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Γραmmική ΄Αλγεβρα Επανάληψη - Βάση και διάσταση χώρων Τmήmα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών Πανεπιστήmιο Θεσσαλίας 5 Νοεmβρίου 2014

18η διάλεξη Γραμμικής Άλγεβρας

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  • 1. Grammik 'AlgebraEpanlhyh - Bsh kai distash qrwnTmma Hlektrolgwn Mhqanikn kai Mhqanikn UpologistnPanepistmio Jessalac5 Noembrou 2014

2. Jerhma'Estw o mn pnakac A. Tte1. O mhdenqwrc tou A enai dianusmatikc upqwroc touRn. 3. Jerhma'Estw o mn pnakac A. Tte1. O mhdenqwrc tou A enai dianusmatikc upqwroc touRn.2. O aristerc mhdenqwrc tou A enai dianusmatikcupqwroc tou Rm. 4. Jerhma'Estw o mn pnakac A. Tte1. O mhdenqwrc tou A enai dianusmatikc upqwroc touRn.2. O aristerc mhdenqwrc tou A enai dianusmatikcupqwroc tou Rm.3. O qroc grammn tou A enai dianusmatikc upqwroctou Rn. 5. Jerhma'Estw o mn pnakac A. Tte1. O mhdenqwrc tou A enai dianusmatikc upqwroc touRn.2. O aristerc mhdenqwrc tou A enai dianusmatikcupqwroc tou Rm.3. O qroc grammn tou A enai dianusmatikc upqwroctou Rn.4. O qroc sthln tou A enai dianusmatikc upqwroctou Rm. 6. Jerhma'Estw o mn pnakac A. Tte1. O mhdenqwrc tou A enai dianusmatikc upqwroc touRn.2. O aristerc mhdenqwrc tou A enai dianusmatikcupqwroc tou Rm.3. O qroc grammn tou A enai dianusmatikc upqwroctou Rn.4. O qroc sthln tou A enai dianusmatikc upqwroctou Rm.5. To snolo twn lsewn tou Ax b den apoteledianusmatik upqwro tou Rn. 7. Epanlhyh diadikasac eplushcAn jsw lec tic elejerec metablhtc sec me 0 ekolaupologzw mia lsh s0 tou mh-omogenoc Ax b :Pardeigma:26666641 0 0 0 1 0 e0 0 0 1 p20 0 0 0 0 00 0 0 0 0 03777775 8. Epanlhyh diadikasac eplushcAn jsw lec tic elejerec metablhtc sec me 0 ekolaupologzw mia lsh s0 tou mh-omogenoc Ax b :Pardeigma:26666641 0 0 0 1 0 e0 0 0 1 p20 0 0 0 0 00 0 0 0 0 03777775Jtw x3 x5 0x1 0x2 00x4 0 0x1 x2 00x4 0 e0x1 0x2 0x4 0 p2 9. Epanlhyh diadikasac eplushcAn jsw lec tic elejerec metablhtc sec me 0 ekolaupologzw mia lsh s0 tou mh-omogenoc Ax b :Pardeigma:26666641 0 0 0 1 0 e0 0 0 1 p20 0 0 0 0 00 0 0 0 0 03777775Jtw x3 x5 0x1 0x2 00x4 0 0x1 x2 00x4 0 e0x1 0x2 0x4 0 p2x1 ,x2 e,x4 p2,x3 x5 0 enai mia lsh.Pc ja brome lec tic llec lseic? 10. H diafor do lsewn'Estw si mia opoiadpote llh lsh, opte Asi b. 11. H diafor do lsewn'Estw si mia opoiadpote llh lsh, opte Asi b.TteA(si s0) Asi As0 bb 0 12. H diafor do lsewn'Estw si mia opoiadpote llh lsh, opte Asi b.TteA(si s0) Asi As0 bb 0Dhlad to (si s0) s enai lsh tou omogenoc Ax 0. 13. H diafor do lsewn'Estw si mia opoiadpote llh lsh, opte Asi b.TteA(si s0) Asi As0 bb 0Dhlad to (si s0) s enai lsh tou omogenoc Ax 0.'Ara si s0 s. 14. H diafor do lsewn'Estw si mia opoiadpote llh lsh, opte Asi b.TteA(si s0) Asi As0 bb 0Dhlad to (si s0) s enai lsh tou omogenoc Ax 0.'Ara si s0 s.SumprasmaTo snolo twn lsewn tou Ax b isotai me to snolo twnlsewn tou Ax 0 sun mia opoiadpote lsh tou Ax b. 15. Ta snola twn lsewn Ax 0 and Ax b san uposnolatou RnAx = bs0 Ax = 0 16. PardeigmaMh-omogenc: Ax bx1 2x2 3x3 2x4 4x5 12x1 4x2 5x3 1x4 6x5 35x1 10x2 13x3 4x4 16x5 7 17. PardeigmaMh-omogenc: Ax bx1 2x2 3x3 2x4 4x5 12x1 4x2 5x3 1x4 6x5 35x1 10x2 13x3 4x4 16x5 7Omogenc: Ax 0x1 2x2 3x3 2x4 4x5 02x1 4x2 5x3 1x4 6x5 05x1 10x2 13x3 4x4 16x5 0 18. Uprqei lsh?Epauxhmnoc pnakac:241 2 3 2 4 12 4 5 1 6 35 10 13 4 16 735 19. Uprqei lsh?Epauxhmnoc pnakac:241 2 3 2 4 12 4 5 1 6 35 10 13 4 16 735L2 L2 2L1, L3 L3 5L1:241 2 3 2 4 10 0 1 3 2 10 0 2 6 4 235 20. Uprqei lsh?Epauxhmnoc pnakac:241 2 3 2 4 12 4 5 1 6 35 10 13 4 16 735L2 L2 2L1, L3 L3 5L1:241 2 3 2 4 10 0 1 3 2 10 0 2 6 4 235L3 L3 2L2:241 2 3 2 4 10 0 1 3 2 10 0 0 0 0 035 21. Uprqei lsh?Epauxhmnoc pnakac:241 2 3 2 4 12 4 5 1 6 35 10 13 4 16 735L2 L2 2L1, L3 L3 5L1:241 2 3 2 4 10 0 1 3 2 10 0 2 6 4 235L3 L3 2L2:241 2 3 2 4 10 0 1 3 2 10 0 0 0 0 035Uprqei lsh. 22. Upolgise mia lsh tou mh-omogenocElejerec metablhtc: x2,x4,x5. 23. Upolgise mia lsh tou mh-omogenocElejerec metablhtc: x2,x4,x5. Tic jtw sec me 0 kai lnw: 24. Upolgise mia lsh tou mh-omogenocElejerec metablhtc: x2,x4,x5. Tic jtw sec me 0 kai lnw:x1 4, x3 1. 'Ara mia lsh enais0 2666664401003777775. 25. Upolgise lec tic lseic tou omogenoc241 2 3 2 4 00 0 1 3 2 00 0 0 0 0 0352x1x2x3x4x5666664377777522x2 7x4 2x5666664x23x4 2x5x4x53777775 x22210006666643777775x42703106666643777775x52202016666643777775 26. 'Olec oi lseic tou omogenoc8>>>>>>>>>:321000777775,2703106666643777775,2666664 202019>>>>>=>>>>>;3777775Span{u,v}uv 27. 'Olec oi lseic tou mh-omogenoc24010066666437777758>>>>>>>>>:2210006666643777775,2703106666643777775,23666664 277777502019>>>>>=>>>>>;vuss + Span{u,v}00 28. Grammik Exrthshx y 29. Grammik Exrthshx yxk c1x1c2x2. . .cnxn 30. Grammik Exrthshx yxk c1x1c2x2. . .cnxn'Ena snolo dianusmtwn lgontai grammikexarthmna an to kajna apo aut mpore nagrafje san grammikc sunduasmc twn upolopwn. 31. 'Ena snolo dianusmtwn x1,x2, . . . ,xk 2 Rn lgontai grammikexarthmna ann uprqoun arijmo c1,c2, . . . ,ck 2 R ek twnopown toulqiston nac den enai mhdn kai gia toucopoouc isqei c1x1c2x2 . . . ,ckxk 0. 32. 'Ena snolo dianusmtwn x1,x2, . . . ,xk 2 Rn lgontai grammikexarthmna ann uprqoun arijmo c1,c2, . . . ,ck 2 R ek twnopown toulqiston nac den enai mhdn kai gia toucopoouc isqei c1x1c2x2 . . . ,ckxk 0.Gia na elgxoume thn exrthsh twn x1,x2, . . . ,xk 2 Rn Sqhmatzoume ton pnaka A opooc qei san stlec tadiansmata aut Upologzoume ton mhdenqwro tou AAn autc perilambnei mnon to mhdenik dinusma toteaut enai grammik anexrthta. 33. OrismoEn nac dianusmatikc qroc V apoteletai aplouc touc grammikoc sunduasmoc twndianusmtwn v1,v2, . . . ,vk tte lme ti autpargoun ton V. 34. OrismoEn nac dianusmatikc qroc V apoteletai aplouc touc grammikoc sunduasmoc twndianusmtwn v1,v2, . . . ,vk tte lme ti autpargoun ton V.'Ena snolo dianusmtwn pargei nandianusmatik qro ann kje dinusma tou qroumpore na grafje san grammikc sunduasmc twnen lgw dianusmtwn. 35. Orismo'Ena snolo dianusmtwn apotele bsh encdianusmatiko qrou ann aut enai grammikanexrthta kai pargoun ton qro. 36. Orismo'Ena snolo dianusmtwn apotele bsh encdianusmatiko qrou ann aut enai grammikanexrthta kai pargoun ton qro.Distash enc upoqrou enai to pljoc twnstoiqewn thc bshc tou.