120: Dimension d`un espace vectoriel. Rang. Exemples

E K
E
Kn
E A ⊂A0E A0
A E B A ⊂B⊂A0
E
p p + 1
Kn
n
n
E L −ev K ⊂L
p E K −ev n ∗p
Knn
Kn[X]n+ 1
n n
Rn
n

F E F
E E E
A B E
f(ei)ei
E F
dim(E) + dim(F)dim(E) + dim(F)−
dim(E∩F)
E F L(E, F )
E F
(n, m)K
E×F(dimE)(dimF )
E/F dim(E)−dim(F)
(e1, . . . , ep)

Kerf f
E F rg(f) = n
U E dimU =dim(Ker(f)) ∩rg(fU)
|rg(f)−rg(g)|6rg(f+g)6rg(f) + rg(g)
E
g◦f6rg(g)rg(f)
Ker(g)Im(f)Ker(g)∩Im(f)
rg(a) + rg(b)6rg(ab) + n
rg(f)6rg(g)
uf =gv
r Ir
A
L K L
K−ev L
L
kZ/pZ
|k|=pn
k⊂K⊂L[K:k][L:K] = [L:k]

L L =K(α)α
T7→ P(α)
µα
α
K[α] = K(α)
dimKK[α]<∞
P K[X]
p(2) QX2−2Q[p(2)]
3p(7) + p(2) : X6−6X4−14X3+ 12X2−84X+ 41
A M
A M
A
A A
A1,...An
A0= (0,0),(1,0)
M∈An
Ai=Ai−1∪MiMiAi−1
x(x, 0)
Q
URnf:U→RpC1
a U
f1fp
V f f
ϕ ϕ(f1(), . . . , fp(x) = 0 V f
C1

URnf:U→Rp
C1a
fRnRpf
C1
r
r
Ir
VRna∈V d V
a d C1U
aRnd F (U∩V) = V0∩F(U)
V0=Rd× {0} ⊂ RnV d RnV
d
CkC∞
URnf1, . . . , fpURC1
V x x ∈U f1(x1,...xn) = 0 . . . fp(x1,...xn) = 0
vRn
a d
a g Rn−p
U∩M=g−1(0)
ARp(a1, . . . , ap)
C∞U∩M
V a d a
d a V
a
P∂jfi(a)vj= 0
SLn(R)n2−1
SLn(C) 2(n2−1)
On, SOn(R)tM=−M
n(n−1)/2
Un(C)t¯ M=−M n2