Algemene lineaire groep

In de wiskunde is de algemene lineaire groep van de orde $n$ over een unitaire ring $R$ , aangeduid door $\mathrm {GL} (n,R)$ of $\mathrm {GL} _{n}(R)$ , de groep van de inverteerbare n×n-matrices met elementen in $R$ , met als groepsbewerking de matrixvermenigvuldiging. In veel gevallen zal $R$ een lichaam NL / veld B zijn. Doordat het product van twee inverteerbare matrices opnieuw inverteerbaar is en de inverse van een inverteerbare matrix ook weer kan worden geïnverteerd, is $\mathrm {GL} (n,R)$ inderdaad een groep. De algemene lineaire groepen $\mathrm {GL} (n,R)$ worden in de groepentheorie bestudeerd, waarbij hun groepsrepresentatie met behulp van matrices goed kan worden gebruikt.

Als de ring $R$ een eindig lichaam $\mathbb {F} _{q}$ is met $q$ een priemgetal of een macht van een priemgetal, schrijft men wel $\mathrm {GL} (n,q)$ in plaats van $\mathrm {GL} (n,R)$ . Als uit de context blijkt dat de ring $R$ het lichaam/veld $\mathbb {R}$ van de reële getallen of $\mathbb {C}$ van de complexe getallen is, wordt ook alleen $\mathrm {GL} (n)$ of $\mathrm {GL} _{n}$ geschreven.

De groep $\mathrm {GL} (n,R)$ is voor alle $n\geq 2$ is niet commutatief. De groep $\mathrm {GL} (1,R)$ is commutatief als $R$ een commutatieve ring is.

De speciale lineaire groep, geschreven als $\mathrm {SL} (n,R)$ of $\mathrm {SL} _{n}(R)$ , is de ondergroep van $\mathrm {GL} (n,R)$ van de matrices met determinant gelijk aan 1.

De groepen $\mathrm {GL} (n,R)$ en hun ondergroepen worden vaak lineaire groepen of matrixgroepen genoemd, onder de voorwaarde dat $\mathrm {GL} (R)$ een groep is. Wanneer dit het geval is, is $\mathrm {GL} (R)$ een lineaire groep, maar geen matrixgroep. De modulaire groep kan als een factorgroep van de speciale lineaire groep $\mathrm {SL} (2,\mathbb {Z} )$ worden geconstrueerd.

Bronnen

(en) M Hazewinkel voor de Encyclopedia of Mathematics. General linear group. ISBN 978-1-55608-010-4
(en) MathWorld. General Linear Group.

Read other articles:

Carlo Muraro

Carlo Muraro Muraro all'Inter nel 1980 Nazionalità Italia Altezza 176 cm Peso 74 kg Calcio Ruolo Allenatore (ex attaccante) Termine carriera 1988 - giocatore Carriera Giovanili 19??-1973 Inter Squadre di club1 1973-1975 Inter6 (0)1975-1976 Varese35 (16)1976-1981 Inter120 (38)1981-1982 Udinese22 (2)1982-1983 Ascoli15 (0)1983-1985 Inter14 (1)1985-1987 Arezzo33 (9)1987-1988 Pistoiese6 (1) Nazionale 1976-1977 Italia militare5 (4)1976-1980 I...

منتخب المجر لهوكي الجليد للسيدات

منتخب المجر لهوكي الجليد للسيدات اللقب Magyar البلد المجر ألوان الفريق رمز IIHF HUN التصنيف العالمي لهوكي الجليد التصنيف الحالي 14 (14 April 2019)[1] أعلى تصنيف 14 (2018 ) أدنى تصنيف 27 (2010 ) مشاركة دولية المجر 6–0 جنوب إفريقيا(سيكشفهيرفار، المجر؛ 19 مارس 1999) أكبر فوز ال�...

2008–09 Manchester City F.C. season

2008–09 season of Manchester City Manchester City 2008–09 football seasonManchester City2008–09 seasonOwnerAbu Dhabi United GroupChairmanThaksin Shinawatra until the sale of club on 1 September 2008 Khaldoon Al Mubarak from 1 September 2008ManagerMark HughesStadiumCity of Manchester StadiumPremier League10thFA CupThird roundLeague CupSecond roundUEFA CupQuarter-finalsTop goalscorerLeague: Robinho (14)All: Robinho (15)Highest home attendance47,331 vs Chelsea 13 September 2008Lowest home ...

Lubiana

Disambiguazione – Se stai cercando altri significati, vedi Lubiana (disambigua). Lubianacittà(SL) Ljubljana Lubiana – VedutaVeduta del centro storico di Lubiana, in cui spiccano la sede dell'università (a sinistra), il fiume Ljubljanica, la chiesa francescana, il castello e, sullo sfondo, le Alpi LocalizzazioneStato Slovenia Regione statisticaSlovenia Centrale AmministrazioneSindacoZoran Janković (PS) dall'11-4-2016 TerritorioCoordinate46°03′19.8″N 14°30′29.88″E...

Wulff series of dry flies

Wulff series of dry fliesArtificial flyBlond WulffTypeDry flyImitatesAdult mayfliesHistoryCreatorLee WulffMaterialsTypical sizes8-18 standard dry flyTypical hooksTMC 100, Firehole 419Thread6/0, 8/0TailBucktail, deer, mooseBodyFur or synthetic dubingWingBucktail, deer, moose, calf tailHackleDry fly hackleUsesPrimary useTrout, Salmon The Wulff series of dry flies evolved from a dry fly style conceived by angler Lee Wulff in the 1930s. Origin In 1930, Lee Wulff designed three innovative dry flie...

Sweeter than Fiction

Sweeter than FictionSingel promosi oleh Taylor Swiftdari album One Chance: The Incredible True Story of Paul Potts: Motion Picture SoundtrackDirilis21 Oktober 2013 (2013-10-21)Genre Bubblegum pop Durasi3:54LabelBig MachinePencipta Taylor Swift Jack Antonoff Produser Taylor Swift Jack Antonoff AudioSweeter than Fiction di YouTube Sweeter than Fiction adalah lagu oleh penyanyi-penulis lagu asal Amerika Serikat Taylor Swift untuk film Inggris One Chance (2013). Ditulis dan diproduksi oleh S...

جامعة محمد البشير الابراهيمي-برج بوعريريج

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (فبراير 2016) جامعة محمد البشير الابراهيمي برج بوعريريج الشعار وخذو الافكار منا وابعثو منها الجزائر معلومات التأسيس س...

Johannes Ringk

Johannes Ringk (Frankenhain, 26 giugno 1717 – Berlino, 24 agosto 1778) è stato un compositore e organista tedesco. Prima pagina del D-B Mus. ms. Bach P 595, Fascicle 8, il più vecchio manoscritto sopravvissuto della Toccata e Fuga in Re minore, BWV 565 Biografia Nato nel 1717 in Turingia, studiò organo con Johann Peter Kellner a Gräfenroda e con Gottfried Heinrich Stölzel a Gotha. Dal 1740 fu insegnante di musica a Berlino, e, nel 1754, venne nominato organista della Marienkirche, post...

ロバート・デ・ニーロ

ロバート・デ・ニーロRobert De Niro 2011年のデ・ニーロ生年月日 (1943-08-17) 1943年8月17日（80歳）出生地アメリカ合衆国・ニューヨーク州ニューヨーク市身長 177 cm職業俳優、映画監督、映画プロデューサージャンル映画、テレビドラマ活動期間 1963年 -配偶者ダイアン・アボット（1976年 - 1988年）グレイス・ハイタワー（1997年 - ）主な作品『ミーン・ストリート』（1973年）...

Nana Jorjadze

Nana Jorjadze ნანა ჯორჯაძეNana JorjadzeLahir24 Agustus 1948 (umur 75)Tbilisi, USSR (sekarang Georgia)PekerjaanSutradara, penulis latar, aktorTahun aktif1978 - sekarang Nana Jorjadze (bahasa Georgia: ნანა ჯორჯაძე; kelahiran 24 Agustus 1948) adalah seorang sutradara, penulis latar dan aktris Georgia. Jorjadze lahir di Tbilisi, dan mula-mula lulus dari sebuah sekolah musikal lokal (1966), dan kemudian dari departemen arsitektural di Akad...

Sikhism in Pakistan

Overview of the role and impact of Sikhism in Pakistan Sikhism in PakistanGurdwara Janam AsthanTotal population~50,0000.01% of Pakistan's total population (2010 survey)Regions with significant populationsHasan Abdal • Lahore • Nankana Sahib • Narowal Peshawar • Swat • Khyber • Karachi • Mithi LanguagesPunjabi • Urdu • Pashto • Sindhi • Hindi • Pakistani English Part of a series onSikhism People Topics Outline History ...

Sukhadeo Thorat

Indian economist Sukhadeo ThoratSukhadeo Thorat speaking at Symbiosis International University, Lavale, PuneBorn (1949-07-12) 12 July 1949 (age 74)Mahimapur, Amravati Maharashtra, IndiaNationalityIndianAcademic careerInstitutionUniversity Grants Commission [Institute for Social and Economic Change]FieldEconomicsAlma materJawaharlal Nehru University (PhD)AwardsPadmashree (2008) Vidyalankara (2008) P.K. Patil Purushottam Award (2007) Bharat Shiromani (2006) Ambedkar Chetna (2001)[...

Palauli

PalauliDistrikLokasi di SamoaNegara SamoaIbu kotaVailoaLuas • Total523 km2 (202 sq mi)Populasi (2016) • Total9.317 • Kepadatan18/km2 (46/sq mi)Kode ISO 3166-2WS-PA Palauli adalah salah satu distrik di Samoa. Kode ISO 3166-2 distrik yang beribu kota di Vailoa ini adalah WS-PA. Menurut sensus 2016, jumlah penduduk distrik yang luasnya 523 kilometer persegi ini adalah 9.317 jiwa. Referensi Census 2016 Preliminary Count Report lbsD...

Dora Maar

French artist and partner of Pablo Picasso (1907–1997) Dora MaarPortrait of Maar by Man RayBornHenriette Theodora Markovitch22 November 1907Paris, France[1]Died16 July 1997(1997-07-16) (aged 89)Paris, FranceEducationSchool of photography, École des Beaux-Arts, Académie JulianKnown forPhotography, PaintingMovementSurrealismPartnerPablo Picasso (1935–1943) Henriette Theodora Markovitch (22 November 1907 – 16 July 1997), known as Dora Maar, was a French photographer, pa...

Deep-sea fish

Fauna found in deep-sea areas Humpback anglerfish Deep-sea fish are fish that live in the darkness below the sunlit surface waters, that is below the epipelagic or photic zone of the sea. The lanternfish is, by far, the most common deep-sea fish. Other deep-sea fishes include the flashlight fish, cookiecutter shark, bristlemouths, anglerfish, viperfish, and some species of eelpout. Only about 2% of known marine species inhabit the pelagic environment. This means that they live in the water co...

Astronomical survey

General map or image of a region of the sky with no specific observational target Composite image of the GOODS-South field, result of a deep survey using two of the four giant 8.2-metre telescopes composing ESO's Very Large Telescope Gamma-ray pulsars detected by the Fermi Gamma-ray Space Telescope An astronomical survey is a general map or image of a region of the sky (or of the whole sky) that lacks a specific observational target. Alternatively, an astronomical survey may comprise a set of...

Comando supremo delle potenze alleate in Europa

Comando supremo delle potenze alleate in EuropaStemma Descrizione generaleAttivadal 1951 ServizioNATO Guarnigione/QGParigi (1951–1967)Mons (dal 1967) MottoVigilia Pretium Libertatis ComandantiComandante in capogenerale Christopher G. Cavoli Degni di notaDwight D. EisenhowerAlexander M. Haig, Jr.Wesley Clark Voci su unità militari presenti su Wikipedia Il Comando supremo delle potenze alleate in Europa[1][2][3][4] (in inglese Supreme Headquarters Allied ...

Kellogg-HKUST Executive MBA Program

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Kellogg-HKUST Executive MBA Program – news · newspapers · books · scholar · JSTOR (October 2020) (Learn how and when to remove this message) Kellogg-HKUST Executive MBA ProgramEstablished1998LocationHKUST, Clear Water Bay, Hong KongWebsitehttps://emba.hkust.edu...

Whittington Hospital

NHS hospital in London Hospital in EnglandWhittington HospitalWhittington Health NHS TrustThe main entrance on Magdala AvenueLocation within IslingtonGeographyLocationUpper Holloway, London, N19, England, United KingdomCoordinates51°33′59″N 0°08′22″W / 51.5665°N 0.1395°W / 51.5665; -0.1395OrganisationCare systemNHS EnglandTypeAcute hospital trustAffiliated universityUniversity College LondonServicesEmergency departmentYes Accident & EmergencyBeds360Hist...

Pilgrims Progress (album)

2010 studio album by Kula ShakerPilgrims ProgressStudio album by Kula ShakerReleased28 June 2010Recorded2009 – early 2010, Lompret, BelgiumGenreRock, psychedelic rockLength41:42LabelStrangeF.O.L.K.ProducerCrispian Mills, Alonza BevanKula Shaker chronology Peasants, Pigs & Astronauts - 10th Anniversary Edition(2010) Pilgrims Progress(2010) K 2.0(2016) Pilgrims Progress is the fourth studio album by Kula Shaker and was released on 28 June 2010. The first single to be taken from t...

/profillengkap.com/index.php/article/John Shea

Cladoraphis

Daftar rumah sakit di Maluku

Rudi Rubiandini

Bangga, Dolo Selatan, Sigi

16th Annual D.I.C.E. Awards

Athletics at the 1996 Summer Olympics – Men's 400 metres hurdles

Берлинская стена

Keuskupan Basse-Terre

1872 United States presidential election in Indiana

Suwon

Riviera Theatre