Colle de Mathématiques : Semaine du 10 / 10 / 2016

RMn,p(R)RnR[X]RNRI
R
R
Rn[X]C0(I, R)Cn(I, R)
In
R Rn
I(e1,· · · , ep)RE
V ect(e1,· · · , ep)E
IE
E
I(e1, e2,...,en)E
(e1, e2,...,en)
(e1, e2,...,en)
(e1, e2,...,en)
(e1, e2,...,en)
E
RnRn[X]Mn,p(R)
E
E
IE n
n
n E
p p > n
n
n E
p p < n

IER−nFE
(FE)⇐⇒ (Fn)⇐⇒ (Fn).
rg(x1, x2,...,xn) = dim (V ect(x1, x2,...,xn))
rg(x1, x2,...,xn)6n
IE F E
F dim(F)6dim(E)
dim(F) = dim(E)F=E
r
V(aX +b) = a2V(X)
99K ((1,0,0),(1,1,0),(1,1,1)) R3
99K X3, X2+X+ 1, X3+X+ 1R[X]
99K f1:x7→ exp(x)f2:x7→ exp(2x)f3:x7→ exp(3x)
(f1, f2, f3)C∞(R,R)
99K F={(x;y;z)∈R3/ x +y+z= 0}R−
99K F={P∈R2[X], P (1) = 0}F F
99K F2= a+b b
a a −b(a, b)∈R2R
99K X X(Ω) = Nn∈NP(X=n) = 1
16 (n+ 1) 3 4nX
99K Z Z(Ω) = NnNP(Z=n) = 1
2n+1 Z
99K Z(Z−1) Z