Elementi di logica Algebra lineare - memotef.uniroma1.it · 4 i mp lic azio n e log i ca Una fr as e s c ritta nella forma Se (pro p osizion e 1) allo ra (prop o sizione 2) Costitui

1

esab osroc acitametaMa.a 102. 41/3

acigol id itnemelEeraenil arbeglA

2

osroc led OVITTEIBO

ilitu icitametam itnemurts erisiuqcAl rep ’ id enoizulos al rep e isilana

itercnoc imelborp

acitametam aL è oiggaugnil nu a atuia ehc ougibma non e osorogir

isselpmoc imelborp ius eranoigar

3

acigol id itnemelE

anU enoizisoporp è is elauq alled esarf anutiugibma aznes erediced òup à es è o arev

è .aslaf

:ipmesE “ angaval al è aren ” )orev(“ angaval al è eralognairt ” )oslaf(“ T è olognatter olognairt nu ”

)odnopsir e T odraug(

4

acigol enoizacilpmi

amrof allen attircs esarf anUeS )1enoizisoporp( arolla )2enoizisoporp(

anu ecsiutitsoC acigol enoizacilpmi

5

ilG itaicnune onos imeroet iedehcigol inoizacilpmi

)2enoizisoporp( arolla )1enoizisoporp( eS

isetopi iset

enoizidnoc enoizidnocetneiciffus airassecen

6

e isetopi eraibmacs onossop is eSiset

ehc ecid is arollA• 1 enoizisoporP è e airassecen enoizidnoc

2 enoizisoporP rep etneiciffus

• 2 enoizisoporP è e airassecen enoizidnoc1 enoizisoporP rep etneiciffus

7

oipmesE )arogatiP id ameroeT(

• T( es è )olognatter olognairt nu• itardauq ied eera elled ammos al( arolla

itetac ius itiurtsoc è lla elaugu ’ led aerallus otiurtsoc otardauq ’ )asunetopi

8

ip imeroet eraredisnoc onossop iS ùisselpmoc

oipmesEeS 3 enoizisoporp e 2 enoizisoporp e 1enoizisoporp( … )

arolla 5 enoizisoporp e 4enoizisoporp( …)

osac otseuq nI• e 2 enoizisoporp e 1enoizisoporp(= isetopI

3 enoizisoporp …)• =iseT 5 enoizisoporp e 4enoizisoporp( …)

9

erartsomiD ameroet nu

erartsomiD al eravort erid louv ameroet nu etrap ehc ehcigol inoizacilpmi id anetac

iset alla eravirra id ettemrep e isetopi ellad rep( acigol enoizacilpmi id eloger etnaidem

etarcoS( imsigollis i oipmese è ingo ,omou omou è etarcoS ,elatrom è ))elatrom

01

ilasrevinu irotacifitnauq I• ingo reP etsisE enoizageN

id enoizagen allus enoizavressO “ ingo rep ” :oipmesE

esarf al eragen “ ehcnaib onos erocep el ettut ”:

non “ ehcnaib onos erocep el ettut ”acnaib non arocep anu etsise

al acifirev non ehc otnemele nu onemla atsab idniuQteirporp à teirporp al ehc erid rep atangessa à non è

.ittut rep atacifirev

11

eraenil arbeglA

• )76.p( irotteV nU erottev è anu n- alp ilaer iremun id atanidro

.elocsunim enital erettel noc acidni iS

321 ,...,,, naaaaa

Sn= n enoisnemid id irottev ied oizaps

21

• :oipmesE 4=n

• titnauQ à idenihccuz ,irodomop ,erep ,elem

5.,5.1,2,1a

31

)azzehgnul assets( irottev eud id ammoS

, ,…, )( ,

321 ,...,,, naaaaa

321 ,...,,, nbbbbb

332211 ,...,,, nn bababababa

41

5.,...,5.1,2,1a

2.,...,.1,7.,3b

7.,...,5.2,7.2,4ba

oipmesE

51

0,...,0,0,00

00 aaa

ollun erottev lI

enoizavressO :

61

321 ,...,,, naaaaa

5.*3,...,5.1*3,2*3,1*33

5.1,...,5.4,6,3

a

=iralacS ilaer iremun5141.3 ,1. ,3 …

ehcerg erettel noc onacidni iS α , β , γ, …

eralacs onu rep erottev nu id ottodorP

oipmesE

71

irottev ied eraenil enoizanibmoC a e b noc itneiciffeoc α e β

321

321

332211

,...,,,

,...,,,

,...,,,

n

n

nn

baaaaa

bbbb

babababa

81

oipmesE

5.1,5.4,6,323

4.,2,4.1,6

9.1,5.6,4.7,9

ba

91

:enoizazzilareneG irottev ied eraenil enoizanibmoc a1 , a2 , … , am noc

itneiciffeoc α1 , α2 , … , αm

21 21 ... mm aaab

b irottev iad etnednepid etnemraenil ecid is a1 , a2 , … , am

:enoizinifeD

02

oipmese eroiretlU irottev eud i itaD e1 )0,1(= de e2 )1,0(=

ehc avresso is

- erottev lI b= )3,2( ad eneitto is 2 e1 3+ e 2 ittafni , *2 e1 *3+ e2 )3,2(=)3,0(+)0,2(=)1,0(*3+)0,1(*2=

- erottev ortla isaislauq ehc ehcna ovressO a= a( 1 a, 2) eneitto is ad e1 de e2 :elimis areinam ni

a1 * e1 +a2 * e2 =a1 +)0,1(* a2 ( =)1,0(* a1 ,0(+)0, a2 (=) a1,a2)

- etnemraenil onos itnenopmoc eud noc irottev i ittut idniuQ ad itnednepid e1 de e2.

12

:enoizazzilareneG irottev ied eraenil enoizanibmoc a1 , a2 , … , am noc

itneiciffeoc α1 , α2 , … , αm

21 21 ... mm aaab

b irottev iad etnednepid etnemraenil ecid is a1 , a2 , … , am

irottev id emeisni nU a1 , a2 , … , am etnemraenil ecid is isse id onu onemla es etnednepid è etnemraenil

.itnenamir iad etnednepid

:enoizinifeD

:enoizinifeD

22

:aigolonimreTitnednepidni = itnednepid noN

irottev id emeisni nU a1 , a2 , … , am etnemraenil ecid is es etnednepid onu onemla isse id è etnemraenil


irottev id emeisni nU a1 , a2 , … , am etnemraenil ecid isNI etnednepid es ONUSSEN isse id è etnemraenil


32

erottev id oipmesE b irottev iad etnednepidni

a1 de a2 :

b= ,)3,2( a1 ,)1,0(= a2 )2,0(=

eneitto is b id etnenopmoc adnoces al ittafnIid itnenopmoc II evittepsir el odnammos a1 de a2 am ,

.orez erpmes enamir itnenopmoc emirp elled ammos al

ehc erid reP b , a1 de a2 id emeisni nu onocsiutitsoc non ehc eracifirev ehcna angosib itnednepidni irottev

erevircs òup is a1 id eraenil enoizanibmoc emoc b dea2 erevircs òup is non ehc e a2 enoizanibmoc emoc

id eraenil b de a1.

42

de aznednepid onazzirettarac is emoC odnauq aznednepidni b è e ollun erottev li

irottev irtla ilg a1 , a2 , … irottev onos id oremun ossets ol noc( isaislauq

)itnenopmoc

52

es :enoizavressO α1 = α 2 = αm arolla 0=

21 00...00 maaab

orevvO ?asreveciv li elav :adnamoD orez a ilaugu itneiciffeoc i ittut erroP è l’ ocinu

?ollun erottev li erenetto rep odom)2,2(*1 :oipmesE - )0,0(=)1,1(*2

ehcna ollun erottev li erenetto òup is isac inucla ni :atsopsiR noc.illun non itneiciffeoc

non idniuQ è l ehc orev erpmes ’ ollun erottev li erenetto id odom ocinuè .orez a ilaugu itneiciffeoc i errop id

i ittut odnenop eneitto is ollun erottev li idniuQ.orez a ilaugu itneiciffeoc

62

:ameroeT irottev i a1 , a2 , … , am etnemraenil onosl es olos e es itnednepidni ’ noc ,erenetto rep odom ocinu

ollun erottev li ,eraenil enoizanibmoc orol anu è id olleuq.orez a ilaugu itneiciffeoc i ittut erednerp

l iuc ni osac lI ’ ollun erottev li erenetto id odom ocinu è orez a ilaugu itneiciffeoc i ittut errop id è etnematterts.aznednepidni alla otagelloc

ip omailosi noN ù irottev ia ottepsir b a1 , a2 am , art irottev ied onu emoc olomairedisnoc a1 , a2, … , am.

72

ilga odnassap ehcna elav aznelaviuqe al odnaredisnoc ,itsoppo

id inoizagen el etnemaenaropmetnoc.inoizisoporp el ebmartne

1 enoizisoporP 2 enoizisoporP

alled enoizageN1 enoizisoporP

alled enoizageN2 enoizisoporP

es olos e es

es olos e es

82

:ameroeT irottev i a1 , a2 , … , am

etnemraenil onositnednepidni

enoizagen

etnemraenil onositnednepid

etneiciffeoc nu onemla è.orez ad osrevid

21 21 0... mm aaa

onos itneiciffeoc i ittut.orez a ilaugu

es olos e es

es olos e es

92

:ameroeT irottev i a1 , a2 , … , am etnemraenil onos enoizanibmoc anu etsise es olos e es itnednepid

ied onu onemla noc irottev m ilged eraenil otatlusir emoc ah ehc ,orez ad osrevid itneiciffeoc

.ollun erottev li

etnedecerp ameroet led enoizartsomid aL:etneuges led enoizartsomid alla elaviuqe

03

:enoizartsomiD ied onu onemla ehc ognoppus :isetopI

.orez ad osrevid ais itneiciffeoc ol non es :omirp li ais ehc ognoppuS è oibmacs

ad idniuQ .imon i

13

21 21 0mm aaa

21 21 mm aaa

21 21 mm aaa

221

11

mmaaa

ehc ah is

ehc idniuq e

idniuq e

23

1a

},,,,{ 321 maaaa

otnatrep e id enoizinifed al acifirev

è hcrep etnednepid é orol id onu onemla è .itnenamir iad etnednepid

iset aL è .atartsomid idniuq

l idniuQ ’ emeisni

eraenil aznednepid

33

l erartsomid reP ’ orrocrepir asrevni enoizacilpmi.osortir a iggassap i

43

:ameroeT irottev i a1 , a2 , … , am etnemraenil onosl es olos e es itnednepidni ’ noc ,erenetto rep odom ocinu

ollun erottev li ,eraenil enoizanibmoc orol anu è id olleuq.orez a ilaugu itneiciffeoc i ittut erednerp

53

:enoizinifeD irottev id emeisni nu otaD a1 , a2 , … , am oizaps omisedem led Sn ognar li è oremun omissam li

.emeisni elat ni itneserp itnednepidni irottev id

63

:ameroeT oizaps onu ni Sn è n eravort elibissop erpmesip o )1+n( am ,itnednepidni etnemraenil irottev ù irottev

itnednepid etnemraenil erpmes onos

73

321 },,,,{ neeee

1 )0,,0,0,1(e

2 )0,,0,1,0(e

:oizicresE l otad ’ id emeisni n irottev

…

)1,,0,0,0(ne

83

321 ,,,, n

21 21 0mm eee

oic ,itnednepidni onos erartsomid è nu etsise ehciralacs id emeisni ocinu

ehc elat

93

21 21

1111

2222

0*,...,0*,0*,1*

0*,...,0*,1*,0*...

1*,...,0*,0*,0*

mm

nnnn

eee

04

1

2

3

0,...,0,0,

0,...,0,,0...

0,...,,0,0

,...,0,0,0 n

321 0,...,,, n

14

ICIRTAM.1 icirtaM )57.p(

ecirtam anU mxn è .ennoloc m de ehgir n noc allebat anu.elocsuiam enital erettel noc acidni iS

opit led ilobmis I a ji l onacidni ’ .j annoloc ,i agir allus otnemele

mnnn

m

m

m

aaa

aaaaaaaaa

A

21

32313

22212

12111

eS m=n .n enidro id e atardauq ecid is ecirtam al

24

3x2 ecirtam

)4.n ,15 .p(3x3 ecirtam

)3.n ,15 .p(2x2 ecirtam

1x1 ecirtam

ipmesE:iciremun

:ipmesE

11aA 5A

2212

2111

aaaa

A4531

A

332313

322212

312111

aaaaaaaaa

A

212101231

A

322212

312111

aaaaaa

A262

131A

34

itnanimreteD )77.p(

olobmis li noc acidni iS ted .|A| eruppo ,)A(

eraloclac òup iS etardauq icirtam rep olos .

44

2x2 ecirtam

→

led oloclac li :enoiznettA NON etnanimreted è oloclac li

otulossa erolav led

1x1 ecirtam

:iciremun ipmesE:ipmesE

11|| aA

5A 55|| A

2212

2111

aaaa

A

122122112212

2111 aaaaaaaa

A

4531

A

11514)5(*)3(4*14531

A

54

ecirtam 3x3

332313

322212

312111

aaaaaaaaa

A212101231

A

64

id alogeR surraS :)77.p( al evircsir is eud emirp el onovircsir is otnaccA .ecirtam

:ennoloc

2313332313

2212322212

2111312111

aaaaaaaaaaaaaaa

122120110131231

74

ilanogaid el onaiccart iS

2313332313

2212322212

2111312111

aaaaaaaaaaaaaaa

122120110131231

84

itnemele ilged ittodorp i erammos a onaizini is e :elanogaid assets al ottos onodac ehc

231231133221332211 aaaaaaaaa8260

)1(*1*22*1*)3(2*0*1

94

ertla el iop onaiccart iS ilanogaid :

2313332313

2212322212

2111312111

aaaaaaaaaaaaaaa

122120110131231

05

ehc itnemele ilged ittodorp i onoggarttos is e:elanogaid assets al ottos onodac

132231233211331221 aaaaaaaaa7016

2*0*2)1(*1*12*1*)3(

15

A id etnanimreted lI è :ad otad 178A

132231233211331221231231133221332211 aaaaaaaaaaaaaaaaaaA

25

A id acitsirettaraC )18.p(

olobmis li noc acidni iS rac òup iS .)A( .isaislauq icirtam rep eraloclac

A id eronim nU è anu id etnanimreted li ilg odneilgecs A ad atunetto ecirtamottos

.ennoloc k e ehgir k a enumoc ni itnemele

35

led enidro id emon li ednerp k oremun lI.eronim

A ecirtam anu id acitsirettarac aL è l’ enidro otseuQ .illun non ironim ious ied omissam

ehc A id ironim i eravort angosib ehc acifingis id onu enraredisnoc e orez ad isrevid onatlusir

ip enidro ù .elibissop ednarg

45

oipmesE al eravorT )a.1.3.3.n,26.p( :1:ecirtam etneuges alled acitsirettarac

262131

A

55

A è idniuq ,ennoloc 3 e ehgir 2 noc ecirtam anuip la etardauq icirtamottos errartse ossop ù id

:2 enidro

65

2,1:ehgir 3,2:ennoloc

2,1:ehgir3,1:ennoloc

2,1:ehgir2,1:ennoloc

:oloclac il e 2 enidro id ironim i oggartse :A id

6231

1M066

)3(*)2(6*1

2211

2M022

)2(*1)2(*1

2613

3M066

6*1)2(*)3(

75

teirporp( :enoizavressO à )2T ah ecirtam anu es ilanoizroporp ennoloc eud eruppo ehgir eud

etnanimreted ous li arolla è .ollunhcioP é 2 enidro id ironim ied onussen è osrevid

:orez ad

85

2 :agir3:annoloc

2 :agir 2:annoloc

2 :agir 1:annoloc

1 :agir 3:annoloc

1 :agir 2:annoloc

1 :agir1:annoloc

:oloclac il e 1 enidro id ironim i oggartse:A id

14M

35M

16M

27M

68M29M

1

31

26

2

1

95

al eravorT )d.1.3.3.n,26.p( :2 oipmesEecirtam alled acitsirettarac

242111121

A

:ironim enoizartsE

l ,1,2,3 enidro id ironim errartse onossop is ’ enidro al ecsinrof ollun non eronim omirp led

.acitsirettarac

06

242111121

1M

422421111121121

0244442)2(*1*14*1*1)2(*1*)2(

4*1*1)2(*1*)2()2(*1*1)(ted A

:3 enidro id eronim olos nu errartse òup iS

id aloger al odnasu oloclac oL surraS :

16


hcioP é nu otavort oh 2 enidro id eronim

ollun non rac .2=)A(c noN è’ id ongosib

irtla ilg eraloclac.ironim



:oloclac il e enidro id ironim i oggartse :A id 2

1121

2M

1111

3M

1112

4M

03)2(*11*1

263,2:ehgir

3,2:ennoloc





3,1:ehgir 2,1:ennoloc 42

215M

2211

6M

2412

7M

4211

8M

2411

9M

2211

01M

36

ecirtam alled acitsirettarac al eravorT :3 oipmesE

024201110121

A )d.1.3.3.n,26.p(

:enoizuloS ittartse illeuq ittut : ironim ied enoizartsE ehc illeuq ittut ertloni de etnemetnedecerp

annoloc al onognetnoc

000

:

46

242111121

1M

042011021

11M

022011011

21M

024011012

31M

:enoizavressO itnemele olos ah annoloc anu es etnanimreted li illun è .orez

56

ig os :2 enidro id ironim enoizartsE à nu etsise ehc ollun non eronim

→ rac 2=)A(

66


424221112121

A

:enoizuloS ittartse illeuq ittut : ironim ied enoizartsE

:itneuges i ertloni de etnemetnedecerp

76

3,2,1:ehgir 4,3,2:ennoloc



:oloclac il e id 2 enidro id ironim i oggartse :A

0

0

0

442211221

41M

422211211

51M

424211212

61M

86

hcrep illun onos 3 enidro id ironim i ittuT é .ilanoizroporp ennoloc eud ah onucsaic

ig os :2 enidro id ironim enoizartsE à nu etsise ehc ollun non eronim

→ rac 2=)A(

96

124211112121

A

142111221

71M421121

071M


:enoizuloS ittartse illeuq ittut : ironim ied enoizartsE

: ertloni de etnemetnedecerp

2+8+4+1 = - 51 = 4+4

≠ 0

→ rac 3=)A(

07


2321

32)m.1.3.3.n,36.p(

17


723521332

)m.1.3.3.n,36.p(

27

},,,,{ 321 maaaa

),,,,( 11312111 naaaaa

2 2232221 ),,,,( naaaaa

3 3333231 ),,,,( naaaaa

321 ),,,,(m mnmmm aaaaa

OGNAR id emeisnI m irottev

…

37

},,,{ 21 maaa

mnnn

m

m

m

aaa

aaaaaaaaa

A

21

32313

22212

12111

irottev ied itnenopmoc elled ecirtaM erottev nucsaic è :ennoloc ellus ottircsart

47

},,,{ 21 maaa

mnnn

m

m

m

aaa

aaaaaaaaa

A

21

32313

22212

12111

id ognar lI

è l’ id illun non ironim ied omissam enidro

57

},,,{ 21 maaa

baxaxax mm2211

11221111

22222112

33223113

2211

...

...

...

...

mm

mm

mm

nmmnnn

bxaxaxabxaxaxabxaxaxa

bxaxaxa

irottev ied eraenil enoizanibmoC

:etnenopmoc rep etnenopmoc

etingocni m ni iraenil inoizauqe n id ametsis

67

“ ametsis ” : enoizauqe alos anu non“ inoizauqe ” : “=”“ iraenil ” : ittodorp e emmos olos etasu onos“ etingocni ” : iton onos non irolav i“ ametsis li erevlosiR ” = id irolav i eravort

mxxx ,,, 21

onacifirev ehc etnemaenaropmetnoc ettut eleznailgaugu

77

ametsis led itneiciffeoc ied ecirtamatelpmocni ecirtam

mnnn

m

m

m

aaa

aaaaaaaaa

A

21

32313

22212

12111

87

atelpmoc ecirtam

nmnnn

m

m

m

baaa

baaabaaabaaa

bA

21

332313

222212

112111

|

97

},,,,{ 321 maaaa

},,,,{ 321 maaaab

},,,,,{ 321 baaaa m

:1 oizicresE id emeisnI m irottev

:isetopI itnednepidlla ognuigga eS ’ emeisni

erottev li emeisni ovoun li

ras à ?on o etnednepid

08

},,,,{ 321 maaaa

mm aaa 221

m,,2

l ehc odrociR ’ emeisni

è orol id onu onemla es etnednepid è etnednepid ais omirp li ehc idniuq ognoppuS .itnenamir iad

ol non es :irtla ilgad etnednepid è :imon i oibmacs

art onu onemla noc .orez ad osreviderevircs erpmes ossoP

baaa mm 0221

id enoizinifed al arocna acifirev otseuq e.aznednepid

18

idniuQ irottev id emeisni nu da erottev nu ognuigga es

irottev id emeisni nu arocna ognetto itnednepiditnednepid

28

},,,,{ 321 maaaa

},,,,{ 321 maaaa

;)0,1(1a ;)1,0(2a;)2,0(3a

},,{ 321 aaa 3a

},{ 21 aa


:isetopI itnednepidllad oglot eS ’ emeisni

ras emeisni ovoun li osac a erottev nu à o etnednepid?on

:oipmesE

llad oglot ’ emeisni erottev li.L’ emeisni è .etnednepidni emeisni nu

38

;)0,1(1a ;)1,0(2a ;)2,0(3a

},,{ 321 aaa .1a

},{ 32 aa

:oipmesE

llad oglot ’ emeisni erottev li

L’ emeisni

lla attase atsopsir al idniuQ ’ oizicrese è : os ol non

è .etnednepid emeisni nu

48

},,,,{ 321 maaaa

},,,,{ 321 maaaa

b },,,,,{ 321 baaaa m

},,,,{ 321 maaaa


ni :isetopI itnednepidlla ognuigga eS ’ emeisni

erottev li emeisni ovoun li

l es ehc odrociR ’ emeisnil arolla etnednepidni ’ li erenetto id odom ocinu

orol anu id otatlusir emoc ollun erottev eraenil enoizanibmoc è itneiciffeoc i ittut errop id

.orez a ilaugu

ras à ?on o etnednepid

è

58

02211 mm axaxax

0,,, 21 m

:idniuQ

noc

68

},{ 21 aa;)0,0,1(1a ;)0,1,0(2a

;)1,0,0(3a

},,{ 321 aaa

)1,0,0()0,1,0()0,0,1( 321332211 aaa),,(),0,0()0,,0()0,0( 321321

)0,0,0(),,( 321

0;0;0 321

l oredisnoc :oipmesE ’ emeisni

itnednepidni onos

ig omaibba à ehc oizicrese nu ni otartsomidè .etnednepidni emeisni nu

es olos e es

erottev li ognuiggA

78

},{ 21 aa;)0,0,1(1a ;)0,1,0(2a

;)0,2,0(3a

)0,2,0()0,1,0()0,0,1( 321332211 aaa)0,2,()0,2,0()0,,0()0,0( 321321

)0,0,0()0,2,( 321

02;0 321 321 2;0

l oredisnoc :oipmesE ’ emeisni

itnednepidni onos

arollA

es

idniuQ

OREZ NON EHCNAlla attase atsopsir al idniuQ ’ oizicrese è : os ol non

erottev li ognuiggA

88

},,,,{ 321 maaaa

},,,,{ 321 maaaa

02211 mm axaxax0,,, 21 m


llad oglot eS ’ emeisniras emeisni ovoun li osac a erottev nu à o etnednepid

?on

noc

ni :isetopI itnednepid

98

0112211 mm axaxax

0,,, 121 m

l oipmese rep ,osac a erottev nu ogloT ’ .omitlu

noc.itnednepidni arocna onoS

irottev id emeisni nu ad erottev nu oglot es id emeisni nu arocna ognetto itnednepidni

itnednepidni irottev

idniuQ

09

oglot itnednepidni

ognuigga itnednepiD

itnednepidni itnednepiD

19

},,,{ 21 maaa

b },,,{ 21 maaa

mxxx ,,, 21

baxaxax mm2211

},,,{ 21 maaa

irottev ied eraenil enoizanibmoC

:enoizavressOad etnednepid

onotsise es ehc ilat

da ognuigga esetnednepid erottev nu arolla

non itnednepidni irottev id oremun omissam liaibmac

.asreveciv li ehcna elav

29

)|( bAracArac

.asreveciv li ehcna elav

},,,,{},,,{ 2121 baaaognaraaaognar mm

39

321 },,,,{ neeee1 )0,,0,0,1(e2 )0,,0,1,0(e

)1,,0,0,0(ne

321 ,,,, n

21 21 mm bexexexnRb

:oizicresE l otad ’ id emeisni n irottev

…

iralacs id emeisni ocinu nu etsise ehc erartsomidehc elat

otlecs euqnumoc

49

b321 321

,,,, n nxxxx

321,,,,

n

21 21 mm beee

21 2121 ),,,(m mm eee

2121 ),,,(),,,( mm bbbb

2211 ,,; mm bbb

:enoizartsomiD“ etsise ”: ig à otad ehc otsiv ereilgecs atsab

è“ ocinu ” enoizulos artla elibissop anu iredisnoc is :idniuQ

ehc ah iS .ortsinis otal li osetse rep omailoclaC

idniuQ

hcioP é orol el odnauq ilaugu onos irottev eud ah is ilaugu etnematanidro onos itnenopmoc

59

nmmnnn

mm

mm

mm

bxaxaxa

bxaxaxabxaxaxabxaxaxa

...

...

...

...

2211

33223113

22222112

11221111

mnnn

m

m

m

aaa

aaaaaaaaa

A

21

32313

22212

12111

nmnnn

m

m

m

baaa

baaabaaabaaa

bA

21

332313

222212

112111

|

bA |

iraenil imetsiS

.1 n omaihc ol e inoizauqe id oremun li otnoC

.2 m omaihc ol e etingocni id oremun li otnoC

.3 ovircS

e

( b noc ”) eggel is “ otailpma A

:enoizulosiR

69

:adnamoD m=n ?.1 s eS ì

→ oloclac ted )A( :adnamoD ted )A(

≠ ? 0s eS ì

→• id ameroet li osU remarC c : è’ nu ’ enoizulos acinu• id aloger al osU remarC enoizulos al oloclac e

on eS

→ 2 otnup la euqnumoc odav on eS .2

→ id ameroet li osu hcuoR é- illepaC : oloclaC rac=p oloclac ; )A( rac ( b|A )adnamoD : A(rac =) b|A(rac ? )

• on eS

→ inoizulos onos ic non ametsis li( è )elibitapmocni

• s eS ì

→ onos ic

∞m- p al osu de inoizulosarudecorp )*( .elravort rep

79

id alogeR remarC iuc ni osac leN m=n e ted )A(

≠ ognop ,0 Δ .)A(ted=

ehc ah iS oloclaC

mnnn

m

m

m

aab

aabaabaab

2

3233

2222

1211

1

11x

89

……

mnnn

m

m

m

aba

abaabaaba

1

3313

2212

1111

22

2x

99

12111

22212

32313

21

n

nnn

baabaabaa

baa

nnx

001

es inoizulos elled oloclaC )*(rac=p )A( rac= ( b|A )

• .acitsirettarac al rep otavort eronim li aredisnoc iS

• .ennoloc k e ehgir k onaudividni itnemele ious I

• ia onodnopsirroc ecirtam alled ehgir eL is inoizauqe ilat :inoizauqe enucla id itneiciffeoc

.onallecnac is ertla el ,onognet

101

• ia onodnopsirroc ecirtam alled ennoloc eL etingocni ilat :etingocni enucla id itneiciffeoc

onangessa is ertla ella ,otsop orol la onognamir.iton inimret ied otal lad onatrop is e ehcerg erettel

• id aloger al ocilppA remarC .ametsis ovoun la

)09 .p orbil idev( ipmesE

201

026203

zyxzyx

262131

A 00

262131

| bA

: ametsis etneuges li erevlosiR :oipmesE

enoizulosiR3 = m ,2=n ottutiznannI

;

301

0201

0603

0201 0 0

ehc ah iS rac )1 oipmese idev( 1 = )A (ertlonI

rac ( b|A 1 = ) ad ittafnI b|A is ehc ,ironim i ittut errartse onossop is

ip ni de ,A ad errartse onossop ù :itneuges i

.illun ittut onos irez id annoloc anu odneva ehc

401

oipmesE ametsis etneuges li erevlosiR :

0242002

zyxzyxzyx

501

242111121

A000

242111121

| bA

enoizulosiR3 = m = n ottutiznannI

601

pm 1

rac )2 oipmese idev( 2 = )A(rac ( b|A hcrep 2 =) é annoloc anu otnuigga oh

3 enidro id ironim i ittut e illun itnemele id idev( 0 a ilaugu onatlusir onognetnoc al ehc

.)3 oipmeseidniuQ = p A(rac = ) b/A(rac 2 = )

ametsis lI è :ettemma de elibitapmoc

= = . inoizulos

701

1121

2M

0242002

zyxzyxzyx

eronim nu eraudividni angosib olrevlosir rePoipmese rep ,ollun non

etingocni 2 e inoizauqe 2 acifitnedi ehc

801

• ia onodnopsirroc ecirtam alled ehgir eL is inoizauqe ilat :inoizauqe enucla id itneiciffeoc

.onallecnac is ertla el ,onognet

• ia onodnopsirroc ecirtam alled ennoloc eL etingocni ilat :etingocni enucla id itneiciffeoc

onangessa is ertla ella ,otsop orol la onognamir inimret ied otal lad onatrop is e ehcerg erettel

.iton

901

lla idniuq ongessA ’ erolav nu z atingocni = z :oirartibra α

ametsis ovoun lI è:

yxyx 2

011

ametsis ovoun lI

yxyx 2

id aloger al odnacilppa evlosir is remarC :

111

32

112112

x

0

1121

11

y

211

:acifireV id enoizutitsos al etnaidem autteffe is acifirev al ametsis led inoizauqe el ettut ni etingocni el ettut

:oiranigiro

00*200

020*4)(2

311

ametsis etneuges li erevlosiR :oipmesE

4242222

zyxzyxzyx

411

242111121

A

422

242111121

/ bA

enoizulosiR3 = m = n ehc avresso is ottutiznannI

511

123pm

1121

2M

rac )2 oipmese idev( 2 = )A(rac )4 oipmese idev( 2 = )b/A( = p rac 2 = )b/A(

ametsis li idniuQ è :ettemma de elibitapmoc

. nu oudividni ametsis led inoizulos el eravort reP

ollun non eronim

eud emirp elled y de x id itneiciffeoc i eneitnoc ehc.inoizauqe

inoizulos

611

4242222

zyxzyxzyx

lla irartibra irolav ongessA ’ :etnenamir atingocni = z α

:ametsis ovoun lI

222

yxyx

711

noc evlosir iS remarC :

23

242

11211222

x

03

)2(2

1121

2121

y

zyx ,0,2 ametsis led enoizulos aL è :ad atad

811

:acifireV

20*2)2(20)2(

420*4)2(2

911

ametsis etneuges li erevlosiR :oipmesE

1242122

zyxzyxzyx

021

enoizulosiR3 = m = n ehc avresso is ottutiznannI

242111121

A

rac )2 oipmese idev( 2 = )A(

121

112

242111121

/ bA

rac )5 oipmese idev( 3 = )A(

ametsis li idniuQ è ic non( elibitapmocni.)inoizulos onos

221

ortemarap ad itnednepid iraenil imetsiS)49.p(

nu id iton inimret ien e itneiciffeoc ien eS ied onoiappa iraenil inoizauqe id ametsis

oic ,irtemarap è eneiv non iuc id itnatsoc elled led otnematropmoc li ,erolav li otacificeps ehc irolav iad elareneg ni ednepid ametsis

de irtemarap ilat onomussa è oirassecen ied isodnevres ilibissop isac i ittut eranimase

id imeroet hcuoR è- illepaC id e remarC .

321

oipmesE : k id irolav ilauq rep eranimreteDametsis li

kzyxzyxzyx

242122

.inoizulos ah

421

242111121

A

kbA 1

2

242111121

|

enoizuloS : ametsis led itneiciffeoc ied ecirtam aL è ad atad

atelpmoc ecirtam aL è :ad atad

eraloclac oveD rac . )b/A(l noc odecorP ’ :ironim ied enoizartse

rac (A 2 = ) )2 oipmese idev(

521



:oloclac il e id 3 enidro id ironim i oggartse :A

AM1

kM

42111221

2 21324484

kkk

0

621


annoloc II e I()ilaugu

3,2,1:ehgir4,3,1:ennoloc 0

kM

22111211

3

kM

24111212

4 21348442

kkk

721

rep ehc ovressO k = - ehc ah is 4 M1 M= 2 M= 3 M= 4 = 0:isac eud ereugnitsid idniuq angosiB

3 = b|A rac

es ecevnI )2 k = - ehc ah is 4 rac b|A . 3 < ertlonI b|A enenitnoc non 2 enidro id eronim nu

ol( ollun eraloclac rep otasu ossets rac .))A(idniuQ rac ( b|A onos ic e 2=)

.inoizulos

k eS )1

≠ -4 A rac

inoizulos onos ic non idniuq e

1pm

821

21321300

42

31

kMkMMM

?04321 MMMM

: odnemussaiR

id ocinu erolav nu etsise :adnamoD k ehc elat

arolla ovort ol non es rac ( b|A li de 3=)inoizulos ah non ametsis

k id erolav leuq rep arolla ovort ol esrac ( b|A .inoizulos ettemma ametsis li de ,3<)

921

1.n eroiretlu oizicresE

031

:oizicresE ametsis etneuges li erevlosiR

)h1.n36p(23

5221933

zyxzyxzyx

:etingocni id oremuN n=3:inoizauqe id oremuN m=3

131

ecirtaM atelpmocni ecirtaMatelpmoc

311122933

A

23115122|1933

bA

idniuq ,ilanoizroporp onos A id 3 e 1 ehgir eL ted idniuq ,0=)A( rac 3<)A(

ollun non 2 enidro id eronim nu ovorT

idniuQ rac 2=)A(

A id acitsirettarac alled oloclaC

3173*21*)1(

12

231


311122933

A

:3 enidro id ironim i oloclaC

ehc ,|A| ognetto 3 ,2 ,1 ennoloc el odnanoizeleS è 0=

23115122|1933

bA

id acitsirettarac alled oloclaC b|A

331


311122933

A23115122|1933

bA


ognetto 4 ,2 ,1 ennoloc el odnanoizeleS


211522133

431

id oloclaC id aloger al noc surraS

211522133

112112252233133

2- 21+51 - 21( - =)2+51 - 0=1+1

ironim irtla ilg eraloclac angosiB

el ehc ehcna ovressO onos 2 e 1 ennoloc

ehc li ,ilanoizroporp etnanimreted li amrefnoc

.ollun

531

atelpmoc ecirtaM

23115122|1933

bA


id acitsirettarac alled oloclaC b|A 3 enidro id ironim :

231512193

631


312311251293193

54+1 - 63 - 54+6( - =)6 - 53

id acitsirettarac aL b|A è 3

rac )A( rac ( b|A ) inoizulos ah non ametsis lI

231512193

731

2.n eroiretlu oizicresE

831

:oizicresE ametsis etneuges li erevlosiR

)i1.n36p( 235226933

zyxzyxzyx

:etingocni id oremuN n=3:inoizauqe id oremuN m=3

931


311122933

A

23115122|6933

bA



idniuQ rac 2=)A(


3173*21*)1(

12

041


311122933

A


ehc ,|A| ognetto 3 ,2 ,1 ennoloc el odnanoizeleS è 0=

23115122|6933

bA


141


311122933

A23115122|6933

bA




211522633

241


211522633

112112252233633

21 - 21+51 - 21( - 0=)21+51


341

ecirtaMatelpmoc

23115122|6933

bA ognetto 4 ,3 ,1 ennoloc el odnanoizeleS


231512693

441


312311251293693

54+6 - 63 - 54+6( - 0=)63


231512693

541

ecirtaMatelpmoc

23115122|6933

bA ognetto 4 ,3 ,2 ennoloc el odnanoizeleS


231512693

641


312311251293693

-6- 63+54 -(-6- 0=)63+54

idniuq ,orez onos 3 enidro id ironim i ittuT rac 3<)b1A(

emocciS b|A non 2 enidro id eronim nu ah A e A eneitnoc arolla ,ollun rac ( b|A .2=)

231512693

741

311122933

A


23115122|6933

bA

rac (rac=2=)A( b|A ah ametsis li idniuq ,) 3- 2 inoizulos

841

:inoizulos el eravorT

23522

6933

zyxzyx

zyxennoloc e ehgir audividni eronim lI

311122933

A

941


235226933

zyxzyxzyx

eronim lad esulcse ehgir el :ennoloc e ehgir audividni eronim lIetallecnac onognev

311122933

A

051


23522

xx

zyzy

el :ennoloc e ehgir audividni eronim lI eronim lad esulcse ennoloc

is iuc ,etingocni a onodnopsirroc noc itacidni( irartibra irolav onangessa ied otal lad onatrop is e )ehcerg erettel

inimet iton

311122933

A

=x

151


23252

zyzyx

etnanimreted e ,etingocni p ,inoizauqe p ah ametsis ovoun lI.orez ad osrevid itneiciffeoc ied ecirtam aus alled

eracilppa òup is idniuQ remarC ametsis ovoun la

251


23252

zyzyx

3173*21*)1(

12

731731177

99277

y

z

x

32731)25(31*)2(1

125

219)2(*2)25(*12

252

351

:acifireV731

797

y

z

x

235226933

zyxzyxzyx

41727317 2377

416241

973177973177

539 522772418129312 6933 9731

7 777

451

oipmesE : eranimreteD li oremun elled inoizulos led etneugesametsis la erairav id :k

23522

933

zyxzyx

kzyx

n=3:inoizauqe id oremuN

m=3:etingocni id oremuN



idniuQ rac 2=)A(


551

23115122|

933bA

k

ecirtam alled acitsirettarac al eraloclac angosiB atelpmoc b|A : 3 enidro id ironim i oloclaC

23115122|

933bA

k

23115122|

933bA

k

ehc ,|A| ognetto 3 ,2 ,1 ennoloc el odnanoizeleS è0=

23115122|

933bA

k

li idniuq ,ilanoizroporp onos 2 e 1 ennoloc eL eronim odnoces led etanimreted è erednicserp a ,0

k ad231

6072454666354512193

keskkkMk

23160724546663545122

93keskkkM

k

itnemirtla ,illun onos 3 enidro id ironim i ittut 6=k es idniuQ .on

651

.on itnemirtla ,illun onos 3 enidro id ironim i ittut 6=k es

6=k es idniuQ rac ( b|A k es ,3<) 6 arolla rac ( b|A 3=)

6=k eS rac ( b|A :3<)

eronim nu oudividnI

ehc eredulcnoc rep atsab otseuq :ollun non 2 enidro id rac ( b|A 2=)

23115122|

933bA

k

751

oipmesE : eranimreteD li oremun elled inoizulos led etneugesametsis la erairav id :k

23522

933

zyxzyx

kzyx

n=3:inoizauqe id oremuN

m=3:etingocni id oremuN

6=k es idniuQ rac (rac=2=)A( b|A onos ic idniuq e ) 3-2 = 1 inoizulos

k eS 6 rac 2=)A( rac ( b|A inoizulos onos ic non idniuq e 3=)

851

951

ortemarap ad itnednepid imetsis us oizicrese eroiretlU

061

212

kyxyxykx

led inoizulos elled oremun li eretucsid :oizicresEk ortemarap led erairav la ametsis etneuges

21111|21

kbA

k

2111

1A

k

3=n

2=m

atelpmocni ecirtaM atelpmoc ecirtaM

161

id e A id acitsirettarac alled oloclaCb|A

21111|21

kbA

k

2111

1A

k

A atelpmocni ecirtaM

atelpmoc ecirtaMb|A

:enoizavressO rac ,2=>)A(

emocciS rac arolla ,2=<)A( rac 2=)A(

rac ( b|A ?3 eresse òuP .2=>)

ehc otad ,erednopsir reP b|A è

eraloclac oved ,atardauq ted ( b|A )

261

361

021 nbbb

0)(...

0...0...)(

0...)(

2211

3223113

2222112

1221111

nxaxaxa

xaxaxaxaxaxa

xaxaxa

nnnn

nn

nn

nn

irolavotuA 59.pp( - )69 n enidro id A atardauq ecirtam anu ataD

oic( oenegomo ametsis li de è ortemarap lad etnednepid λ :

)

461

id erolav ingo λ orez a elaugu itlusir ehc elat ametsis led ecirtam alled etnanimreted li

nnnn

n

n

n

aaa

aaaaaaaaa

21

32313

22212

12111

id emon li ednerp erolavotua ametsis led

561

01332

01332

1121901202

)3(*)3()01(*)2(013

32ted

22

ilg eraloclaC )2.n,69.p( :oipmesE irolavotua:ecirtam alled

:enoizulosiRecirtam alled etnanimreted li eraloclac angosiB

661

02 cxbxa acabbx

242

2,1

011212

211*1*444121

2,1

id irolav ilauq rep erepas oilgoV λ etnanimreted elat è.orez a elaugu

etnanimreted otseuQ è id oimonilop nu ad osserpse ni odarg odnoces λ is iuc ni irolav i eravort rep e

l erevlosir errocco allunna ’ id acirbegla enoizauqe ni odarg odnoces λ rep avitulosir alumrof al ocilppA .

eud odarg id inoizauqe el

inoizulos eud el ognetto de λ1 e 1= λ2 11= .

761

ilg onos inoizulos ilaT irolavotua alled elaizini ecirtam

861

irottevotuA 69.pp( - )79 ingo reP erolavotua λ i onaloclac is otavort

itnednopsirroc irottevotua odneutitsos λ len.olodnevlosir e ametsis

)2.n,69.p( oipmesEig oH à ilg otaloclac irolavotua ecirtam alled

01332

ilg eraloclac aro oveD irottevotua itnednopsirroceud ied onucsaic a

onos ehc λ1 e 1= λ2 .11=

961

yx 3

3

onos errartse ossop ehc 2 enidro id ironim i ittut idniuQ .illun

l e enoizauqe I al audividni eronim elaT ’ .x atingocni ongessa ,enoizauqe adnoces al ollecnaC =y α otrop ol e

llad ’ atnevid ametsis ovoun lI .etrap artla

.opit led irottev I

ilg onos irottevotua id erolavotua λ1 1=

1oipmese rep ,ollun non eronim nu ocreC .

071

00

)01(33)2(

yxyx

ilged oloclaC irottevotua a itnednopsirroc λ1 11= :.agolana areinam nI

32013

xyxyyx

adbmal

;2=m;2=n

171

3193

2231001999)3()3()9(*)1(

93

onos inoizulos eL e 0= .01=

l eratelpmoC ’ ilg odnavort oizicrese irottevotua.itnednopsirroc

Documents

Elementi di logica Algebra lineare - memotef.uniroma1.it · 4 i mp lic azio n e log i ca Una fr as e s c ritta nella forma Se (pro p osizion e 1) allo ra (prop o sizione 2) Costitui