Espaces vectoriels

K=R C
K−E
G+
K∀x, y ∈E, x +y∈E
K∀x, y ∈E, x +y=y+x
K∀x, y, z ∈E, x + (y+z) = (x+y) + z
K∃x0∈E, ∀x∈E, x +x0=x0+x=x
E0E
K∀x∈E, ∃y∈E, x +y=y+x= 0E
x−x
GK×E→E
(λ, x)7→ λ.x
K∀x∈E, ∀λ∈K, λ.x ∈E
K∀λ∈K,∀x, y ∈E, λ.(x+y) = λ.x +λ.y
K∀λ, µ ∈K,∀x∈E, (λ+µ).x =λ.x +µ.x
K∀λ, µ ∈K,∀x∈E, (λµ).x =λ.(µ.x)
K∀x∈E, 1.x =x
λ.x =λx
E
K
©
GR R−
GC C−R
GKnK
GKNK
GFKX
FXX F K
GMn,p(K)K
GK[X]K
EK
G∀λ∈K,∀x∈E, λx = 0E⇐⇒ λ= 0Kx= 0E
G∀λ, µ ∈K,∀x∈E, (λ−µ)x=λx −µx
G∀λ∈K,∀x, y ∈E, λ(x−y) = λx −λy

EK−F⊂E
F
F6=∅
∀x, y ∈F, ∀λ∈K, λx +y∈F
F E 0E∈F
(E, +, .)KF
(F, +, .)K
©
GF={(x, y, z)∈R3, x +y+z= 0}
GG={(x, x, x)∈R3, x ∈R}
G
GKn[X]
EK−(Fi)1≤i≤nE
F=∩
1≤i≤nFi
F1∪F2
F1⊂F2
F= (x1, x2, . . . , xn)E
FVect(F)
Vect(F) = (n
X
i=1
λixi,(λ1, λ2, . . . , λn)∈Kn)
F
©R2= Vect(e1, e2)e1= (1,0) e2= (0,1)

F= (x1, x2, . . . , xn)E
Vect(F)F
FFVect(F)⊂F
EF= (xi)1≤i≤nE
GF
∀(λ1, . . . , λn)∈Kn,
n
X
i=1
λixi= 0E=⇒ ∀1≤i≤n, λi= 0K
GF
∃(λ1, . . . , λn)∈Kn,(λ1, . . . , λn)6= (0K,...,0K),
n
X
i=1
λixi= 0E
©
GF= (x)
Fx6= 0
GF= (x1, x2)
F(x1, x2)
R2x1= (1,1), x2= (2,1), x3= (−1,0)
G
G
(xi)1≤i≤n
x∈E(λ1, . . . , λn)∈Kn
x=
n
X
i=1
λixi(λi)
â(0,...,0)
â(0,...,0)
EF= (xi)1≤i≤n
xn+1 /∈Vect(x1, . . . , xn)
(x1, . . . , xn, xn+1)

G(Pi)i∈I
G
(P0, P1, . . . , Pn)∀0≤i≤n, deg(Pi) = i
EF= (xi)1≤i≤nE
FEVect(F) = E
∀x∈E, ∃(λ1, . . . , λn)∈Kn, x =
n
X
i=1
λixi
©
G((1,0),(0,1))
G((1,0),(0,1),(1,1))
G((1,0))
G(1, i)
(x1, . . . , xn)
âx∈E(x1, . . . , xn)
x=
n
X
i=1
λixi
âλi
Kn
â(λi)1≤i≤nx
EB= (xi)1≤i≤nE
BEB
©
R2
G((1,0),(0,1))
G((1,0),(0,1),(1,1))
G((1,0))

EB= (xi)1≤i≤nE
BE⇐⇒ ∀x∈E, ∃!(λ1, . . . , λn)∈Kn, x =
n
X
i=1
λixi
(λ1, . . . , λn)xB
©
R2(x, y)
G((1,0),(0,1))
G((1,1),(0,2))
Rn
©
1≤i≤n, ei= (0,...,1,0. . . , 0)
(e1, . . . , en)Rn
x= (x1, x2, . . . , xn)x=
n
X
i=1
xiei
Kn[X]
©(1, X, X2, . . . , Xn)Kn[X]
Mn,p(K)
©
1≤i≤n, 1≤j≤p
Ei,j =
j
↓
0. . . 0
1
0. . . 0
←i
(Ei,j )1≤i≤n
1≤j≤p
Mn,p(K)
A= (ai,j )1≤i≤n
1≤j≤p
A=X
1≤i≤n
1≤j≤p
ai,j Ei,j