Click here to load reader

19η διάλεξη Γραμμικής Άλγεβρας

  • View
    3.593

  • Download
    0

Embed Size (px)

DESCRIPTION

Βάση και διάσταση θεμελειωδών χώρων

Text of 19η διάλεξη Γραμμικής...

  • 1. Grammik 'AlgebraBseic jemeleiwdn dianusmatikn qrwnTmma Hlektrolgwn Mhqanikn kai Mhqanikn UpologistnPanepistmio Jessalac10 Noembrou 2014

2. Grammik Anexarthsa'Ena snolo dianusmtwn v1,v2, . . . ,vk 2 Rn lgontai grammikexarthmna ann uprqoun arijmo c1,c2, . . . ,ck 2 R ek twnopown toulqiston nac den enai mhdn kai gia toucopoouc isqei c1v1c2v2 . . . ,ckvk 0. 3. Grammik Anexarthsa'Ena snolo dianusmtwn v1,v2, . . . ,vk 2 Rn lgontai grammikexarthmna ann uprqoun arijmo c1,c2, . . . ,ck 2 R ek twnopown toulqiston nac den enai mhdn kai gia toucopoouc isqei c1v1c2v2 . . . ,ckvk 0.Gia na elgxoume thn exrthsh twn v1,v2, . . . ,vk 2 Rn Sqhmatzoume ton pnaka A opooc qei san stlec tadiansmata aut Upologzoume ton mhdenqwro tou AAn autc perilambnei mnon to mhdenik dinusma tte autenai grammik anexrthta. 4. Orismo En nac dianusmatikc qroc V apoteletai ap louctouc grammikoc sunduasmoc twn dianusmtwnv1,v2, . . . ,vk tte lme ti aut pargoun ton V. 5. Orismo En nac dianusmatikc qroc V apoteletai ap louctouc grammikoc sunduasmoc twn dianusmtwnv1,v2, . . . ,vk tte lme ti aut pargoun ton V. 'Ena snolo dianusmtwn pargei nan dianusmatikqro ann kje dinusma tou qrou mpore na grafjesan grammikc sunduasmc twn en lgw dianusmtwn. 6. Orismo En nac dianusmatikc qroc V apoteletai ap louctouc grammikoc sunduasmoc twn dianusmtwnv1,v2, . . . ,vk tte lme ti aut pargoun ton V. 'Ena snolo dianusmtwn pargei nan dianusmatikqro ann kje dinusma tou qrou mpore na grafjesan grammikc sunduasmc twn en lgw dianusmtwn. 'Ena snolo dianusmtwn apotele bsh encdianusmatiko qrou ann aut enai grammikanexrthta kai pargoun ton qro. 7. Orismo En nac dianusmatikc qroc V apoteletai ap louctouc grammikoc sunduasmoc twn dianusmtwnv1,v2, . . . ,vk tte lme ti aut pargoun ton V. 'Ena snolo dianusmtwn pargei nan dianusmatikqro ann kje dinusma tou qrou mpore na grafjesan grammikc sunduasmc twn en lgw dianusmtwn. 'Ena snolo dianusmtwn apotele bsh encdianusmatiko qrou ann aut enai grammikanexrthta kai pargoun ton qro. Distash enc upoqrou enai to pljoc twn stoiqewnthc bshc tou. 8. Parathrseic Uprqoun peirec diaforetikcbseic enc upqwrou. 9. Parathrseic Uprqoun peirec diaforetikcbseic enc upqwrou. Do opoiesdpote bseic encqrou apotelontai ap to diopljoc dianusmtwn. 10. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m n 11. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm 12. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVA 13. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0 14. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0Pollaplasizoume thn parapnw exswsh me to c aut kaiqoume )Wc VAc 15. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0Pollaplasizoume thn parapnw exswsh me to c aut kaiqoume )Wc VAcTte Ac 0 16. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0Pollaplasizoume thn parapnw exswsh me to c aut kaiqoume )Wc VAcTte Ac 0)VAc 0 17. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0Pollaplasizoume thn parapnw exswsh me to c aut kaiqoume )Wc VAcTte Ac 0)VAc 0)Wc 0,c6 0 18. JerhmaAn ta snola v1,v2, . . . ,vm kai w1,w2, . . . ,wn apotelonbseic tou qrou V tte m n.Apdeixh:'Estw m nwj a1,jv1a2,jv2. . .am,jvm)WVAEpeid m n tte o A qei ston mhdenqwr toutoulqiston na mh-mhdenik dinusma stw to c6 0Pollaplasizoume thn parapnw exswsh me to c aut kaiqoume )Wc VAcTte Ac 0)VAc 0)Wc 0,c6 0'Atopo giat ta wi, i 1, . . . ,m enai grammik anexrthta. 19. Parathrseic Kje grammik anexrthto snolo dianusmtwnenc qrou V mpore, en aut enai aparathto,na epektaje (me prosjkh llwn dianusmtwn)tsi ste na gnei bsh tou V. 20. Parathrseic Kje grammik anexrthto snolo dianusmtwnenc qrou V mpore, en aut enai aparathto,na epektaje (me prosjkh llwn dianusmtwn)tsi ste na gnei bsh tou V. Kje snolo dianusmtwn pou pargei nanqro V mpore, en aut enai aparathto, nasurriknwje (afairntac diansmata) tsi stena gnei bsh tou V. 21. ParathrseicA241 3 0 2 10 0 1 4 31 3 1 6 435, U241 3 0 2 10 0 1 4 30 0 0 0 035 22. ParathrseicA241 3 0 2 10 0 1 4 31 3 1 6 435, U241 3 0 2 10 0 1 4 30 0 0 0 0351. O qroc grammn tou A tautzetai (ra qei thn diadistash r kai thn dia bsh) me ton qro grammn touU. 23. ParathrseicA241 3 0 2 10 0 1 4 31 3 1 6 435, U241 3 0 2 10 0 1 4 30 0 0 0 0351. O qroc grammn tou A tautzetai (ra qei thn diadistash r kai thn dia bsh) me ton qro grammn touU.2. O qroc sthln tou A den tautzetai me ton qrogrammn tou U. 24. ParathrseicA241 3 0 2 10 0 1 4 31 3 1 6 435, U241 3 0 2 10 0 1 4 30 0 0 0 0351. O qroc grammn tou A tautzetai (ra qei thn diadistash r kai thn dia bsh) me ton qro grammn touU.2. O qroc sthln tou A den tautzetai me ton qrogrammn tou U.3. O mhdenqwroc tou A tautzetai me ton mhdenqwro touU. 25. ParathrseicA241 3 0 2 10 0 1 4 31 3 1 6 435, U241 3 0 2 10 0 1 4 30 0 0 0 0351. O qroc grammn tou A tautzetai (ra qei thn diadistash r kai thn dia bsh) me ton qro grammn touU.2. O qroc sthln tou A den tautzetai me ton qrogrammn tou U.3. O mhdenqwroc tou A tautzetai me ton mhdenqwro touU. 26. Bseic kai diastseic jemeleiwdn upqwrwn tou A2 Rm,nqroc grammn Bsh: to snolo twn mh-mhdenikngrammn tou U. Distash: r (to pljoc twnodhgn tou A). 27. Bseic kai diastseic jemeleiwdn upqwrwn tou A2 Rm,nqroc grammn Bsh: to snolo twn mh-mhdenikngrammn tou U. Distash: r (to pljoc twnodhgn tou A).qroc sthln Bsh: to snolo twn sthln tou A pouantistoiqon se stlec tou U pou frounodhg. Distash: r. 28. Bseic kai diastseic jemeleiwdn upqwrwn tou A2 Rm,nqroc grammn Bsh: to snolo twn mh-mhdenikngrammn tou U. Distash: r (to pljoc twnodhgn tou A).qroc sthln Bsh: to snolo twn sthln tou A pouantistoiqon se stlec tou U pou frounodhg. Distash: r.mhdenqwroc Bsh: oi (grammik anexrthtec) lseic touomogenoc. Distash: nr. 29. Bseic kai diastseic jemeleiwdn upqwrwn tou A2 Rm,nqroc grammn Bsh: to snolo twn mh-mhdenikngrammn tou U. Distash: r (to pljoc twnodhgn tou A).qroc sthln Bsh: to snolo twn sthln tou A pouantistoiqon se stlec tou U pou frounodhg. Distash: r.mhdenqwroc Bsh: oi (grammik anexrthtec) lseic touomogenoc. Distash: nr.aristerc mhdenqwroc Bsh: to snolo twn grammn touL1 pou antistoiqon se mhdenikc grammc touU. Distash: mr. 30. Bseic kai diastseic jemeleiwdn upqwrwn tou A2 Rm,nqroc grammn Bsh: to snolo twn mh-mhdenikngrammn tou U. Distash: r (to pljoc twnodhgn tou A).qroc sthln Bsh: to snolo twn sthln tou A pouantistoiqon se stlec tou U pou frounodhg. Distash: r.mhdenqwroc Bsh: oi (grammik anexrthtec) lseic touomogenoc. Distash: nr.aristerc mhdenqwroc Bsh: to snolo twn grammn touL1 pou antistoiqon se mhdenikc grammc touU. Distash: mr.A LU)L1AU 31. PardeigmaA241 3 0 2 10 0 1 4 31 3 1 6 435 32. PardeigmaA241 3 0 2 10 0 1 4 31 3 1 6 435 !241 0 00 1 01 1 135241 3 0 2 10 0 1 4 30 0 0 0 035