majhmatik'a gia qhmiko'us · idiìthtec orizous¸n an mÐa orÐzousa èqei ìla ta...

M�jhma 160

24 NoembrÐou 2011(9h ebdom�da)

S. Malef�kh Genikì Tm ma Majhmatik� gia QhmikoÔc

Idiìthtec orizous¸n

An mÐa orÐzousa èqei ìla ta stoiqeÐa mÐac gramm c miac

st lhc 0 tìte aut isoÔtai me to 0.

H tim thc orÐzousac den all�zei an all�xoume ìlec tic

grammèc me tic antÐstoiqec st lec.

To prìshmo thc orÐzousac all�zei an all�xoume mia

gramm me mÐa st lh metaxÔ touc.

An pollaplasi�soume ta stoiqeÐa mÐac gramm c mÐac

st lhc me ènan arijmì tìte h tim thc orÐzousac ja

pollaplasi�zetai me ton par�gonta autìn.

An dÔo grammèc dÔo st lec miac orÐzousac eÐnai Ðsec

an�logec tìte h tim thc orÐzousac ja isoÔtai me mhdèn.

An ta stoiqeÐa miac gramm c pollaplasiastoÔn me ènan

arijmì kai prostejoÔn afairejoÔn me ta stoiqeÐa miac

�llhc gramm c tìte h tim thc orÐzousac den all�zei

dhlad ∣∣∣∣ a+ kc b + kdc d

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣

IsqÔei ∣∣∣∣ a+ a′ b + b′

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣+ ∣∣∣∣ a′ b′

∣∣∣∣H orÐzousa enìc tetragwnikoÔ enìc diag¸niou pÐnaka

isoÔtai me to ginìmeno twn diag¸niwn stoiqeÐwn tou.

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣IsqÔei ∣∣∣∣ a+ a′ b + b′

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣+ ∣∣∣∣ a′ b′

∣∣∣∣

H orÐzousa enìc tetragwnikoÔ enìc diag¸niou pÐnaka

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣IsqÔei ∣∣∣∣ a+ a′ b + b′

∣∣∣∣ = ∣∣∣∣ a bc d

∣∣∣∣+ ∣∣∣∣ a′ b′

∣∣∣∣H orÐzousa enìc tetragwnikoÔ enìc diag¸niou pÐnaka

H orÐzousa tou I isoÔtai me 1.

H orÐzousa enìc mhdenikoÔ pÐnaka isoÔtai me to 0.

|AB| = |A||B|

|Ak | = |A|k

|A−1| = 1|A|

|AB| = |A||B|

|Ak | = |A|k

|A−1| = 1|A|

|AB| = |A||B|

|Ak | = |A|k

|A−1| = 1|A|

|AB| = |A||B|

|Ak | = |A|k

|A−1| = 1|A|

|AB| = |A||B|

|Ak | = |A|k

|A−1| = 1|A|

Ask seic

Na upologÐzete tic orÐzousec twn parak�tw pin�kwn

a x 2a− xb y 2b − yc z 2c − z

1 + a b c da 1 + b c da b 1 + c da b c 1 + d

Ask seic

Na upologÐzete tic orÐzousec twn parak�tw pin�kwn

a x 2a− xb y 2b − yc z 2c − z

1 + a b c da 1 + b c da b 1 + c da b c 1 + d

AntÐstrofoc tou pÐnaka A

Kataskeu�zoume to pÐnaka S , pou san stoiqeÐo tou sij èqei tonsumpopollaplasiast Aij tou stoiqeÐou aij tou pÐnaka A,

dhlad ton pÐnaka

A11 A12 . . . A1n

A21 A22 . . . A2n

. . . . . . . . . . . .An1 An2 . . . Ann

'Epeita brÐskoume ton prosarthmèno sumplhrwmatikì pÐnaka

adj(A) o opoÐoc isoÔtai me ton an�strofo tou S , dhlad

adj(A) =

A11 A21 . . . An1

A12 A22 . . . An2

. . . . . . . . . . . .A1n A2n . . . Ann

Kataskeu�zoume to pÐnaka S , pou san stoiqeÐo tou sij èqei tonsumpopollaplasiast Aij tou stoiqeÐou aij tou pÐnaka A,

dhlad ton pÐnaka

A11 A12 . . . A1n

A21 A22 . . . A2n

. . . . . . . . . . . .An1 An2 . . . Ann

'Epeita brÐskoume ton prosarthmèno sumplhrwmatikì pÐnaka

adj(A) o opoÐoc isoÔtai me ton an�strofo tou S , dhlad

adj(A) =

A11 A21 . . . An1

A12 A22 . . . An2

. . . . . . . . . . . .A1n A2n . . . Ann

opìte o antÐstrofoc pÐnakac A−1 tou pÐnaka A ja isoÔtai me

A−1 =1

|A|adj(A) =

A11 A21 . . . An1

A12 A22 . . . An2

. . . . . . . . . . . .A1n A2n . . . Ann

T�xh enìc pÐnaka

T�xh enìc pÐnaka A rank(A) onom�zoume to mègisto arijmì

gramm¸n sthl¸n pou eÐnai anex�rthtec.

isodÔnama

Th di�stash thc megalÔterhc mh mhdenik c orÐzousac.

'Enac tetragwnikìc pÐnakac An×n eÐnai mh idi�zon an kai

mìno an h t�xh tou eÐnai Ðsh me n.

Genikìtera t�xh tou pÐnaka Am×n onom�zetai h di�stash

tou megalÔterou mh idi�zontoc tetragwnikoÔ pÐnaka pou

sqhmatÐzetai ìtan epilèxoume merikèc ( ìlec tic grammèc)

tou A kai tic antÐstoiqec st lec tou A.

isodÔnama

Sust mata Grammik¸n exis¸sewn

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2

. . . = . . .

am1x1 + am2x2 + . . .+ amnxn = bm

SÔsthma m exis¸sewn me n agn¸stouc.

'Ena sÔsthma lègetai tetragwnikì an o arijmìc twn

exis¸sewn tou eÐnai Ðsoc me ton arijmì twn agn¸stwn.

'Ena sÔsthma lègetai omogenèc an ìloi oi stajeroÐ ìroi

eÐnai 0, diaforik� to sÔsthma lègetai mh omogenèc.

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2

. . . = . . .

am1x1 + am2x2 + . . .+ amnxn = bm

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2

. . . = . . .

am1x1 + am2x2 + . . .+ amnxn = bm

'Ena sÔsthma grammik¸n exis¸sewn mporeÐ na grafeÐ me th

morf pin�kwn wc ex c

A · x = b

a11 a12 . . . a1na21 a22 . . . A2n

. . . . . . . . . . . .am1 am2 . . . amn

o pÐnakac twn suntelest¸n

tou sust matoc

x to di�nusma twn agn¸stwn kai

b to di�nusma twn stajer¸n ìrwn.

O pÐnakac

a11 a12 . . . a1n b1. . . . . . . . . . . . . . .am1 am2 . . . amn bm

onom�zetai epauxhmènoc pÐnakac tou sust matoc.

Je¸rhma

'Otan to pedÐo arijm¸n eÐnai aperiìristo (dhlad eÐnai oi

pragmatikoÐ kai oi migadikoÐ arijmoÐ) tìte èna sÔsthma

grammik¸n exis¸sewn ja èqei mÐa lÔsh �peirec lÔseic

kamÐa lÔsh.

O pÐnakac

a11 a12 . . . a1n b1. . . . . . . . . . . . . . .am1 am2 . . . amn bm

onom�zetai epauxhmènoc pÐnakac tou sust matoc.

Je¸rhma

'Otan to pedÐo arijm¸n eÐnai aperiìristo (dhlad eÐnai oi

pragmatikoÐ kai oi migadikoÐ arijmoÐ) tìte èna sÔsthma

grammik¸n exis¸sewn ja èqei mÐa lÔsh �peirec lÔseic

kamÐa lÔsh.

Sust mata se trigwnik morf

a11x1 + a12x2 + . . .+ a1nxn = b1

a22x2 + . . .+ a2nxn = b2

. . . = . . .

annxn = bn

Ο πρώτος άγνωστος της κάθε εξίσωσης ονομάζεται

ηγετικός άγνωστος.

΄Ενα τέτοιο σύστημα έχει πάντα μια μοναδική λύση!

Η λύση αυτού του συστήματος μπορεί να βρεθεί με

αντικατάσταση προς τα πίσω.

Na lujeÐ to parak�tw sÔsthma

2x1 + 3x2 + 5x3 − 2x4 = 9

5x2 − x3 + 3x4 = 1

7x3 − x4 = 3

4x4 = 8

Sust mata se klimakwt morf .

2x1 + 6x2 − x3 + 4x4 − 2x5 = 7

x3 + 2x4 + 2x5 = 5

3x4 − 9x5 = 6

Οι ηγετικοί άγνωστοι του συστήματος ονομάζονται βασικές

μεταβλητές. Οι υπόλοιποι άγνωστοι ονομάζονται ελεύθερες

μεταβλητές.

Je¸rhma

'Ena sÔsthma exis¸sewn se klimakwt morf me r exis¸seic kain agn¸stouc

An n = r (trigwnikì sÔsthma) tìte to sÔsthma èqei mÐa kai

monadik lÔsh.

An r < n (perissìteroi �gnwstoi apì ìti exis¸seic) tìte

to sÔsthma ja èqei �peirec lÔseic.

Τότε μπορούμε αυθαίρετα να αναθέσουμε τιμές στις n − r

ελεύθερες μεταβλητές και να λύσουμε μοναδικά ως προς τις

υπόλοιπες r βασικές μεταβλητές.

Apaloif Gauss

KÔria mèjodoc epÐlushc grammik¸n susthm�twn

Stadiak el�ttwsh tou sust matoc pou odhgeÐ eÐte se èna

trigwnikì klimakwtì sÔsthma eÐte se exis¸seic pou den

èqoun lÔsh.

Stadiak antikat�stash proc ta pÐsw gia na brejeÐ h

lÔsh tou aploÔsterou sust matoc.

Ask seic

Na lujoÔn ta parak�tw sust mata

x − 3y − 2z = 6

2x − 4y − 3z = 8

−3x + 6y + 8z = −5

x − 2y − 3z = 1

2x + 5y − 8z = 4

3x + 8y − 13z = 7

Ask seic

Na lujoÔn ta parak�tw sust mata

x − 3y − 2z = 6

2x − 4y − 3z = 8

−3x + 6y + 8z = −5

x − 2y − 3z = 1

2x + 5y − 8z = 4

3x + 8y − 13z = 7

'Estw ìti èqoume to sÔsthma

Ax = b

Je¸rhma

'Ena tetragwnikì sÔsthma grammik¸n exis¸sewn èqei mÐa

monadik lÔsh an kai mìno an o pÐnakac A eÐnai antistrèyimoc.

Se aut n thn perÐptwsh h lÔsh eÐnai o pÐnakac

x = A−1b.

Omogen sust mata grammik¸n exis¸sewn

Ax = 0

EÐnai fanerì ìti èna tètoio sÔsthma èqei p�nta lÔsh to 0. H

lÔsh aut onom�zetai mhdenik tetrimènh.

An o pÐnakac A eÐnai tetragwnikìc (dhlad èqei Ðso arijmì

exis¸sewn kai agn¸stwn) kai mh idi�zon tìte to sÔsthma èqei

monadik lÔsh to 0.

'Ena omogenèc sÔsthma me perissìterec agn¸stouc apì ìti

exis¸seic èqei mh mhdenikèc lÔseic. Ta sust mata aut� èqoun

�peirec lÔseic

Je¸rhma

JewroÔme èna sÔsthma grammik¸n exis¸sewn me n agn¸stouc,

èstw A o pÐnakac twn suntelest¸n kai M o epauxhmènoc

pÐnakac.

To sÔsthma èqei lÔsh an kai mìno an

rankA = rank(M)

To sÔsthma èqei mÐa kai monadik lÔsh an kai mìno an

rankA = rank(M) = n.

An rankA 6= rank(M) to sÔsthma den èqei lÔsh.

majhmatik'a gia qhmiko'us · idiìthtec orizous¸n an mÐa orÐzousa èqei ìla ta...

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