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(F, , ) V + · F (V, +, ·) (F, , ) (V, +) λ, μ F x, y V (i) λ · (x + y)= λ · x + λ · y (ii) (λ μ) · x = λ · x + μ · x (iii) (λ μ) · x = λ · (μ · x) (iv) 1 · x = x 1

12 Vector Spaces I

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Page 1: 12 Vector Spaces I

Dianusmatiko� q¸roiOrismo�JewroÔme èna s¸ma (F,⊕,⊙), me monadia�o stoi-qe�o to 1 kai èna mh kenì sÔnolo V efodiasmèno memia eswterik  pr�xh + kai mia exwterik  pr�xh · mesÔnolo telest¸n to F pro ta arister�. H algebri-k  dom  (V, +, ·) onom�zetai pro ta arister� dianu-smatikì q¸ro (  grammikì q¸ro ) ep� tous¸mato (F,⊕,⊙) an kai mìno an(1) H dom  (V, +) e�nai abelian  om�da.(2) Gia k�je λ, µ ∈ F , x,y ∈ V isqÔoun:

(i) λ · (x + y) = λ · x + λ · y.(ii) (λ ⊕ µ) · x = λ · x + µ · x.(iii) (λ ⊙ µ) · x = λ · (µ · x).(iv) 1 · x = x.

1

Page 2: 12 Vector Spaces I

Parathr sei :(1) An h exwterik  pr�xh · me sÔnolo telest¸n pro ta arister� antikatastaje� ston parap�nw ori-smì apì exwterik  pr�xh · me sÔnolo telest¸npro ta dexi�, kai epomènw k�je ìro th mor-f  λ·x sthn idiìthta 2 antikatastaje� apì x·λ,tìte prokÔptei o orismì tou pro ta dexi� dia-nusmatikoÔ q¸rou.Epeid  up�rqei pl rh analog�a metaxÔ twn dÔ-o aut¸n ennoi¸n, ìpw kai twn apotelesm�twnpou ti dièpoun, ja perioristoÔme sth melèth twnpro ta arister� dianusmatik¸n q¸rwn, tou o-po�ou ja onom�zoume apl� dianusmatikoÔ q¸rou (  grammikoÔ q¸rou ).(2) Ta stoiqe�a tou sunìlou F onom�zontai tele-stè kai shmei¸nontai me ellhnik� gr�mmata (su-n jw ta α, β, γ, κ, λ, µ), en¸ ta stoiqe�a tou Vonom�zontai dianÔsmata kai shmei¸nontai melatinik� gr�mmata (sun jw ta x,y, z,u,v,w).Gia na mhn up�rqei sÔgqush, ta dianÔsmata jad�dontai me èntonh graf  (ìpw fa�netai kai stonorismì tou dianusmatikoÔ q¸rou).2

Page 3: 12 Vector Spaces I

(3) Oi eswterikè pr�xei ⊕, ⊙ tou s¸mato , giaaplìthta ja sumbol�zontai me + kai · ant�stoiqa.(4) H exwterik  pr�xh tou sunìlou twn dianusm�-twn onom�zetai bajmwtì (  arijmhtikì ) pol-laplasiasmì . Sta epìmena, ja parale�petai tosÔmbolo · tou bajmwtoÔ pollaplasiasmoÔ, dh-lad  gia λ ∈ F kai x ∈ V , ja gr�foume λx ant�λ · x.

K�je dianusmatikì q¸ro ep� tou s¸mato (R, +, ·)(ant. (C, +, ·)) onom�zetai pragmatikì (ant. mi-gadikì ) dianusmatikì q¸ro .O dianusmatikì q¸ro tou opo�ou o forèa periè-qei èna mìno stoiqe�o (to 0) onom�zetai mhdenikì dianusmatikì q¸ro .

3

Page 4: 12 Vector Spaces I

Parade�gmata:(1) To sÔnolo twn migadik¸n arijm¸n C or�zei prag-matikì dianusmatikì q¸ro. Oi pr�xei or�zontaiw ex  : 'Estw α, β, γ, δ, λ ∈ R me x = α + iβ,y = γ + iδ. Tìte

x + y = α + iβ + γ + iδ = α + γ + i(β + δ) ∈ C,

λx = λ(α + iβ) = λα + iλβ ∈ C,

4

Page 5: 12 Vector Spaces I

(2) 'Estw Pn to sÔnolo twn poluwnÔmwn, me prag-matikoÔ suntelestè , bajmoÔ mikrìterou   �soume n. To sÔnolo Pn, efodiasmèno me ti akì-louje pr�xei or�zei pragmatikì dianusmatikìq¸ro: 'Estw p(x), q(x) ∈ Pn, mep(x) = anxn + an−1x

n−1 + · · · + a1x + a0kaiq(x) = bnxn + bn−1x

n−1 + · · · + b1x + b0.Tìte, oi pr�xei th prìsjesh kai tou bajmw-toÔ pollaplasiasmoÔ or�zontai ant�stoiqa apìti akìlouje sqèsei (p + q)(x) = (an + bn)xn + (an−1 + bn−1)x

n−1+

· · · + (a1 + b1)x + a0 + b0

(λp)(x) = λanxn+λan−1xn−1+· · ·+λa1x+λa0, λ ∈ R

(3) K�je s¸ma (F, +, ·) e�nai dianusmatikì q¸ro ep� tou eautoÔ tou (ìpou h pr�xh · tou s¸mato jewre�tai kai w bajmwtì pollaplasiasmì ).5

Page 6: 12 Vector Spaces I

(4) Gia k�je s¸ma (F, +, ·), sto sÔnolo Fn, n ≥ 2,or�zontai oi pr�xei (x1, x2, . . . , xn) + (y1, y2, . . . , yn)

= (x1 + y1, x2 + y2, . . . , xn + yn)kaiλ(x1, x2, . . . , xn) = (λx1, λx2, . . . , λxn), λ ∈ F.Ja apodeiqje� ìti e�nai dianusmatikì q¸ro ep� tou s¸mato (F, +, ·), me mhdenikì stoiqe�oto 0 = (0, 0, . . . , 0).Pr�gmati, gia k�je

x = (x1, x2, . . . , xn),y = (y1, y2, . . . , yn),

z = (z1, z2, . . . , zn),e�nai1. (Antimet�jesh)x + y = (x1 + y1, x2 + y2, . . . , xn + yn)

= (y1 + x1, y2 + x2, . . . , yn + xn)

= y + x

6

Page 7: 12 Vector Spaces I

(Proseta�rish)x + (y + z)

=(x1, x2, . . . , xn) + (y1 + z1, y2 + z2, . . . , yn + zn)

=(x1 + (y1 + z1), x2 + (y2 + z2), . . . , xn + (yn + zn))

=((x1 + y1) + z1, (x2 + y2) + z2, . . . , (xn + yn) + zn)

=(x1 + y1, x2 + y2, . . . , xn + yn) + z

=(x + y) + z(Oudètero stoiqe�o)x + 0 = (x1, x2, . . . , xn) + (0, 0, . . . , 0)

= (x1 + 0, x2 + 0, . . . , xn + 0)

= (x1, x2, . . . , xn) = x(Summetrikì stoiqe�o)(x1, x2, . . . , xn) + (−x1,−x2, . . . ,−xn)

= (x1 − x1, x2 − x2, . . . , xn − xn)

= (0, 0, . . . , 0) = 0Opìte, to (−x1,−x2, . . . ,−xn) e�nai to ant�-jeto tou x = (x1, x2, . . . , xn). 'Ara h dom (V, +) e�nai abelian  om�da.

7

Page 8: 12 Vector Spaces I

Epiplèon, an λ, µ ∈ F , e�nai2.i)

λ(x + y) = λ(x1 + y1, x2 + y2, . . . , xn + yn)

= (λ(x1 + y1), λ(x2 + y2), . . . , λ(xn + yn))

= (λx1 + λy1, λx2 + λy2, . . . , λxn + λyn)

= (λx1, λx2, . . . , λxn) + (λy1, λy2, . . . , λyn)

= λx + λy

2.ii)(λ + µ)x = ((λ + µ)x1, (λ + µ)x2, . . . , (λ + µ)xn)

= (λx1 + µx1, λx2 + µx2, . . . , λxn + µxn)

= (λx1, λx2, . . . , λxn) + (µx1, µx2, . . . , µxn)

= λx + µx

2.iii)(λµ)x = ((λµ)x1, (λµ)x2, . . . , (λµ)xn)

= (λ(µx1), λ(µx2), . . . , λ(µxn))

= λ(µx1, µx2, . . . , µxn)

= λ(µx)

2.iv)1x = (1x1, 1x2, . . . , 1xn) = (x1, x2, . . . , xn) = x

8

Page 9: 12 Vector Spaces I

'Ara, h dom  (Fn, +, ·) e�nai èna dianusmatikì q¸ro ep� tou s¸mato (F, +, ·).Parat rhsh: Genikìtera, an(V1,+,·), (V2,+,·), . . . , (Vn, +, ·), n ≥ 2,e�nai dianusmatiko� q¸roi ep� tou s¸mato (F, +, ·),tìte sto sÔnolo V = V1×V2×· · ·×Vn, or�zontaioi pr�xei

(x1,x2, . . . ,xn) + (y1,y2, . . . ,yn)

=(x1 + y1,x2 + y2, . . . ,xn + yn)kaiλ(x1,x2, . . . ,xn) = (λx1, λx2, . . . , λxn),ìpou λ ∈ F kai xi,yi ∈ Vi, gia k�je i ∈ [n].ApodeiknÔetai an�loga ìti h dom  (V, +, ·) e�naidianusmatikì q¸ro ep� tou s¸mato (F, +, ·),o opo�o onom�zetai dianusmatikì q¸ro ginìmeno twn dianusmatik¸n q¸rwn (Vi, +, ·),

i ∈ [n].9

Page 10: 12 Vector Spaces I

(5) Sto sÔnolo M2 ìlwn twn 2× 2 mhtr¸n me stoi-qe�a sto R, dhlad  th morf  [α β

γ δ

], ìpouα, β, γ, δ ∈ R, or�zontai oi parak�tw dÔo pr�-xei [

α1 β1

γ1 δ1

]

+

[

α2 β2

γ2 δ2

]

=

[

α1 + α2 β1 + β2

γ1 + γ2 δ1 + δ2

]

kaiλ

[

α β

γ δ

]

=

[

λα λβ

λγ λδ

]

.

Ja apodeiqje� ìti h dom  (M2, +, ·) e�nai èna pragmatikì dianusmatikì q¸ro . Pr�gmati, giathn pr�xh + tou M2 e�nai[

α1 β1

γ1 δ1

]

+

[

α2 β2

γ2 δ2

]

=

[

α1 + α2 β1 + β2

γ1 + γ2 δ1 + δ2

]

=

[

α2 + α1 β2 + β1

γ2 + γ1 δ2 + δ1

]

=

[

α2 β2

γ2 δ2

]

+

[

α1 β1

γ1 δ1

]

,

opìte h pr�xh e�nai antimetajetik .10

Page 11: 12 Vector Spaces I

[

α1 β1

γ1 δ1

]

+

([

α2 β2

γ2 δ2

]

+

[

α3 β3

γ3 δ3

])

=

[

α1 β1

γ1 δ1

]

+

[

α2 + α3 β2 + β3

γ2 + γ3 δ2 + δ3

]

=

[

α1 + α2 + α3 β1 + β2 + β3

γ1 + γ2 + γ3 δ1 + δ2 + δ3

]

=

[

α1 + α2 β1 + β2

γ1 + γ2 δ1 + δ2

]

+

[

α3 β3

γ3 δ3

]

=

([

α1 β1

γ1 δ1

]

+

[

α2 β2

γ2 δ2

])

+

[

α3 β3

γ3 δ3

]

,

opìte h pr�xh + e�nai prosetairistik .Epiplèon,[

α β

γ δ

]

+

[

0 0

0 0

]

=

[

α β

γ δ

]

,

opìte h m tra [0 0

0 0

] e�nai to mhdenikì stoiqe�otou M2 w pro thn pr�xh +.11

Page 12: 12 Vector Spaces I

Tèlo , epeid [

α β

γ δ

]

+

[

α′ β′

γ′ δ′

]

=

[

0 0

0 0

]

[

α + α′ β + β′

γ + γ′ δ + δ′

]

=

[

0 0

0 0

]

⇔α + α′ = β + β′ = γ + γ′ = δ + δ′ = 0

α′ = −α

β′ = −β

γ′ = −γ

δ′ = −δ

èpetai ìti k�je [α β

γ δ

]

∈ M2 èqei ant�jeto stoi-qe�o to [−α −β

−γ −δ

].'Ara, h dom  (M2, +) e�nai abelian  om�da.12

Page 13: 12 Vector Spaces I

Epiplèon, gia λ, µ ∈ R e�naiλ

([

α1 β1

γ1 δ1

]

+

[

α2 β2

γ2 δ2

])

= λ

[

α1 + α2 β1 + β2

γ1 + γ2 δ1 + δ2

]

=

[

λ(α1 + α2) λ(β1 + β2)

λ(γ1 + γ2) λ(δ1 + δ2)

]

=

[

λα1 + λα2 λβ1 + λβ2

λγ1 + λγ2 λδ1 + λδ2

]

=

[

λα1 λβ1

λγ1 λδ1

]

+

[

λα2 λβ2

λγ2 λδ2

]

[

α1 β1

γ1 δ1

]

[

α2 β2

γ2 δ2

]

,

(λ + µ)

[

α β

γ δ

]

=

[

(λ + µ)α (λ + µ)β

(λ + µ)γ (λ + µ)δ

]

=

[

λα + µα λβ + µβ

λγ + µγ λδ + µδ

]

=

[

λα λβ

λγ λδ

]

+

[

µα µβ

µγ µδ

]

[

α β

γ δ

]

+ µ

[

α β

γ δ

]

,

λ

(

µ

[

α β

γ δ

])

= λ

[

µα µβ

µγ µδ

]

=

[

λ(µα) λ(µβ)

λ(µγ) λ(µδ)

]

=

[

(λµ)α (λµ)β

(λµ)γ (λµ)δ

]

= (λµ)

[

α β

γ δ

]

13

Page 14: 12 Vector Spaces I

kai1

[

α β

γ δ

]

=

[

1α 1β

1γ 1δ

]

=

[

α β

γ δ

]

.'Ara, h dom  (M2, +, ·) e�nai èna pragmatikì dianusmatikì q¸ro .Parat rhsh: Genikìtera, apodeiknÔetai ìti to sÔ-nolo ìlwn twn n × m mhtr¸n, n,m ∈ N

∗, apì stoi-qe�a enì s¸mato (F, +, ·), efodiasmèno me ti pr�-xei th prìsjesh kai tou bajmwtoÔ pollaplasia-smoÔ mhtr¸n, e�nai èna dianusmatikì q¸ro ep� tous¸mato (F, +, ·).

14

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Idiìthte dianusmatik¸n q¸rwn'Estw (V, +, ·) èna mh mhdenikì dianusmatikì q¸-ro ep� tou s¸mato (F, +, ·). Tìte, gia k�je λ, µ ∈

F , x,y ∈ V , isqÔoun ta akìlouja:(1) λ0 = 0, ìpou 0 e�nai to mhdenikì stoiqe�o th om�da (V, +).(2) 0x = 0, ìpou 0 e�nai to mhdenikì stoiqe�o th om�da (F, +).(3) An λx = 0, tìte λ = 0   x = 0.(4) λ(−x) = −(λx).(5) (−λ)x = −(λx).(6) (−1)x = −x.(7) (−λ)(−x) = λx.(8) An λ ∈ F ∗, tìte λx = λy ⇒ x = y. (Nìmo diagraf  gia tou telestè .)(9) An x ∈ V ∗, tìte λx = µx ⇒ λ = µ. (Nìmo diagraf  gia ta dianÔsmata.)(10) λ(x − y) = λx − λy.(11) (λ − µ)x = λx − µx.15

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Dianusmatiko� upìqwroiAn (V, +, ·) e�nai èna dianusmatikì q¸ro ep� tous¸mato (F, +, ·) kai W èna mh kenì uposÔnolo touV , kleistì w pro ti pr�xei + kai · tou V , tìteoi periorismo� twn pr�xewn aut¸n sto sÔnolo W ×

W kai F ×W ant�stoiqa onom�zontai epagìmene pr�xei sto W apì to V kai, q�rin aplìthta ,sumbol�zontai ep�sh me + kai · ant�stoiqa.An epiplèon h dom  (W, +, ·) e�nai èna dianusma-tikì q¸ro , tìte onom�zetai dianusmatikì ( grammikì ) upìqwro (  apl� upìqwro ) tou(V, +, ·).K�je dianusmatikì q¸ro mpore� na jewrhje� w upìqwro tou eautoÔ tou. An 0 e�nai to oudèterostoiqe�o th om�da (V, +), tìte h dom  ({0}, +, ·)e�nai upìqwro tou (V, +, ·) kai onom�zetai mhdeni-kì upìqwro .K�je forèa upoq¸rou (W, +, ·) perièqei to 0, a-foÔ gia x ∈ W e�nai 0 = 0x kai W kleistì w pro thn pr�xh ·.

16

Page 17: 12 Vector Spaces I

Sta epìmena, gia lìgou aplìthta , ja taut�zoumeton forèa enì upìqwrou me ton upìqwro, dhlad  jagr�foume W upìqwro tou (V, +, ·) ant� W forèa upoq¸rou tou (V, +, ·).Protash 1. 'Estw (V, +, ·) dianusmatikì q¸ro ep�tou s¸mato (F, +, ·) kai W èna mh kenì uposÔnolotou V . Tìte, ta parak�tw e�nai isodÔnama:(1) λx ∈ W , kai x + y ∈ W , gia k�je λ ∈ F kaix,y ∈ W .(2) to W e�nai upìqwro tou (V, +, ·).(3) λx + µy ∈ W , gia k�je λ, µ ∈ F kai x,y ∈ W .(4) λx + y ∈ W , gia k�je λ ∈ F kai x,y ∈ W .

17

Page 18: 12 Vector Spaces I

Parade�gmata:(1) JewroÔme ton pragmatikì dianusmatikì q¸ro(R3, +, ·) (dhlad  ton trisdi�stato q¸ro) kaiv,w ∈ R

3. Tìte, ta sÔnolaW1 = {αv : α ∈ R} kai W2 = {αv+βw : α, β ∈ R}or�zoun dianusmatikoÔ upoq¸rou tou (R3, +, ·),afoÔ

α1v + α2v = (α1 + α2)v ∈ W1

λ(αv) = (λα)v ∈ W1kai (α1v + β1w) + (α2v + β2w) = (α1 + α2)v +

(β1 + β2)w ∈ W2

λ(αv + βw) = (λα)v + (λβ)w ∈ W2.

18

Page 19: 12 Vector Spaces I

Gewmetrik�, o upìqwro W1, ìtan v 6= (0, 0, 0),parist�nei thn euje�a pou pern� apì thn arq twn axìnwn kai perièqei to di�nusma v.x

y

z

O

bv

An epiplèon ta dianÔsmata w,v den e�nai sug-grammik�, (dhlad  isqÔei ìti w 6= λv, gia k�jeλ ∈ R) tìte o upìqwro W2 parist�nei to ep�-pedo pou perièqei thn arq  twn axìnwn kai tadianÔsmata v kai w.

x

y

z

O

b

v

bw

19

Page 20: 12 Vector Spaces I

Askhsh 1.Na apodeiqjoÔn oi idiìthte twn dianu-smatik¸n q¸rwn:(i) λ0 = 0, (iii) λx = 0 ⇒ λ = 0   x = 0,(ii) 0x = 0, (iv) λ(−x) = −(λx),gia k�je λ ∈ F , x ∈ V , ìpou (V, +, ·) dianusmatikì q¸ro ep� tou s¸mato (F, +, ·).Lush. (i) Gia λ ∈ F , x ∈ V , e�nai

λx + λ0 = λ(x + 0) = λx,opìte, apì ton nìmo diagraf  th om�da (V, +),prokÔptei ìti λ0 = 0.(ii) Gia λ ∈ F , x ∈ V , e�nai

λx + 0x = (λ + 0)x = λx,opìte, apì ton nìmo diagraf  th om�da (V, +),prokÔptei ìti 0x = 0.(iii) An λ 6= 0, ja deiqje� ìti x = 0. Pr�gmati,tìte ja up�rqei to ant�strofo λ−1 ∈ F kai

x = 1x = (λ−1λ)x = λ−1(λx) = λ−10 = 0.

(iv) Epeid  λ(−x) + λx = λ(−x + x) = λ0 = 0,èpetai ìti to λ(−x) e�nai to ant�jeto tou λx, dhlad λ(−x) = −(λx). �

56

Page 21: 12 Vector Spaces I

Askhsh 2 (bl. Prìtash 1). 'Estw (V, +, ·) dianu-smatikì q¸ro ep� tou s¸mato (F, +, ·) kai W ènamh kenì uposÔnolo tou V . Na apodeiqje� ìti to We�nai upìqwro tou (V, +, ·) an kai mìno an(1) λx + µy ∈ W,gia k�je λ, µ ∈ F kai x,y ∈ W.Lush.Arqik�, upot�jetai ìti to W e�nai upìqwro tou (V, +, ·) kai ja deiqje� ìti isqÔei h sunj kh 1.An λ, µ ∈ F kai x,y ∈ W , tìte, epeid  to W e�naikleistì w pro thn pr�xh ·, èpetai ìti λx, µy ∈ W ,opìte, epeid  to W e�nai kleistì w pro thn pr�xh+, èpetai ìti λx + µy ∈ W .Ant�strofa, upot�jetai ìti isqÔei h sunj kh (1)kai ja deiqje� ìti to W e�nai upìqwro tou (V, +, ·).Arke� na deiqje� ìti to sÔnolo W e�nai mh kenì, klei-stì w pro thn pr�xh · kai ìti h dom  (W, +) e�-nai upoom�da th (V, +). (H sunj kh 2 tou orismoÔtou dianusmatikoÔ q¸rou, profan¸ isqÔei kai giata stoiqe�a tou W , afoÔ W ⊆ V .)'Etsi, an x,y ∈ W , tìte, efarmìzonta th sunj kh(1) gia λ = µ = 0, èpetai ìti

0x + 0y ∈ W ⇒ 0 + 0 ∈ W ⇒ 0 ∈ W,57

Page 22: 12 Vector Spaces I

�ra to W e�nai mh kenì kai m�lista perièqei to mhde-nikì stoiqe�o th om�da (V, +). Epiplèon, efarmì-zonta th sunj kh (1) gia µ = 0, èpetai ìtiλx + 0y ∈ W ⇒ λx + 0 ∈ W ⇒ λx ∈ W,�ra to W e�nai kleistì w pro thn pr�xh ·.Tèlo , gia na e�nai h dom  (W, +) upoom�da th

(V, +), arke� na isqÔeix,y ∈ W ⇒ x − y ∈ W,to opo�o ìmw prokÔptei �mesa, efarmìzonta thsunj kh (1) gia λ = 1 kai µ = −1.Epomènw , h dom  (W, +) e�nai abelian  om�da kai�ra h dom  (W, +, ·) e�nai upìqwro tou (V, +, ·). �

58