aplicatiiliniarerom

Cluj Napoca, 2012 Radu

Aplicatii liniare

In acest capitol vom studia aplicatii liniare, aplicatii compatibile cu structura de spatiu

vectorial.

Definitia 22. Fie V si W s.v. pesteF. O aplicatie liniara de la V la W este o aplicatief : V W cu prop.fpv ` uq fpvq ` fpuq @ v, u P V and , P F.

Multimea aplicatiilor liniare de la V la W o vom nota cu LFpV,W q or HomFpV,W qObsfp0V q 0W si

fpni1iviq

ni1ifpviq, @ i P F, @vi P V, i 1, . . . , n

ker f f1p0W q tv P V |fpvq 0wu, andimf fpV q tw P W |D v P V, fpvq wu

Algebra liniara 29


Definitia 23. Multimile ker f si fpV q se numeasc nucleul, resp. imaginea lui f .

Propositia 24. Nucleul si imaginea unei aplicatii liniare f : V W sunt subspatii alelui V , resp. W .

dem. gata dem.

In cazul finit dimensional avem:

Teorema 25. Fie f : V W linara, V si W s.v. peste F, V finit dimensional.

dimV dim ker f ` dim fpV q.

Dem Fie n si m dimensiunile lui V resp. ker f , m n. Fie te1, . . . emubaza n ker f . Sistemul de vectori l.i. e1, . . . , em poate fi completat la o baza

te1, . . . em, em`1, . . . enu a lui V .Aratam ca fpem`1q, . . . , fpenq formeaza o baza a lui fpV q. este suficient sa aratam

ca fpem`1q, . . . , fpenq sunt l. ind., deoarece genereaza fpV q.

Algebra liniara 30


Pp. ca fpem`1q, . . . , fpenq nu sunt l.i. Exista am`1, . . . an P F a..

nkm`1

kfpekq 0W ,

iar din liniaritatea lui f ,

fpn

km`1kekq 0W ,

adica

v1

nkm`1

kek P ker f.

si deci se sunt comb. liniara de e1, . . . em. Dar e1, . . . en formeaza baza in V deci

km`1 kn 0, adica fpem`1q, . . . , fpenq sunt l.i.. gata dem.

Algebra liniara 31


Teorema 26. Fie F : V W liniara V si W s.v., si dimV dimW 8. Atunci,fpV q W d.s.n.d ker f t0V u. In particular f is surj. d.s.n.d. A bij.

Dem Pp. ker f t0vu. fpV q subspatiu a luiW , deci dimV dim fpV q dimW ,adica dim fpV q dimW , adica fpV q W .

fpV q W implica ker f t0V u analog.sf. dem

Propositia 27. Fie f : V W liniara V,W s.v. peste F. Daca f este bijectie ,rezulta ca inversa f1 : W V este liniara.Dem f bij. @w1, w2 P W , D! v1, v2 P V ,a.i. fpviq wi, i 1.2. Deoarece f liniara, avem

1w1 ` 2w2 a1fpv1q ` a2fpv2q fpa1v1 ` 2v2q.

Deci a1v1 ` 2v2 f1p1w1 ` 2w2q, sof1p1w1 ` 2w2q 1f1pw1`q ` 2f1pw2q

Algebra liniara 32


sf. dem

Definitia 28. O aplicatie liniara bij. f : V W intre s.v. V,W peste F s.n.izomorfism intre V si W

Un s.v. V s.n. izomorf cu un s.v. W daca exista un izomorfism f : V W . Dacas.v. V si W sunt izomorfe, vom nota acest fapt cu V W .

Example

Fie V un F s.v. si V1, V2 subspatii suplementare adica V V1 V2. @v P V avemdescompunerea unica v v1 ` v2, cu v1 P V1 and v2 P V2. Aplicati a

p : V V1, ppvq v1, @v P V

s.n. proiectia lui V pe V1, paralela cu V2,.

Aplicatia s : V V, spvq v1 v2, @v P V s.n.simetria lui V relativa la V1,paralela V2.

Algebra liniara 33


Avem v P V1, v2 0, adica ppvq v and spvq v, iar pt. v P V2, v1 0, soppvq 0 and spvq v.

Algebra liniara 34


Proprietati ale lui LpV,W q

Propositia 29. Fie f : V W lniara intre s.v. V,W peste F.

1. Daca V1 V subsp. a lui V , atunci fpV1q subspatiu a lui W .2. Daca W1 subspatiu a lui W , atunci f

1 subspatiu a lui V .

Dem 1. Let w1, w2 be in fpV q. It follows that there exists v1, v2 P V such thatfpviq wi, i 1, 2. The, for every , P F we have

w1 ` w2 fpv1q ` fpv2q fpv1 ` v2q P fpV q.2. For v1, v2 P f1pW1q we have that fpv1q, fpv2q P W , so @ P F, fpv1q `

fpv2q P W . Because F is linear fpv1q ` fpv2q fpv1 ` v2q v1 ` v2 Pf1pW q.

sf. dem

Algebra liniara 35


Propositia 30. Fie f : V W apl. liniara intre sp. vect. V,W .

1. f este injectiva ker f t0u.2. f surjectiva fpV q W .3. f bijectiva ker f t0u and fpV q W .

Dem 1 PP. ca f inj. Deoarece fp0q 0 P W rezuta ca ker f t0u V . Ptincluziunea inversa ker f t0u. Let v1, v2 P V with fpv1q fpv2q. Rezulta cafpv1 v2q 0 si deoarece ker f t0u it follows that v1 v2.

2 and 3 similar (pt voi).

sf. dem

Propositia 31. Fie f : V W liniara intre sp vect V,W iar S tvi|i P Iu un sistemde vectori in V .

1. daca f inj. S este liniar ind., atunci fpSq este lin. independent.

Algebra liniara 36


2. daca f surj S sist. de generatori, atunci fpSq este sist. de gen.3. daca t bij. si S baza V , atunci fpSq este baza W .

Dem 1. Fie tw1, . . . wnu o parte finita din fpSq, and i P F with ni1 aiwi 0.Exista vi P V a.i. fpviq wi, @ i 1, . . . n. Atunci ni1 aiwi ni1 aifpviq fpni1 aiviq 0, so ni1 aivi 0. Deoarece S este lin. ind. i 0@ i 1, . . . , n,deci fpSq este lin. ind..

2. Fie w P W . exista v P V a.i. fpvq w. Deoarece S este sistem de generatoriexista in S, vi, si scalarii i P F, i1, . . . , n a.i. ni1 aivi v. rezulta ca:

w fpvq fpni1aiviq

ni1aifpviq,

3. Deoarece f este bij. si S este baza V , rezulta ca 1 si 2 au loc, deci fpSq is a bazapt. W . sf. dem

Algebra liniara 37


Definitia 32. Let f, g : V W be linear maps between the linear spaces V.W overF, and inF. We define

1. f ` g : V W by pf ` gqpvq fpvq ` gpvq.@ v P V , the sum of the linear maps,and

2. f : V W by pfqpvq fpvq, @ v P V, @ P F, the scalar multiplication of alinear map.

Propositia 33. With the operations above defined LpV,W q becomes a vector spaceover F.

In the next we specialize the study of the linear maps, namely we consider the case

V W .Definitia 34. The set of endomorphisms of e linear space pV q is the set:

EndpLq tf : V V |f linearu

By the results from the previous section, EndpV q is an F linear space.

Algebra liniara 38


LetW,U another linear spaces over the same field F, f P LpV,W q and g P LpW,Uq.We define the product (composition) of f and g by h g f : V U ,

hpvq gpfpvqq, @ v P V.Propositia 35. The product of two linear maps is a linear map.

Moreover if f and g as above are isomorphisms, the product h g f is anisomorphism.

Dem

hpv1 ` v2q gpfpv1 ` v2qq gpfpv1q ` fpv2qq gpfpv1qq ` gpfpv2qq hpv1q ` hpv2q, @ v1, v2 P V, @ , P F

The last statement follows from the fact that h is a linear bijection. sf. dem

It can be shown that the composition is distributive with respect to the sum of linear

maps, so EndpV q becomes an unitary ring.

Algebra liniara 39


Propositia 36. The isomorphism between two F linear spaces is an equivalence relation.

Definitia 37. Let V be an F linear space. The set

AutpV q tf P EndpV q|f isomorphismu

is called the set of automoprhisms of the vector space V .

Propositia 38. AutpV q is a group with respect to the composition.

Dem It is only needed to list the properties.

1. the identity map 1V is the unit element.

2. g f is an automorphism for f and g automorphisms.3. the inverse of an automorphism is an automorphism.

sf. dem

Algebra liniara 40


The group of automoprhisms of a linear space is called the general linear group and is

denoted by GlpV q.

Example

Projectors endomorphisms. An endomorphismp : V V is called projector of thelinear space F is p2 p (p2 p p). If p is a projector, then:1. ker p ppV q V2. the endomorphism q 1V p s again a projector.Denote v1 ppvq and v2 v v1, it follows that ppv2q ppvq ppv1q ppvq p2pvq 0v, so v2 P ker f . It follows that

v v1 ` v2, @ v P V,where v1 P fpV qand v2 P fpV q,

and moreover the decomposition is unique, so we have the direct sum decomposition

ker pppV q V . For the last assertion simply compute q2 p1V pq p1V pq 1V p p ` p2 1V p q, because p is a projector. It can be seen thatqpV q ker p and ker q qpV q. Denote by v1 ppV q and V2 ker p. It follows

Algebra liniara 41


that p is the projection of V on V1, parallel with V2, and q is the projection of V on

V2 parallel with V1.

Involutive automorphisms. An operator s : V V is called involutive iff s2 1V .From the definition and the previous example one have:

1. an involutive operator is an automorphism

2. for every involutive automorphism, the linear operators:

ps : V V, pspvq 12pv ` spvqq

qs : V V, qspvq 12pv spvqq

are projectors and satisfy the relation ps ` qs 1V .3. reciprocally, for a projector p : V V , the operator sp : V V , given bysppvq 2ppvq v is an involutive automorphism.

From the previous facts it follows that ps s s ps p, sp p p sp p.An involutive automorphism s ia a symmetry of V with respect to the subspace pspV q,parallely with the subspace ker ps.

Algebra liniara 42


Linear maps between finite dimensional subspaces

Up to now we studied the global properties of linear maps. In this section we are

interested to see how they look like locally, for example how they acts on basis, or what

objects characterize the transformation of vectors by linear maps.

All this section is related to finite dimensional spaces. We will write V n for a vector

space of dimension n.

Let V n a linear space over F, of dimension n, and E te1, . . . , enu a basis. Avector v P V can be uniquely written as v ni1 viei. We will call pv1, . . . , vnq thecoordinates (components) of v in the basis E.

We will start this section with the change of basis. Consider two basis E te1, . . . , enu and F tf1, . . . , fnu, of the vector space V . It follows that there existsthe scalars aij P F, i, j 1, . . . , n such that

Algebra liniara 43


fj ni1aijei, j 1, . . . n

For e vector v which has the components pv1, . . . , vnq and pv11, . . . , v1nq resp. wehave its representation in the basis E and F

v nj1v1jfj

j1v1i

ni1aijei

ni1

nj1aijv

1jei, (1)

that is

ni1viei

ni1pnj1aijv

1jqei,

and from the fact that the coordinates in a basis are unique we have that

Algebra liniara 44


vi nj1aijv

1j, i 1, n

We denote

rvsE v1...

vn

and A paijqi,j1,n, we can write

rvsE ArvsF ,

relation which express the change from the basis E to the basis F . Let G tg1, . . . , gnu be third coordinate system, and rvsG the coordinates of v in G, we have

Algebra liniara 45


rvsF A1rvsG,for some matrix A1.We deduce that

rvsE ArvsF AA1rvsG.

Particularly it is true for G E, so in this case we obtain that In AA1.

Algebra liniara 46

aplicatiiliniarerom

Documents

Transcript of aplicatiiliniarerom