(2) Jewrome ton dianusmatik qro (R3,+, ) kai

ta snola

W1 = {(x1, x2, x3) R3 : 2x1 4x2 + 2x3 = 5}

kai

W2 = {(x1, x2, x3) R3 : 3x1 + 2x2 4x3 = 0}.

Ja meletsoume poi ap ta parapnw snola

enai dianusmatiko upqwroi tou (R3,+, ).

Arqik, parathrome ti to mhdenik stoiqeo

(0, 0, 0) th omda (R3,+) den ankei sto W1,

afo 2 0 4 0 + 2 0 = 0 6= 5, opte to W1 den

enai upqwro tou (R3,+, ).

Antjeta, (0, 0, 0) W2, opte W2 6= . Epi-

plon, an , R kai x,y W2, me

x = (x1, x2, x3), y = (y1, y2, y3), tte

3x1 + 2x2 4x3 = 0 kai 3y1 + 2y2 4y3 = 0.

Pollaplasizonta ti parapnw do isthte

me kai antstoiqa kai prosjtonta ti is-

thte pou prokptoun, petai ti

(3x1 + 2x2 4x3) + (3y1 + 2y2 4y3) = 0,20

opte

3(x1 + y1) + 2(x2 + y2) 4(x3 + y3) = 0

kai epomnw,

x + y = (x1 + y1, x2 + y2, x3 + y3) W2,

ra, bsei th Prtash 1, toW2 enai upqwro

tou (R3,+, ).

21

(3) 'Estw (M2,+, ) o pragmatik dianusmatik

qro twn 2 2 mhtrn me pragmatik stoiqea

kai

W1 = {

[

2 1

]: , , R},

W2 = {

[4 3

0

]: , , R}.

Ja meletsoume poi ap ta parapnw snola

enai upqwroi tou (M2,+, ).

Arqik, parathrome ti to mhdenik stoiqeo[0 0

0 0

]den ankei sto W1, opte to W1 den enai

upqwro tou (M2,+, ).

Antjeta,

[0 0

0 0

]W2, opte W2 6= .

22

Epiplon, an , R kai[41 31

1 0

],

[42 32

2 0

]W2,

tte enai

[41 31

1 0

]+

[42 32

2 0

]

=

[41 31

1 0

]+

[42 32

2 0

]

=

[4(1 + 2) 3(1 + 2)

(1 + 2) 0

]W2.

'Ara, bsei th Prtash 1, to W2 enai upqw-

ro tou (M2,+, ).

23

Tom dianusmatikn upoqrwn

H tom do perissterwn upoqrwn en dianu-

smatiko qrou enai upqwro. Sugkekrimna, an

V1, V2 enai upqwroi tou (V,+, ), tte V1 V2 enai

upqwro tou (V,+, ).

Prgmati, epeid 0 V1 kai 0 V2, petai ti

0 V1 V2, opte V1 V2 6= .

An , F kai x,y V1 V2, tte epeid ta V1,

V2 enai upqwroi kai x,y V1 kai x,y V2, petai

ti

x + y V1 kai x + y V2.

Opte, x+y V1V2 kai smfwna me thn Prta-

sh 1, petai ti V1 V2 enai upqwro tou (V,+, ).

24

'Enwsh dianusmatikn upoqrwn

H nwsh do perissterwn upoqrwn en dianu-

smatiko qrou den enai en gnei upqwro. Sugke-

krimna, an V1, V2 enai upqwroi tou (V,+, ), tte

V1 V2 enai upqwro tou (V,+, ) an kai mno an

V1 V2 V2 V1.

Prgmati, an V1V2 enai upqwro tou (V,+, ) kai

V1 * V2, ja apodeiqje ti V2 V1. 'Estw, x V2.

Epeid V1 * V2, ja uprqei y V1, me y / V2.

Epeid x,y V1V2, petai ti x+y V1V2, opte

x+y V1 x+y V2. An x+y V2, tte epeid

x V2 (afo x V2), petai ti y = x+(x+y)

V2, to opoo enai topo. 'Ara, x + y / V2, opte

x+y V1 kai epeid y V1 (afo y V1), petai

ti x = (x + y) + (y) V1. 'Ara, V2 V1.

25

'Ajroisma dianusmatikn upoqrwn

To jroisma do perissterwn upoqrwn en

dianusmatiko qrou enai upqwro. Sugkekrimna,

an V1, V2 enai upqwroi tou (V,+, ), tte to snolo

V1 + V2 = {x1 + x2 : x1 V1,x2 V2}

onomzetai jroisma twn upoqrwn V1, V2 kai enai

upqwro tou (V,+, ).

Prgmati, epeid 0 V1 kai 0 V2, petai ti

0 = 0 + 0 V1 + V2, opte V1 + V2 6= . Epiplon,

an , F kai x,y V1 + V2, tte x = x1 + x2kai y = y1 + y2, pou x1,y1 V1 kai x2,y2 V2.

Epeid oi V1, V2 enai upqwroi, petai ti

x1 + y1 V1 kai x2 + y2 V2.

Tte mw, epeid

x + y = (x1 + x2) + (y1 + y2)

=(x1 + y1) + (x2 + y2) V1 + V2

kai smfwna me thn Prtash 1, petai ti V1 + V2

enai upqwro.

26

Shmeinoume ti V1 V2 V1 + V2. Prgmati,

epeid gia kje x V1 isqei ti x = x+0 V1+V2,

petai ti V1 V1 + V2. Omow apodeiknetai ti

V2 V1 + V2, opte telik V1 V2 V1 + V2.

27

Grammik sunduasm

'Estw (V,+, ) dianusmatik qro ep tou sma-

to (F,+, ) kai

x1,x2, . . . ,xn V.

Tte, kje stoiqeo x V pou mpore na grafe sth

morf

x = 1x1 + 2x2 + + nxn,

pou 1, 1, . . . , n F , onomzetai grammik

sunduasm twn x1,x2, . . . ,xn.

Oi telest 1, 2, . . . , n onomzontai suntele-

st tou grammiko sunduasmo. Profan, kje

grammik sunduasm enai stoiqeo tou V .

28

Pardeigma: 'Estw o pragmatik dianusmatik

qro (R3,+, ). Na exetasje an to dinusma x =

(10,5, 5) R3 enai grammik sunduasm twn dia-

nusmtwn

x1 = (3,1, 2), x2 = (1, 0, 4), x3 = (1, 1, 1).

Ja exetsoume an uprqoun 1, 2, 3 R, me x =

1x1 + 2x2 + 3x3. Enai

x = 1x1 + 2x2 + 3x3

(10,5, 5)=1(3,1, 2) + 2(1, 0, 4) + 3(1, 1, 1)

(10,5, 5)=(31,1, 21)+(2, 0, 42)+(3, 3, 3)

(10,5, 5)=(31+23,1+3, 21+42+3)

31 + 2 3 = 10

1 + 3 = 5

21 + 42 + 3 = 5

Epilonta to parapnw ssthma, prokptei ti

1 = 2, 2 = 1 kai 3 = 3, opte x = 2x1+x23x3kai epomnw to x enai grammik sunduasm twn

x1,x2,x3.

29

Grammik jkh sunlou

'Estw (V,+, ) dianusmatik qro ep tou sma-

to (F,+, ) kai stw S = {x1,x2, . . . ,xn} na pe-

perasmno uposnolo tou V . To snolo lwn twn

grammikn sunduasmn twn x1,x2, . . . ,xn onomze-

tai grammik jkh ( apl jkh ) tou S kai sum-

bolzetai me < S > < x1,x2, . . . ,xn >, dhlad

< x1,x2, . . . ,xn >

={1x1 + 2x2 + + nxn : 1, 1, . . . , n F}.

An U =< x1,x2, . . . ,xn >, tte lgetai ti to s-

nolo U pargetai genntai ap ta diansmata

x1,x2, . . . ,xn ( ap to snolo S). Mia llh isod-

namh kfrash enai ti ta diansmata x1,x2, . . . ,xn

( to snolo S) pargoun gennon to snolo

U .

30

To snolo < x1,x2, . . . ,xn > enai upqwro tou

(V,+, ), afo gia , F kai

x = 1x1 + 2x2 + + nxn,

y = 1x1 + 2x2 + + nxn,

to stoiqeo

x + y = (1 + 1)x1+(2 + 2)x2+

+(n + n)xn

ankei sto < x1,x2, . . . ,xn >.

Epiplon xi < x1,x2, . . . ,xn >, gia kje i [n],

afo

xi = 0x1 + + 0xi1 + 1xi + 0xi+1 + + 0xn.

Tlo, an W upqwro tou (V,+, ), me xi W ,

gia kje i [n], tte < x1,x2, . . . ,xn > W , a-

fo kje grammik sunduasm ap stoiqea en

dianusmatiko qrou (tou W ) enai stoiqeo tou.

Ap ta prohgomena prokptei h epmenh prtash.

Protash 2.H grammik jkh < x1,x2, . . . ,xn >

enai o mikrtero dianusmatik qro pou periqei

ta x1,x2, . . . ,xn.31

Paratrhsh: Ap thn prohgomenh prtash pro-

kptei ti an ta diansmata y1,y2, . . . ,yn enai gram-

miko sunduasmo twn dianusmtwn x1,x2, . . . ,xn, t-

te isqei ti

< x1,x2, . . . ,xn,y1,y2, . . . ,yn >=< x1,x2, . . . ,xn > .

Pardeigma: Ston pragmatik dianusmatik qro

(Rn,+, ) an

ei = (0, 0, . . . , 0, 1i, 0, . . . , 0), i [n],

tte kje x Rn enai grammik sunduasm twn

e1, e2, . . . , en, afo, an x = (x1, x2, . . . , xn), tte

x = x1(1, 0, 0, . . . , 0) + x2(0, 1, 0, . . . , 0)+

+ xn(0, . . . , 0, 1)

= x1e1 + x2e2 + + xnen.

'Ara, isqei ti Rn =< e1, e2, . . . , en >. Exllou,

ap thn prohgomenh paratrhsh, prokptei ti

Rn =< e1, e2, . . . , en >=< e1 + e2, e1, e2, . . . , en >

=< e1 + e2, e2, . . . , en >,

dhlad gia nan dianusmatik qro mpore na upr-

qoun perisstera ap na snola, me to dio pljo

stoiqewn, pou ton pargoun.

32

Grammik anexarthsa

'Estw (V,+, ) dianusmatik qro ep tou sma-

to (F,+, ). Ta diansmata x1,x2, . . . ,xn V enai

grammik exarthmna tan uprqoun

1, 2, . . . , n F,

qi la sa me to mhdenik stoiqeo tou F , ttoia

ste

1x1 + 2x2 + + nxn = 0.

An ta x1,x2, . . . ,xn den enai grammik exarthmna,

tte enai grammik anexrthta.

Isodnama, ta x1,x2, . . . ,xn enai grammik ane-

xrthta an kai mno an, gia kje 1, 2, . . . , n F ,

isqei

1x1+2x2+ +nxn = 0 1=2= =n=0.

33

Paradegmata:

(1) Ta diansmata

ei = (0, 0, . . . , 0, 1i, 0, . . . , 0), i [n],

tou dianusmatiko qrou (Rn,+, ) enai grammi-

k anexrthta, afo, gia kje 1, 2, . . . , n

R, enai

1e1 + 2e2 + + nen = 0

1(1, 0, 0, . . . , 0) + 2(0, 1, 0, . . . , 0)+

+ n(0, . . . , 0, 1) = (0, 0, . . . , 0)

(1, 0, 0, . . . , 0) + (0, 2, 0, . . . , 0)+

+ (0, . . . , 0, n) = (0, 0, . . . , 0)

(1, 2, . . . , n) = (0, 0, . . . , 0)

1 = 2 = = n = 0.

34

(2) Ta diansmata

x1 = (2,1, 0), x2 = (1, 3, 1), x3 = (1, 0, 2)

tou dianusmatiko qrou (R3,+, ) enai grammi-

k anexrthta, afo gia kje 1, 2, 3 R,

isqei ti

1x1 + 2x2 + 3x3 = 0

1(2,1, 0) + 2(1, 3, 1) + 3(1, 0, 2) = (0, 0, 0)

(21,1, 0)+(2, 32, 2)+(3, 0, 23)=(0, 0, 0)

21 + 2 3 = 0

1 + 32 = 0

2 + 23 = 0

1 = 2 = 3 = 0.

35