Espaces vectoriels de dimension finie 1 Familles de vecteurs

K R C
EK(x1, x2, . . . , xp)p E p ∈N∗
(x1, x2, . . . , xp)E
E E = Vect(x1, x2, . . . , xp)E
Kx1, x2,...xp
(x1, x2, . . . , xp)E
x1, x2, . . . , xp
∀(λ1, λ2, . . . , λp)∈Kp, p
X
k=1
λkxk= 0E!⇒(λ1=λ2=· · · =λp= 0)
∃(λ1, λ2, . . . , λp)∈Kp,
p
X
k=1
λkxk= 0E(λ1, λ2, . . . , λp)6= (0,0,...,0)
(x1, x2, . . . , xp) Vect(x1, x2, . . . , xp)
x1, x2, . . . , xp
(x1, x2, . . . , xp)E
(x1, x2, . . . , xp)E E
Kx1, x2,...xp
∀y∈E, ∃!(λ1, λ2,...λp)∈Kp, y =
p
X
i=1
λixi

y∈E
y(x1, x2, . . . , xp)
Kn

x1= (1,0,2,3), x2= (1,1,0,1), x3= (2,−1,1,2) ∈R4(x1, x2, x3)
u1=(−1,1,1), u2= (1,−1,1), u3= (1,1,−1) ∈R3(u1, u2, u3)
R3t= (1,2,1)
(e1, e2,...ep)Kp(x1, x2, . . . , xp)
p E KpE
u:Kp→E k ∈[[1, p]], u(ek) = xk
(x1, x2, . . . , xp)E
(x1, x2, . . . , xp)E
(x1, x2, . . . , xp)E
x1, x2,...xp∈E p ∈NE= Vect(x1, x2,...xp)

EKL
EGEL
G B E
E E
Edim E
E p p+1

{0E}
1
2
E
n∈NS p E
S p ≤n S
S p ≥n S
S n E
n S
x1= (1,0,1,0), x2= (0,1,0,1), x3= (−1,0,0,1), x4= (0,0,−2,0)
(x1, x2, x3, x4)R4