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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hietala, Roope", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-05-27T06:52:20Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-05-27T06:52:20Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/81300", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 ty\u00f6ss\u00e4 tutkitaan erilaisia representaatioita vektoriavaruuksille sek\u00e4 Hilbertin avaruuden rakennetta. Hilbertin avaruudet ovat t\u00e4ydellisi\u00e4 sis\u00e4tuloavaruuksia, jotka ovat yleistys euklidiselle avaruudelle. Tavoitteena on n\u00e4ytt\u00e4\u00e4, ett\u00e4 \u00e4\u00e4ret\u00f6nulotteiselle Hilbertin avaruudelle l\u00f6ydet\u00e4\u00e4n aina ortogonaalinen kanta. Lis\u00e4ksi tarkastellaan miten t\u00e4m\u00e4 eroaa Hamelin kannasta, joka on lineaarialgebrallinen kanta avaruudelle. T\u00e4m\u00e4 ty\u00f6 koostuu nelj\u00e4st\u00e4 osasta, joista ensimm\u00e4inen on pedagoginen katsaus vektoriavaruuksiin. Toisessa k\u00e4yd\u00e4\u00e4n l\u00e4pi metrisi\u00e4 avaruuksia koskevat esitiedot, kolmannessa rakennetaan abstraktien vektoriavaruuksien teoriaa ja tutustutaan jonoavaruuksiin sek\u00e4 viimeisess\u00e4 osassa tutkitaan Banachin ja Hilbertin avaruuksia.\n\nVektoreiden oppimisessa ja opettamisessa on monenlaisia haasteita, jotka ovat per\u00e4isin representaation hy\u00f6dynt\u00e4misen puutteista. Representaatiossa saatetaan painottua liikaa esimerkiksi geometrisiin malleihin, jotka saattavat aiheuttaa h\u00e4mmennyst\u00e4 milloin tarkastellaan yleist\u00e4 tai spesifi\u00e4 tapausta. Toisaalta ilman visualisointia siirtym\u00e4 algebralliseen representaatioon voi olla liian haastavaa. Oppilaiden konseptikuvan muodostuminen vektoreista riippuu my\u00f6s siit\u00e4, mink\u00e4laisina objekteina opettaja k\u00e4sittelee vektoreita ja miten niit\u00e4 tarkastellaan oppimateriaalissa. Lukion oppimateriaali antaa hyvin geometrisen katsauksen vektoreihin, mutta yliopistossa niit\u00e4 k\u00e4sitell\u00e4\u00e4n huomattavasti abstraktimmin. Aineenopettajalla on syyt\u00e4 olla vahva ymm\u00e4rrys abstraktista vektoriavaruudesta, jotta osaa kehitt\u00e4\u00e4 pedagogisia ratkaisuja joilla antaa selke\u00e4n ja rikastetun kuvan vektoreista. \n\nVektoriavaruudella on aina olemassa kanta, mutta t\u00e4m\u00e4n kannan l\u00f6yt\u00e4minen ei ole aina helppoa. Kanta on t\u00e4rke\u00e4 vektoriavaruuden rakenteen kannalta, koska voimme esitt\u00e4\u00e4 mink\u00e4 tahansa avaruuden vektorin lineaarikombinaationa kannan vektoreista. T\u00e4m\u00e4n vuoksi on t\u00e4rke\u00e4\u00e4 perehty\u00e4 t\u00e4ydellisiin sis\u00e4tuloavaruuksiin, joissa l\u00f6yd\u00e4mme aina ortonormaalin Hilbertin kannan. Ortonormaalilla kannalla on paljon k\u00e4ytt\u00f6\u00e4 matematiikkaa soveltavilla aloilla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2022-05-27T06:52:20Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-05-27T06:52:20Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Vektoriavaruudet ja niiden representaatiot", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202205272923", "language": "", "element": 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