Linear Algebra. Lecture 1 September 19, 2011

Transcription

1 Lier Algebr Lecture September 9,

2 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems. Vectors d Mtrices

3 Course iformtio Istructor: Professor Gwobo Horg Tetbook Lier Algebr : Theory d Applictios, Wrd Cheey & Dvid Kicid, Joes d Brtlett st ed. 9 (76 pges) / d ed. (648 pges) Errt List:

4 Course iformtio Course web pge schedule.htm Pssword: ilb 4

5 Course iformtio Grdig (Tettive) Homework/Quiz % (You my collborte whe solvig the homework. However, whe writig up the solutios you must do so o your ow. Hdwritte oly.) Midterm em % Fil em % Clss prticiptio % 5

6 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems 6

7 Purposes of the course To mke the studets become fmilir with the bsic cocepts of lier lgebr. Uderstd mtrices, vector spces, lier trsformtios. To ehce the studets' bility to reso mthemticlly. Uderstd proofs, bstrct otios. To mke the studets wre of the crucil importce of lier lgebr to my fields i egieerig, sttistics d computer sciece. Nolier mthemtics re hrd. Lieriztios re good pproimtios. 7

8 Some pplictio res Computer etwork Network flow Computer grphics Trsformtios of the ple Codig theory Error-correctig codes Cryptogrphy Hill cipher More pplictios 8

9 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems 9

10 Outlie of the course (/) Chpter. Systems of Lier Equtios. Solvig Lier Systems. Vectors d Mtrices. Homogeeous Lier Systems Chpter. Vector Spces d Trsformtios. Euclide Vector Spces. Lie, Ples, d More. Lier Trsformtios.4 Geerl Vector Spces Chpter. Mtri Opertios. Mtrices. Mtri Iverses Chpter 4. Determits 4. Determits: Itroductio 4. Determits: Properties d Applictios

11 Outlie of the course (/) Chpter 5. Vector Subspces 5. Colum, Row, d Null Spces 5. Bses d Dimesio 5. Coordite Systems Chpter 6. Eigesystems 6. Eigevlues d Eigevectors Chpter 7. Ier Product Vector Spces 7. Ier Product Spces 7. Orthogolity Chpter 8. Additiol Topics 8. Hermiti Mtrices d Spectrl Theorem 8. Mtri Fctoriztios d Block Mtrices 8. Itertive Methods

12 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems

13 Algebric structures A lgebric structure cosists of oe or more sets, closed uder oe or more opertios, stisfyig some ioms. Group-like structures Z uder dditio () is beli group. Field Ech of Q, R, d C, uder dditio d multiplictio, is field.

14 Wht is Lier & Algebr? y Cosider lie through the origi A directed rrow from the origi (v) o the lie, whe scled by costt (c) remis o the lie Two directed rrows (u d v) o the lie c be dded to crete loger directed rrow (u v) i the sme lie. y v u cv v u v This is othig but rithmetic with symbols! Algebr : geerliztio d etesio of rithmetic. Lier opertios: dditio d sclig. Cotiued i the et slide. 4

15 Wht is Lier & Algebr? y Abstrct d Geerlize! Lie vector spce hvig N dimesios Poit vector with N compoets i ech of the N dimesios (bsis vectors). Vectors hve: Legth d Directio. Bsis vectors: sp or defie the spce & its dimesiolity. Lier fuctio trsformig vectors mtri. The fuctio cts o ech vector compoet d scles it Add up the resultig scled compoets to get ew vector! I geerl: f(cu dv) cf(u) df(v) cv v y v u u v 5

16 Wht is lier lgebr? A brch of mthemtics tht studies vectors, mtrices, vector spces, d systems of lier equtios (p. ) Vectors d mtrices c produce systems of lier equtios Systems of lier equtios c ofte model pplied problem from the rel world 6

17 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems. Vectors d Mtrices 7

18 Some dvise Tke otes Red i dvce Do eercises Mke use of the web resources 8

19 Lier equtios Emple 7 -y A sigle lier equtio otiig two vribles The word lier derives from the word lie. (,-7) 9

20 Lier equtios The poit-slope form of lie is y m b where m is the slope d b is the itercept o the y-is. Emple : 7 -y y 7/ 7 For two poits (,y ) d (,y ), the two-poit form of the lie through these poits is y - y m (- ) where m(y -y )/( - )

21 Lier equtios Eted sigle lier equtio cotiig two vribles to system of m lier equtios cotiig vribles (ukows)

22 Lier equtios A lier equtio i the vribles b,,, d b re rel costts,,, re clled vribles d sometimes clled ukows which do ot ivolve y products or roots of vribles. Tht is, ll vribles occur oly to the first power d do ot pper s rgumets for logrithmic, trigoometric, or epoetil fuctios.

23 Emple: Emple: Lier equtios y y si 7, 8z 7, 9 4 Nottio: vribles:, y vribles:, y, z vribles:,,, 4 y 5, yz z 4, d y si

24 Solvig Systems of Lier Equtios A solutio of lier equtio is sequece of umbers s, s,, s such tht the equtio is stisfied. The set of ll solutios of the equtio is clled its solutio set. Emple: Fid the solutio to Solutio: 4 y we ssig rbitrry vlue to d solve for y, or choose rbitrry vlue for y d solve for. t, y solutio 4t set {( t, or 4t ) t t, y t 4 t t R} or {(, 4 t) t R} rbitrry umber t is clled prmeter. 4

25 5 Solvig Systems of Lier Equtios system of lier equtios (lier systems) A system of m lier equtios i ukows m m m m b b b A sequece of umbers s, s,, s is clled solutio of the system if m m m m b s s s b s s s b s s s

26 Solvig Systems of Lier Equtios Defiitio A system of equtios is cosistet if it hs t lest oe solutio, d icosistet if it hs o solutio. Every system of lier equtios hs either o solutios, ectly oe solutio, or ifiitely my solutios. Figure. Differet cses of two lies i R². 6

27 7 Solvig Systems of Lier Equtios to solve lier system Emple: z y z y z y z y z y z y dd - times the first equtio to the secod dd - times the first qutio to the third dd - times the first row to the secod dd - times the first row to the third Cotiued i the et slide.

28 Solvig Systems of Lier Equtios y z 9 9 y 7z y z 7 7 multiple the secod equtio by / multiple the secod row by / y y z 7 y z dd - times the secod equtio to the third z dd - times the secod row to the third Cotiued i the et slide. 8

29 9 Solvig Systems of Lier Equtios z z y z y z z y z y multiple the third equtio by - dd - times the secod equtio to the first multiple the third equtio by - dd - times the secod equtio to the first Cotiued i the et slide.

30 Solvig Systems of Lier Equtios z y z z y z dd -/ times the third equtio to the third d 7/ times the third equtio to the secod solutio dd -/ times the third equtio to the third d 7/ times the third equtio to the secod 7 7 5

31 Solvig Systems of Lier Equtios Mtri form of Lier System m m m m b b b m m m m b b b m m m m m b b b m m m m b b b [A b] ugmeted mtri A b A: coefficiet mtri : ukow vector b: righthd side Cotiued i the et slide.

32 Solvig Systems of Lier Equtios m m m m b b b m m m m b b b Elemetry Row Opertios reduced row-echelo form or row-echelo form

33 Solvig Systems of Lier Equtios Elemetry Row Opertios. Multiply equtio through by ozero costt. i m i m i m ki m k i m k i m Cotiued i the et slide.

34 4 Solvig Systems of Lier Equtios. Add multiple of oe equtio to other. m m m j i j i j i i i i m m m j j j i i i k k k

35 5 Solvig Systems of Lier Equtios. Iterchge two equtio. m m m i i i j j j m m m j j j i i i Cotiued i the et slide.

36 Elemetry Row Opertios Three types of elemetry row opertios:. (scle) Multiple row by ozero fctor.. (replcemet) Add multiple of oe row to other.. (swp) Iterchge pir of rows. Let r i d r j be two rows.. r i kr i (scle k ). r j r j kr i (i j, k is scle). r i r j Note tht the third type of opertio is redudt (sice it c be chieved by sequece of opertios of the first two types). 6

37 Elemetry Row Opertios 7

38 Elemetry Row Opertios 8

39 Elemetry Row Opertios Whe we trsform system by elemetry row opertios, we do ot itroduce spurious solutios or lose geuie solutios. We c dd equl qutities to equl qutities to obti further equlities, d the process c be reverse. 9

40 Equivlet Lier Systems Defiitio (Equivlet Lier Systems) Two lier systems re sid to be equivlet if oe c be obtied from the other by fiite umber of elemetry row opertios. THEOREM Two equivlet systems hve the sme set of solutios. 4

41 Equivlet Lier Systems LEMMA Let C d be the lier system obtied from the lier system A b by sigle elemetry row opertio. The the lier systems A b d C d hve the sme set of solutios. 4

42 Equivlet Lier Systems 4

43 Solvig Systems of Lier Equtios row-echelo form (ref) A mtri is i row-echelo form if it stisfies. All zero rows hve bee moved to the bottom.. The ledig ozero elemet i y row is fther to the right th the ledig ozero elemet i the row just bove it.. I ech colum cotiig ledig ozero elemet, the etry below tht ledig ozero elemet re o. 4

44 Solvig Systems of Lier Equtios reduced row-echelo form (rref) A mtri is i reduced row-echelo form if it stisfies. does ot cosist etirely of zeros. the the first ozero umber i the row is. We cll this ledig or pivot. If there re y rows tht cosist etirely of zeros, the they re grouped together t the bottom of the mtri.. ech row ot ll zeros 44

45 Solvig Systems of Lier Equtios. I y two successive rows tht do ot cosist etirely of zeros, the ledig i the lower row occurs frther to the right th the ledig i the higher row.. 4. Ech colum tht cotis ledig hs zeros everywhere else.. 45

46 Emple: reduced row-echelo form 46

47 47 Solvig Systems of Lier Equtios Emple: row-echelo form: 6,, Emple: reduced row-echelo form:, 7,. If row does ot cosist etirely of zeros the the first ozero umber i the row is.. If there re y rows tht cosist etirely of zeros, the they re grouped together t the bottom of the mtri.. I y two successive rows tht do ot cosist etirely of zeros, the ledig i the lower row occurs frther to the right th the ledig i the higher row. 4. Ech colum tht cotis ledig hs zeros everywhere else

48 48 Solvig Systems of Lier Equtios Emple: row-echelo form,,,, Emple: reduced row-echelo form Theorem (will be proved lter). Every mtri hs oe d oly oe reduced row echelo form.

49 RREF vs REF 49

50 5 Solvig Systems of Lier Equtios Gussi Elimitio Emple: Augmeted Mtri row-echelo form Cotiued i the et slide.

51 5 Solvig Systems of Lier Equtios ,, 6 : ledig vribles, 4, 5 : free vribles - Substitutig 6 / ito the d equtio Cotiued i the et slide.

52 5 Solvig Systems of Lier Equtios Substitutig 4 ito the st equtio Assig free vribles, the geerl solutio is give by the formuls ,,,,, 4 t s s r t s r

53 5 Solvig Systems of Lier Equtios Gussi-Jord Elimitio Emple: Augmeted Mtri reduced row-echelo form 4 Cotiued i the et slide.

54 54 Solvig Systems of Lier Equtios ,,,,, 4 t s s r t s r

55 Algorithm for the reduced row Use blckbord echelo form (p. 6) 55

56 Importt Terms Pivot positio: positio of ledig etry i echelo form of the mtri. Pivot: ozero umber tht either is used i pivot positio to crete s or is chged ito ledig, which i tur is used to crete s. Pivot colum: colum tht cotis pivot positio. 56

57 Algorithm for the reduced row echelo form Row reduce the mtri A below to echelo form d locte the pivot colums of A. A Step : Iterchge rows if ecessry to plce ll zero rows o the bottom. 57

58 Algorithm for the reduced row echelo form Step (): Begi with the leftmost ozero colum. This is pivot colum. The pivot positio is t the top. Pivot Positio Pivot Colum Cotiued i the et slide. 58

59 Algorithm for the reduced row echelo form Step (b): Select ozero etry i the pivot colum s pivot. If ecessry iterchge rows to move this etry ito the pivot positio. Pivot Positio R R 4 Pivot Colum Cotiued i the et slide. 59

60 Algorithm for the reduced row echelo form Step (b): Select ozero etry i the pivot colum s pivot. If ecessry iterchge rows to move this etry ito the pivot positio. Pivot Pivot Colum Cotiued i the et slide. 6

61 Algorithm for the reduced row echelo form Step (c): Use elemetry row opertios to crete zeros i ll positios below the pivot. Pivot R R R R R R Cotiued i the et slide. 6

62 Algorithm for the reduced row echelo form After few computtios we get Cotiued i the et slide. 6

63 Algorithm for the reduced row echelo form Step : Cover (or igore) the row cotiig the pivot positio d cover ll rows, if y, bove it. Apply steps - to the remiig submtri. Repet the process util there re o more ozero rows to modify Possible R Pivots R 5 R R R R Cotiued i the et slide. 6

64 Algorithm for the reduced row echelo form R 5 R R 4 4 R R R Cotiued i the et slide. 64

65 Algorithm for the reduced row echelo form R R Pivot Colums Cotiued i the et slide. 65

66 Algorithm for the reduced row echelo form Step 4: Begiig with the rightmost pivot d workig upwrd d to the left, crete zeros bove ech pivot. If pivot is ot, mke it by sclig opertio R 5 R Cotiued i the et slide. 66

67 Algorithm for the reduced row echelo form R R 9R R R R Step 5: Repet step 4, edig with the uique reduced row echelo form of the give mtri. Cotiued i the et slide. R R 67

68 Algorithm for the reduced row echelo form R R R 4 5 Reduced Row Echelo Form 68

69 More emples TRANSFORMING MATRIX TO ROW ECHELON FORM emple-.htm TRANSFORMING MATRIX TO THE REDUCED ROW ECHELON FORM emple-.htm 69

70 Outlie Course iformtio Motivtio Outlie of the course Wht is lier lgebr? Chpter. Systems of Lier Equtios. Solvig Lier Systems. Vectors d Mtrices 7

71 Vectors d Mtrices vectors [,,, ] T (,,, ). compoet i : ith etry zero vector: (,,, ) vector dditio: (,,, ), y (y, y,, y ) y ( y, y,, y ) (Figure.) sclr multiplictio: (,,, ) (Figure.4) -spce R {(,,, ) i R} 7

72 Vectors d Mtrices Figure. Additio of pirs of vectors i R². 7

73 Vectors d Mtrices Figure.4 Sclr multiples of vectors i R². 7

74 Vectors d Mtrices lier combitios of vectors A vector w is lier combitio of the vectors v, v,, v if it c be epressed i the form w k v k v k v, where k i is sclr. Emple: Emple (p. 44/6): Every poit i R is lier combitio of e (, ) d e (, ). Reso: for y poit (, y) (, ) y(, ) 74

75 Vectors d Mtrices Emple (p. 45/6): Is every poit i R lier combitio of (5, ) d (7, ). Solutio: for y poit b (b, b ), we try to solve the equtio (5, ) y(7, ) (b, b ) 5 7 b b ~ b b 7b 5b b -7b y -b 5b 75

76 Vectors d Mtrices lier combitios of vectors A vector w is lier combitio of the vectors v, v,, v if it c be epressed i the form w k v k v k v, where k i is sclr. Emple (p. 45/7): Is the w (-,, 7) lier combitio of (4,,7) d (,, 4)? to solve the lier system (4,, 7) y(,, 4) (-,, 7) 4 y - y 7 4y 7 76

77 emple-.htm 77

78 Vectors d Mtrices 78

79 Vectors d Mtrices 79

80 Vectors d Mtrices 8

81 Vectors d Mtrices 8

82 Vectors d Mtrices Defiitio Let S {v, v,, v }. The set of ll lier combitios, deoted s sp(s), of set of vectors is clled the sp of S. Emple: (Emple d Emple ) R sp({(, ), (,)}) sp({(5, ), (7, )}) Emple 7 (p. 49/4): Is (4, 6, 76) i the sp of (,, ), (,, 4), d (7, -4, )? solutio: The questio is whether solutio eists for the equtio b 4 c 7 4 Use progrm. emple-.htm 8

83 Vectors d Mtrices Emple 6 (p. 49/4): Give simple descriptio for the sp({(4,, 7), (,, 4)}). solutio: for y (b, b, b ) try to solve for sclrs d y i (4,, 7) y(,, 4) (b, b, b ) b b b ~ b 4b b b b 5b b for the cosistecy, we require b 5b -b sp({(4,, 7), (,, 4)}) {(b, b, b ) b 5b -b } By simple descriptio, we me simple test tht c be pplied to vector to determie whether it is or is ot i the sp of give set. 8

84 Iterpretig lier systems m m m b b b m. m. m. m b b. b m A m m m. b b b. b m A b is cosistet b is lier combitio of. m,,,. m. m 84

85 85 Iterpretig lier systems Equivlet forms of A b (pp. 5/4) () mtri form A b () compct summtio, i m () lier equtios i complete detil j i j ij b m m m m b b b Cotiued i et slide.

86 Iterpretig lier systems (4) mtri with vectors (5) ugmeted mtri m m m m b b b m (6) lier combitio of the colums of A m m m b b b m. m. m. m b b. b m (7) lier combitio of the colum vectors of A 86

87 row equivlet systems Defiitio (p. 5/44) Two mtrices re row equivlet to ech other if ech c be obtied from the other by pplyig sequece of elemetry (permitted) row opertios. Emple: r (-)r r r (/)r r r 5 87

88 row equivlet systems 88

89 row equivlet systems Emple: 5 7 ~ 5 4 ~ 5 ~ 5 ~ ~ 4 5 y 7 y y Emple 8 (p. 5/45) Use blckbord. 89

90 Emple 8 9

91 Emple 8 9

92 Emple 8 9

93 Vectors d Mtrices The sp of the set of colums i mtri A is clled the colum spce of A d is writte Col(A). If A is m, the the sp of the set of its colums is Col(A) Sp{,,, } {A : R } 9

94 Vectors d Mtrices Theorem (p. 56/48). A system of lier equtios A b is cosistet if d oly if the vector b is i the sp of the set of colums of A. 94

95 95 Vectors d Mtrices Theorem (p. 56/48). Let A be m mtri. The system of equtios A b is cosistet for ll b i R m if d oly if the colums of A sp R m. Tht is, col(a) R m. m m m m b b b m m m m b b b,,

96 Vectors d Mtrices Theorem 4 (p. 56/48). Let A be m mtri. The system of equtios A b is cosistet for ll b i R m if d oly if ech row of the coefficiet mtri A hs pivot positio. 96

97 97 Vectors d Mtrices Theorem 5 (p. 57/48). A system of lier equtios is icosistet if d oly if its ugmeted mtri hs pivot positio i the lst colum.

98 Vectors d Mtrices A resttemet of Theorem 5 Theorem 6 (p. 58/5). A system of lier equtios is cosistet if d oly if the reduced row echelo form of its ugmeted mtri does ot hve pivot positio i the lst colum. 98

99 Questios? 99