1 Espaces vectoriels et sous

K=R C
K
K(E, +, .)
+
∀(x, y)∈E2, x +y∈E
∀(x, y, z)∈E3,(x+y) + z=x+ (y+z)
0E∀x∈E, x + 0E= 0E+x=x
E+∀x∈E, ∃y∈E;x+y=y+x= 0E
−x
∀(x, y)∈E2, x +y=y+x
(E, +)
.
∀λ∈K,∀x∈E, λ.x ∈E
∀x∈E, 1K.x =x
∀(λ, µ)∈K2,∀x∈E, (λ+µ).x =λ.x +µ.x
∀λ∈K,∀(x, y)∈E2, λ.(x+y) = λ.x +λ.y
∀(λ, µ)∈K2,∀x∈E, λ.(µ.x)=(λµ).x
EK
λ∈Kx∈E λ.x = 0E⇐⇒ (λ= 0Kx= 0E)
x∈E−x= (−1).x
•R R C C R
•n∈N KnK
•KNK K
•RII⊂R R R
•K[X]K
•K K
EKu1, . . . , unE u1, . . . , un
n
X
k=1
λkuk(λ1, . . . , λn)∈Kn

EKF E F E
F
F∀(x, y)∈F2, x +y∈F
F∀λ∈K,∀x∈F, λ.x ∈F
(E, +, .)KF E F K(+, .)E
EKF E F E
•0E∈F
•F∀(λ, µ)∈K2,∀(x, y)∈F2, λ.x +µ.y ∈F
•R2{(0,0)}R2
•R3{(0,0,0)}R3
•nRn
•Kn[X]nK[X]
n
•KN
• C0(I, R),C1(I, R),...,C∞(I, R)RI
•I
RI
KE E
EK(x1, x2, . . . , xn)∈EnVect (x1, x2, . . . , xn)
x1, x2, . . . , xn
Vect (x1, x2, . . . , xn) = {λ1x1+λ2x2+. . . +λnxn; (λ1, λ2, . . . , λn)∈Kn}
Vect (x1, x2, . . . , xn)E
(x1, x2, . . . , xn)
EK(x1, x2, . . . , xn)∈EnVect (x1, x2, . . . , xn)
x1, x2, . . . , xn
x1, x2, . . . , xn
k∈J1, nKxkx1, x2, . . . , xn
xk

Vect (x1, x2, x3, x4) = Vect (x1, x2, x2+x3−x4, x4)
n∈N(P0, P1, . . . , Pn)n+ 1 K[X] deg Pi=i
i∈J0, nK
Vect (P0, P1, . . . , Pn) = Kn[X]
(P0, P1, . . . , Pn)Kn[X]
EK(x1, x2, . . . , xn)∈En(x1, x2, . . . , xn)
E E Vect (x1, x2, . . . , xn) = E E
x1, x2, . . . , xn
•
Kn→E
(λ1, . . . , λn)→
n
X
k=1
λkxk
•
R2−→ i , −→ i+ 3−→ j , −→ j −→ i , −→ j
EK(x1, x2, . . . , xn)∈En
•(x1, x2, . . . , xn)x1, x2, . . . , xn
∀(λ1, λ2, . . . , λn)∈Kn,(λ1x1+λ2x2+. . . +λnxn= 0) =⇒(λ1=λ2=. . . =λn= 0)
•(x1, x2, . . . , xn)x1, x2, . . . , xn
(x1, x2, . . . , xn)x1, x2, . . . , xn

•(x1, x2, . . . , xn)
Kn→E
(λ1, . . . , λn)→
n
X
k=1
λkxk
•
(x, y)x y (x, y)
EK(x1, x2, . . . , xn)∈En(x1, x2, . . . , xn)E
E
∀x∈E, ∃!(λ1, . . . , λn)∈Kn;x=
n
X
k=1
λkxk
Kn→E
(λ1, . . . , λn)→
n
X
k=1
λkxk
•e1= (1,0,0,...,0), e2= (0,1,0,...,0), . . . , en= (0,0,0,...,1) (e1, e2, . . . , en)Kn
•(1, X, X2, X3, . . . , Xn)Kn[X]
• Mnp(K) (i, j)∈J1, nK×J1, pKEij
(i, j) 1 (Eij )(i,j)∈J1,nK×J1,pKMnp(K)
EK(e1, e2, . . . , en)x∈E x
(e1, e2, . . . , en) (λ1, λ2, . . . , λn)∈Knx=
n
X
k=1
λkekk∈J1, nK
λkk x (e1, e2, . . . , en)

EKE E
EK(x1, x2, . . . , xn) (y1, . . . , yp)
E(y1, . . . , yp)E xkk∈J1, nK
EK
EKE
K
EK
E6={0E}E E dim E
E={0E}dim E= 0
E1E2
KEdim E
Edim E
EKnFEcardF=n= dim E
FE⇐⇒ F ⇐⇒ F E
{0}