. This definition of \"projection\" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object."@en .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/1999/02/22-rdf-syntax-ns#type>	<http://dbpedia.org/ontology/Album> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Rank_of_a_linear_operator> .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"Rzut lub projekcja \u2013 uog\u00F3lnienie poj\u0119cia rzutu znanego z geometrii elementarnej: idempotentny endomorfizm liniowy okre\u015Blony na danej przestrzeni liniowej, czyli operator liniowy zachowuj\u0105cy sw\u00F3j obraz, tzn. dla kt\u00F3rego ka\u017Cdy element obrazu jest punktem sta\u0142ym tego przekszta\u0142cenia. Rzuty/projekcje ortogonalne s\u0105 uog\u00F3lnieniem poj\u0119cia rzutu prostok\u0105tnego z geometrii euklidesowej (zob. ); w przestrzeniach unitarnych (tzn. z iloczynem skalarnym, np. przestrzeniach euklidesowych) s\u0105 to ni mniej, ni wi\u0119cej operatory samosprz\u0119\u017Cone."@pl .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/property/proof>	"Let  be a complete metric space with an inner product, and let  be a closed linear subspace of  .\n\nFor every  the following set of non-negative norm-values  has an infimum, and due to the completeness of  it is a minimum. We define  as the point in  where this minimum is obtained.\n\nObviously  is in . It remains to show that  satisfies  and that it is linear.\n\nLet us define . For every non-zero  in , the following holds:\n\nBy defining  we see that  unless  vanishes. Since  was chosen as the minimum of the aforementioned set, it follows that  indeed vanishes. In particular, : .\n\nLinearity follows from the vanishing of  for every :\n\n\nBy taking the difference between the equations we have \n\nBut since we may choose   it follows that . Similarly we have  for every scalar ."@en .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its image unchanged. This definition of \"projection\" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object."@en .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://www.w3.org/2000/01/rdf-schema#comment>	"\u7DDA\u578B\u4EE3\u6570\u5B66\u304A\u3088\u3073\u51FD\u6570\u89E3\u6790\u5B66\u306B\u304A\u3051\u308B\u5C04\u5F71\u4F5C\u7528\u7D20\u3042\u308B\u3044\u306F\u5358\u306B\u5C04\u5F71\uFF08\u3057\u3083\u3048\u3044\u3001\u82F1: projection\uFF09\u3068\u306F\u3001\u3044\u308F\u3086\u308B\u5C04\u5F71\uFF08\u6295\u5F71\uFF09\u3092\u4E00\u822C\u5316\u3057\u305F\u6982\u5FF5\u3067\u3042\u308B\u3002\u6709\u9650\u6B21\u5143\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593 V \u306E\u5834\u5408\u306F\u3001V \u4E0A\u306E\u7DDA\u578B\u5909\u63DB P: V \u2192 V \u3067\u3042\u3063\u3066\u3001\u51AA\u7B49\u5F8B P2 = P \u3092\u6E80\u305F\u3059\u3082\u306E\u3092\u8A00\u3046\u3002\u30D9\u30AF\u30C8\u30EB v \u306E\u50CF Pv \u3092 v \u306E\u5C04\u5F71\u3068\u3044\u3046\u3002\u5C04\u5F71\u4F5C\u7528\u7D20\u306F\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593 V \u3092 U\u2295W \u3068\u76F4\u548C\u5206\u89E3\u3057\u305F\u3068\u304D\u306B\u3001V \u306E\u5143 v = u + w (u \u2208 U, w \u2208 W) \u3092 u \u306B\u5199\u3059\u3088\u3046\u306A\u5909\u63DB\u3067\u3042\u308B\u3002\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306E\u6B21\u5143\u304C\u7121\u9650\u6B21\u5143\u306E\u5834\u5408\u306B\u306F\u3001\u9023\u7D9A\u6027\u3092\u8003\u616E\u3057\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u3002\u4F8B\u3048\u3070\u30D2\u30EB\u30D9\u30EB\u30C8\u7A7A\u9593 \u306B\u304A\u3051\u308B\u5C04\u5F71\u4F5C\u7528\u7D20\u3068\u306F\u3001 \u4E0A\u306E\u6709\u754C\u7DDA\u578B\u4F5C\u7528\u7D20 \u3067\u3042\u3063\u3066\u3001\u51AA\u7B49\u5F8B P2 = P \u3092\u6E80\u305F\u3059\u3082\u306E\u3092\u8A00\u3046\u3002\u3053\u306E\u3068\u304D\u3055\u3089\u306B\u81EA\u5DF1\u5171\u5F79\u6027 P\u2217 = P \u3092\u6301\u3064\u3068\u304D\u306B\u306F\u76F4\u4EA4\u5C04\u5F71\uFF08\u3061\u3087\u3063\u3053\u3046\u3057\u3083\u3048\u3044\u3001\u82F1: orthogonal projection\uFF09\u3068\u3044\u3046\u3002\u76F4\u4EA4\u5C04\u5F71\u306E\u3053\u3068\u3092\u5358\u306B\u5C04\u5F71\u3068\u547C\u3076\u3053\u3068\u3082\u3042\u308B\u3002 \u3053\u306E\u5B9A\u7FA9\u306F\u62BD\u8C61\u7684\u3067\u306F\u3042\u308B\u304C\u3001\u6295\u5F71\u56F3\u6CD5\u306E\u8003\u3048\u65B9\u3092\u4E00\u822C\u5316\u3057\u3001\u5B9A\u5F0F\u5316\u3057\u305F\u3082\u306E\u306B\u306A\u3063\u3066\u3044\u308B\u3002 \u4E0A\u306E\u5C04\u5F71\u306E\u5F71\u97FF\u306F\u3001\u305D\u306E\u5BFE\u8C61\u306E\u5404\u70B9\u306B\u304A\u3051\u308B\u5C04\u5F71\u306E\u5F71\u97FF\u3092\u8ABF\u3079\u308B\u3053\u3068\u3067\u308F\u304B\u308B\u3002"@ja .
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<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Euclidean_vector> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/property/wikiPageUsesTemplate>	<http://dbpedia.org/resource/Template:Linear_algebra> .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/abstract>	"\u0412 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0435 \u0438 \u0444\u0443\u043D\u043A\u0446\u0438\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C \u0430\u043D\u0430\u043B\u0438\u0437\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 , \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435, \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u0440\u043E\u0435\u0301\u043A\u0442\u043E\u0440\u043E\u043C (\u0430 \u0442\u0430\u043A\u0436\u0435 \u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440\u043E\u043C \u043F\u0440\u043E\u0435\u0446\u0438\u0301\u0440\u043E\u0432\u0430\u043D\u0438\u044F \u0438 \u043F\u0440\u043E\u0435\u043A\u0446\u0438\u043E\u0301\u043D\u043D\u044B\u043C \u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440\u043E\u043C) \u0435\u0441\u043B\u0438 . \u0422\u0430\u043A\u043E\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442 \u0438\u0434\u0435\u043C\u043F\u043E\u0442\u0435\u043D\u0442\u043D\u044B\u043C. \u041D\u0435\u0441\u043C\u043E\u0442\u0440\u044F \u043D\u0430 \u0441\u0432\u043E\u044E \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u043E\u0441\u0442\u044C, \u044D\u0442\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u0431\u043E\u0431\u0449\u0430\u0435\u0442 \u0438\u0434\u0435\u044E \u043F\u043E\u0441\u0442\u0440\u043E\u0435\u043D\u0438\u044F \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043F\u0440\u043E\u0435\u043A\u0446\u0438\u0438. \u0412 \u043A\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u044F \u043C\u043E\u0436\u043D\u043E \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u0442\u044C \u0441\u043B\u0435\u0434\u0443\u044E\u0449\u0435\u0435 \u0441\u0432\u043E\u0439\u0441\u0442\u0432\u043E \u043F\u0440\u043E\u0435\u043A\u0442\u043E\u0440\u0430: \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u043F\u0440\u043E\u0435\u043A\u0442\u043E\u0440\u043E\u043C \u0442\u043E\u0433\u0434\u0430 \u0438 \u0442\u043E\u043B\u044C\u043A\u043E \u0442\u043E\u0433\u0434\u0430, \u043A\u043E\u0433\u0434\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044E\u0442 \u0442\u0430\u043A\u0438\u0435 \u043F\u043E\u0434\u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0438 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 , \u0447\u0442\u043E \u0440\u0430\u0441\u043A\u043B\u0430\u0434\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u0432 \u0438\u0445 \u043F\u0440\u044F\u043C\u0443\u044E \u0441\u0443\u043C\u043C\u0443, \u0438 \u043F\u0440\u0438 \u044D\u0442\u043E\u043C \u0434\u043B\u044F \u043B\u044E\u0431\u043E\u0439 \u043F\u0430\u0440\u044B \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u043E\u0432 \u0438\u043C\u0435\u0435\u043C . \u041F\u043E\u0434\u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0438 \u2014 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043D\u043D\u043E \u043E\u0431\u0440\u0430\u0437 \u0438 \u044F\u0434\u0440\u043E \u043F\u0440\u043E\u0435\u043A\u0442\u043E\u0440\u0430 , \u0438 \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u044E\u0442\u0441\u044F \u0438 . \u0412 \u043E\u0431\u0449\u0435\u043C \u0441\u043B\u0443\u0447\u0430\u0435, \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u0435 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u0432 \u043F\u0440\u044F\u043C\u0443\u044E \u0441\u0443\u043C\u043C\u0443 \u043D\u0435 \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u043E. \u041F\u043E\u044D\u0442\u043E\u043C\u0443, \u0434\u043B\u044F \u043F\u043E\u0434\u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 , \u0432\u043E\u043E\u0431\u0449\u0435 \u0433\u043E\u0432\u043E\u0440\u044F, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043C\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0435\u043A\u0442\u043E\u0440\u043E\u0432, \u043E\u0431\u0440\u0430\u0437 \u0438\u043B\u0438 \u044F\u0434\u0440\u043E \u043A\u043E\u0442\u043E\u0440\u044B\u0445 \u0441\u043E\u0432\u043F\u0430\u0434\u0430\u0435\u0442 \u0441 ."@ru .
<http://dbpedia.org/resource/Projection_(linear_algebra)>	<http://dbpedia.org/ontology/wikiPageWikiLink>	<http://dbpedia.org/resource/Kernel_(linear_algebra)> .