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vChapitreLinear2relativit?.Espacessvetectoriels:etgebratensoriels2.1On40insisteraensurR?f?rencesl'utilit?Blythdesalcestomenotions1,2.enEspacesm?caniqueectorielsquantiqueR E
u2 E u;v;w2 E
; 2R
+
(u;v)2 (EE)! (u +v)2E
u + (v +w) = (u +v) +w
u +v =v +u
0
u + 0 =u
u ( u)
u + ( u) = 0
:
( ;u )2RE! ( :u )2E
: (u +v) = :u + :v
( +):u = :u + :u
():u = : ( :u )
1:u = u
C
2R 2C
n 1 nE := R = R:::R n2 N u = (u ;:::;u )2
n 1 1 n nR u +v := (u +v ;:::;u +v )
1 n :u := ( u ;:::; u ) 2R dim (E) =n
nE :=C dim (E) =n
1E := C (R) R
u :x2R!u (x)2R
(u +v) (x) := u (x) +v (x) ( :u ) (x) := u (x)
dim (E) =1
1 3E :=C (R ) u (x;y;z)
u2E
nX
i iu = ue; u 2Ri
i=1
1 n(u ;:::u ) u

iu ubasee i=1;:::;n
nR n
e = (1; 0;::: 0)1
e = (0; 1; 0;::: 0)2
e = (0; 0; 0;::: 1)n
nC
jui2E
X
ijui = ujeii
i
E E =fu;vg (E)
u + v ; 2 R
Vect (E) E
E n = (E) (E) =1
(e :::e ) E (f ;:::f ) mn1 n 1 m
m =n
f = e +::: + e1 1 1 n n
f1
= 01
1e = (f e ::: e )1 1 2 2 n n1
E =Vect (f ;e ;:::;e )1 2 n
f = f + e ::: + e2 1 2 2 2 n n
::: f f2 n 1 2
= 02
E =Vect (f ;f ;e ;:::;e )1 2 3 n
E =Vect (f ;:::f )1 n
m > n f 2 Vect (f ;:::f ) (f )n+1 1 n j j
m =n
ndim (R ) =n
n n(C ) =n C
nC
e = (1; 0;::: 0) f = (i; 0;::: 0) :::e =1 1 n
(0;::: 0; 1);f == (0;::: 0;i) z = a + ib 2 Cn
n(z; 0;::: 0) =ae +bf dim (C ) = 2n1 1 R
d2 N E p (x) x2 R dd
jE e (x) = x j = 0:::d p (x)d jP P
i i ip (x) = px = pe (x) (E ) =d + 1i di i
1(C (R)) =1 Ed
E A NN aij
yA =A a =a 8i;j Eji ij
E
particulierundoncespace(etvainsiectoriel?compl5exe.edimL'espacecarr?escommeeutest44paeuts'?crit?treacconsid?r?,cp.83)ommeExerciceuntsespacequivqueectorielbaserpar?belp(i.e.onaSivel?ecd?duitdesqueco-(Sinonefs(refr?el).conDansescel'espacecas,uneetbasebaseestvnoncolin?aires.nespaceul,TalorsdimensionunestcomonomeseLfnauecmoinsuniqueest,,elnonfa?onnleul,unique,oserded?compveeutsuite,pOnonetdoncdimdonc.onD?monstr,Blythp(careuxtsupplesoseosonsrSoitquematrices6une.coDoncAlorsdoncc'eseraien.ec,.queetsuppOnpvMoncommeestsinonectorielcarpastserultan?mend?duiresimdeulsPropnform??trelespasositiont1..aPasear,exemplevsi'estenaseuvmaispauraitetcarosersitleconsid?randeonuniqued?compexisteosenombrleestenapp.dimension:l'espd?compeeutqueaonvdeec.pd?duitdimdimctoriel.not?On..dimquetrer).monation.a:.T2,On.d?duitilquetienquientousExemplesespac.Suppne).CHAPITRE52.2.baseDoncdesbase.de.etm?meSoit?uneecienestdeESPcomplexesethermitiennes,AstCESdirel'ensemautreble6descepsignieolynomesar?elosersdoncVECTORIELS,ETOncar.ossibletrer,timpunestvquir?el,nondecomplexe).degr?rouvceuneTENSORIELSet.leUnedebasevecteur2(E) =N
d2N
E :=f p (x ;:::;x ) m dgd 1 m
X
1 2 mp (x ;:::;x ) = p x x :::x1 m 1 2 m

= ( ;::: ) p 2 R +::: d1 m 1 m
Ed
d2N
E :=f p (x ;:::;x ) m dgd 1 m
dp ( x ;:::; x ) = p (x ;:::;x ) 2R1 m 1 m
+::: =d E1 m d
E;F
EF EF (u;v)2EF
0 0 0 0(u;v) + (u;v ) := (u +u;v +v )
: (u;v) := ( :u; :v )
ExerciceariablesUnetl'espacedeectorieletpunet,adegr?EspaceUnSoittelvp?rietsolynomeecienhomog?neseropcoTENSORIELSdesp,l'espaceolynomeExercices'?critdes(2.2)Onrouvdirecteacommeptourteltout2.vvececT?.VECTORIELSIlSolutions'?critectorieldoncdimcommeectoriel(2.2)etaSoitvolynomesec:.espacessaectoriels.vd?nitvsommel'espace:devd?duire?,ansoitl'espace?pvdescouplesTCHAPITRErouvdegr?erariablesuneESPbasevdelesl'espace?rationsvAectorielCESectorielETbase45,:d?duirevsaolynomesdimension.r?el.2.1.1.Somme:directevd'espacessoitv53.ectorielsetD?nition54.55.dimension.Soitl'espace.(e ;:::e ) E (f ;:::f ) F1 n 1 m
EF (e ; 0)2EF1
e (0;f )2EF f EF (e ;:::;e ;f ;:::f )1 1 1 1 n 1 m
dim (EF ) =n +m
G E;FGT S
E F =f0g Vect (E F ) = G G
E F G EF G
EF
E F E
E
uv (u v)2F
[u] u2E [u] = [v] uv
u [u]
E=F
[u] + [v] := [u +v]; [u] := [ u ]
0 0 0uu vv u =u +f
0 0 0v = v +g f;g2 F [u +v ] = [u +v +f +g] = [u +v]
0[ u ] = [ u + f ] = [ u ]
dim (E=F ) =dim (E) dim (F )
F (f ;:::f ) (e ;:::;e )1 m 1 n
E dim (F ) = m dim (E) = m +n
[e ];::: [e ] E=F u2E1 n P P P
i i iu = uf + U e [u] = U [e ] dim (E=F ) =ni i i
E =FG
G (E=F )
(f ) F (g ) Gi ji=1!m j=1!n
E=F [g ];::: [g ]1 n
E F
A :E!F
u;v2E ; 2R
A ( u + v ) = A (u) + A (v)
Ker (A) :=fu2E; A (u) = 0g : : A
Im (A) :=fv2F; 9u2E; v =A (u)g : : A
L (E;F )
E F
A (u)
E E
I (u) =uE
A :E!F (e ) E (f )i ji=1!n j=1!m
F A (e )2F Fi
mX
jA (e ) = A fi ji
j=1
j
A 2Ri
u2E E
X
iu = uei
i
v =A (u)2F F
X
jv = v fj
j
u
nX
jj iv = A ui
i=1

jA = Ai
i(u ) 0 1
1 0 1v 1 ! uB C
1B C B CA1=B C @ Aj@ A ::: Ai nu| {z }mv
A;mn
A
A E F
A
!
X X
i iv = A (u) =A ue = uA (e ) Ai i
i i
m mX X X
j ji i= u A f = A u fj ji i
i j=1 j=1
Pn jj iv = A u (f )ji=1 i j
v
L (E;F ) :=fA :E!Fg
dim (L (E;F )) =dim (E) dim (F )