Espaces vectoriels

. . .
f(X+λY) = f(X) + λf(Y)
Rn
Rn

Rn
`eme
Rn
Rnn
(1,−1,0,0,1 + √2,−1)
n(x1, x2, . . . , xn)
(xi)1≤i≤n
~x = (x1, x2, . . . , xn)
= (y1, y2, . . . , yn)R1
R
C R2
x3x= (x1, x2, x3, x4)`eme
Rnk`eme k`eme
Rn
Rn
(x1, x2, . . . , xn)+(y1, y2, . . . , yn) = (x1+y1, x2+
y2, . . . , xn+yn)p(x1, x2, . . . , xn) = (√x1,√x2, . . . , √xn)
Rn
α(x1, x2, . . . , xn) = (αx1, αx2, . . . , αxn)
αx+βy=α(x1, x2, . . . , xn) + β(y1, y2, . . . , yn) =
(αx1+βy1, αx2+βy2, . . . , αxn+βyn)

α(x+y) = αx+αy
~
0
~
0 = (0,0, . . . , 0) x Rn
~
0 + x=xx3(0,0,1,0) = (0,0, x3,0)
(x1, x2, x3) = (x1,0,0) + (0, x2,0) + (0,0, x3) = x1(1,0,0) + x2(0,1,0) + x3(0,0,1)
(v1,v2, . . . , vk)
αi
k
P
i=1
αivi
w
(vi)1≤i≤k
v1= (a11, a12, . . . , a1n); v2= (a21, a22, . . . , a2n); . . . ;vk= (ak1, ak2, . . . , akn)
w= (b1, b2, . . . , bn)
a11x1+a21x2+··· +ak1xk=b1
a12x1+a22x2+··· +ak2xk=b2
a1nx1+a2nx2+··· +aknxk=bn
F= (vi)1≤i≤pRnc`F
RpRnc`F((ai)1≤i≤p) =
p
P
i=1
aivi
Rnc`Fc`F
c`F
v Rn
c`Fc`F

(vi)1≤i≤k
Rn
hviiVect(vi)
(vi)S
(vi)S
¡(1,0,0); (0,1,0); (0,0,1)¢R3
S
SFRn
RnF
F= (vi)
bi
k
P
i=1
αivi=
k
P
i=1
βivi⇐⇒
k
P
i=1
(αi−βi)vi=0
(v)i
αi=βi
k
P
i=1
xivi=0⇐⇒ x1=x2=. . . =xk= 0
(vi)
vix1v1+··· +xkvk=0xi
x3
v3viv3=−(x1/x3)v1−(x2/x3)v2− ···
S S
S
SRn
(ei)1≤i≤ne1= (1,0,0, . . . , 0) e2= (0,1,0, . . . , 0)
. . . en= (0,0,0, . . . , 1)
v= (x1, x2, . . . , xn)v=
n
P
i=1
xiei
(ei)
(e3,e1,e2)
R3(bi)Rn
biRn
bi
B= (bi)RnRn
B
v= (x1, x2, . . . , xn)(B)v=
n
P
i=1
xibi

Rn
v= (a1, a2, . . . , an)B= (bi)
Bn(xi)
(x0
i)
B
R2R3
Rnn
n
Rn
FRn
F
GRn
G G n
k≤n
n
Rnn
n
F
F
Rn
`eme