Chapitre 3 Espaces vectoriels réels

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E
(E, +, .)
+
∀x, y ∈E, x +y=y+x
∀x, y, z ∈E, (x+y) + z=x+ (y+z)
0E∈E x ∈E x + 0E(= 0E+x) = x
x∈E y ∈E x +y= 0E
−x
x∈E λ µ λ.(µ.x) = (λµ).x =µ.(λ.x)
x∈E1.x =x
.+
∀x, y ∈E, ∀λ∈R, λ(x+y) = λx +λy
∀x∈E, ∀λ, µ ∈R,(λ+µ)x=λx +µx
x∈E0.x = 0E
λ∈Rλ.0E= 0E

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Rn=Mn,1(R)M1,n(R)
x1
xn
+
y1
yn
=
x1+y1
xn+yn
λ
x1
xn
=
λx1
λxn
RnnR
n n
1
0
0
,...,
0
0
1
n
Mn,p(R)
a1,1−−− a1,p
−−− 
an,1−−− an,p
+
b1,1−−− b1,p
−−− 
bn,1−−− bn,p
=
a1,1+b1,1−−− a1,p +b1,p
−−− 
an,1+bn,1−−− an,p +bn,p
λ
a1,1−−− a1,p
−−− 
an,1−−− an,p
=
λa1,1−−− λa1,p
−−− 
λan,1−−− λan,p
Mn,p(R)R
n×p n ×p
1−−− 0
0−−− 0
−−− 
0−−− 0
,
0 1 −−− 0
0 0 −−− 0
−−− 
0 0 −−− 0
,...,
0−−− 0
0−−− 0
−−− 
0−−− 1
n×p
R[X]
(P+Q)(x) = P(x) + Q(x) (λP )(x) = λP (x)
R[X]R
[1 = X0, X =X1, X2,− − −, Xn,− − −]n∈N
Xi=PiPi(x) = xi
Rn[X]
Rn[X]nR[X]
n+ 1 n+ 1
[1 = X0, X =X1, X2,−−−, Xn]
RN
RN
F(A, R)
F(A, R)AR
Card A

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FR
E
F E
F⊂E
0E∈F F 6=∅)
∀x, y ∈F, ∀λ, µ ∈R, λx +µy ∈F
λ+.
f:E→F A Mn(R)
Ker fKer A)E
Im fIm A)F
(f1, . . . , fp)E
p
P
i=1
λifi
(λ1, . . . , λp)∈Rp(f1, . . . , fp)
E(f1, . . . , fp)
Vect(f1, . . . , fp)
F= (f1, . . . , fp)EF
rg(F)F
rg(F) = rg [(f1, . . . , fp)] = dim (Vect[f1, . . . , fp])

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(f1, . . . , fp)E
λ1f1+· · · +λpfp=
p
P
i=1
λifiλ1, . . . , λp
f1, . . . , fp
Vect(f1,...,fp) (f1,...,fp)
u(f1, . . . , fp)
u=λf1+· · · +λpfpλ1, . . . , λp
(f1, . . . , fp)E
Vect(f1, . . . , fp) = E
E(f1, . . . , fp)
∀x∈E, ∃λ1, . . . , λp∈R, x =
p
P
i=1
λifi
(f1, . . . , fp)fp=
p−1
P
i=1
λififp∈Vect(f1, . . . , fp−1)
p−1
Vect(f1, . . . , fp) = Vect(f1, . . . , fp−1)
Vect(f1,...,fp−1)⊂Vect(f1,...,fp)
x∈Vect(f1,...,fp−1)x=
p−1
P
i=1
µifi=
p−1
P
i=1
µifi+ 0fp∈Vect[f1,...,fp]
Vect(f1,...,fp)⊂Vect(f1,...,fp−1)
x∈Vect(f1,...,fp)x="p−1
P
i=1
µifi#+ [µpfp] Vect(f1,...,fp−1)
x∈Vect(f1,...,fp−1) Vect(f1,...,fp)⊂Vect(f1,...,fp−1) Vect(f1,...,fp) = Vect(f1,...,fp−1)

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Vect 
1
1
2
,
2
1
−1
,
−4
−1
7
Vect 
1
1
2
,
2
1
−1
,
−4
−1
7
= Vect 
C1
1
1
2
,
C2←C2−2C1
0
−1
−5
,
C3←C3+ 4C1
0
3
15
= Vect 
1
1
2
,
C2
0
−1
−5
,
C3←C3+ 3C2
0
0
0
= Vect 
1
1
2
,
0
−1
−5
(f1, f2, f3)
(f1, . . . , fp)E
(f1, . . . , fp) 0f1+· · · + 0fp
p
P
i=1
λifi= 0E⇒λ1=· · · =λp= 0
(f1,...,fp)
x=
p
P
i=1
λifi=
p
P
i=1
µifi
p
P
i=1
(λi−µi)fi= 0 i λi−µi= 0 λi=µi
R3
1
1
2
,
2
1
−1
,
−4
−1
7
a
1
1
2
+b
2
1
−1
+c
−4
−1
7
=
0
0
0
⇔
a+ 2b−4c
a+b−c
2a−b+ 7c
=
0
0
0
⇔
a+2b−4c= 0
a+b−c= 0
2a−b+7c= 0
⇔
a+2b−4c= 0 L1
−b+3c= 0 L2←L2−L1
−5b+15c= 0 L3←L3−2L1
⇔
a+2b−4c= 0
−b+3c= 0 L2
0 = 0 L3←L3−5L2
⇔a= 4c−2b= 4c−6c=−2c
b= 3c⇔(a, b, c) = (−2c, 3c, c) = c(−2,3,1)