Espaces vectoriels

RnRn
E+ : E×E→
E . :R×E→E
• ∀x, y, z ∈E3(x+y)+z=x+(y+z)∀x, y ∈E2
x+y=y+x
•E0∀x∈E
0 + x=x+ 0 = 0
•x E E y x +y=
y+x= 0 y=−x
• ∀λ, µ ∈R2∀x∈E λ.(µ.x) = (λµ).x
•1∀x∈E1.x =x
• ∀λ, µ ∈R2∀x∈E(λ+µ).x =
λ.x +µ.x ∀λ∈R∀x, y ∈E2λ.(x+y) = λ.x +λ.y
ER
E E E
E
•(un) (vn)
(un+vn) (un)λ(λun)
R R
• Mn,p(R)n p

•Rn[X]n
R[X]
•n
Rnn
•
E F E
E
F E
•0∈F
• ∀x, y ∈F x +y∈F
• ∀λ∈R∀x∈F λ.x ∈F
F E
∀λ∈R∀(x, y)∈F2λx +y∈F
M3(R)
R3F={(x y z)|x+y+z= 0}G={(x y z)|x+
y+z= 1}0
n
n
n n
ER R
E
lim
x→+∞f(x) = 2
lim
x→+∞f(x) = 0
+∞+∞
2
C2∀x∈Rf00(x)+3f0(x)−2f(x) = 0
]− ∞; 0]

−1
f(x) =
+∞ax+b+o(1) g(x) = +∞cx+d+o(1) λf (x)+g(x) = +∞
(λa +c)x+ (b+d) + o(1) λf +g
1√2
k
nRn
Rn
2x+y−z= 0
x−3y+ 5z= 0
R3
E k (e1, . . . , ek)
E
E
n
(e1, . . . , ek)E
x∈E x =
i=k
X
i=1
λieik(λ1, . . . , λk)
R3(2 5 −3)
(1 −2 6) (−1 11 −21) 3(1 −2 6) + (−1 11 −21) = (2 5 −3)
(0 9 2)
3 3 M3(R)A=
1 12 −1
−3 2 0
−6 2 1
B=
0 24 −2
−6 2 0
−12 4 0
I A =1
2B+I
(e1, . . . , ek)
V ect(e1, . . . , ek)

(e1, . . . , ek)E V ect(e1, . . . , ek)
E e1e2ek
k
X
i=1
λiei+
k
X
i=1
µiei=
k
X
i=1
(λi+µi)eiρ
k
X
i=1
λiei=
k
X
i=1
(ρλi)ei
V ect(e1, . . . , ek)E
V ect(e1, . . . , ek)
R3(2x+y3x−2y−x)x y
(2 3 −1) (1 −2 0) (2x+y3x−2y−x) = x(2 3 −1) + y(1 −2 0)
R3F x −y+ 2z= 0
G=V ect((1 0 −1),(−2 1 1)) R3
G F
x∈G⇔x= (λ−2µµµ−λ) (λ, µ)x
F λ −2µ−µ+ 2µ−2λ= 0 −λ−µ= 0 µ=−λ x = (3λ−λ−2λ)
F∩G=V ect((3 −1−2))
(e1, . . . , ek)E
(e1, . . . , ek)∀x∈E∃λ1, . . . , λkx=
k
X
i=1
λiei
V ect(e1, . . . , ek) = E
x=
k
X
i=1
λiei
λi
R3((0 2 0),(2 −1 2),(1 −3 0),(1 −2 3))
x= 2b+c+d
y= 2a−b−3c−2d
z= 2b+ 3d3x
2+y 2−5z 4,z 2, x −z, 0
d= 0
E

(e1, . . . , ek)
k
X
i=1
λiei= 0 ∀i∈ {1, . . . , k}λi= 0
k
X
i=1
λiei= 0
R3((2 1 0),(1 −1−1),(0 3 −1))
2x+y= 0
x−y−z= 0
3y−z= 0
(0,0,0)
E
E
R3F(x, y, z) 3x−2y=z
R3(x, y, 3x−2y) =
x(1,0,3) + y(0,1,−2) ((1,0,3); (0,1,−2)) F
F
(e1, . . . , en)E x =
n
X
i=1
λiei∈E λi
x(e1, . . . , en)λieix
((1,0, . . . , 0); (0,1,0, . . . , 0); . . . ; (0, . . . , 0,1)) Rn
ei= (0; . . . ; 0; 1; 0; . . . ; 0) x= (x1, . . . , xn)Rn
x=
n
X
i=1
xiei
n
X
i=1
λiei=
n
X
i=1
µiei
(λ1, . . . , λn) = (µ1, . . . , µn)λi=µi
E
E
E
Rnn
(e1, . . . , ek)E
n k 6n ek+1, . . . , en(e1, . . . , en)
E