(1) Dans un espace vectoriel, donnez la définition d`une famille libre

{ei}i∈IE
{i1, . . . , in} ⊂ I λ1, . . . , λnPn
k=1 λkeik= 0
λ1, . . . , λn= 0 x∈E
{i1, . . . , in} ⊂ I λ1, . . . , λnx=Pn
k=1 λkeik
{e1, . . . , en}
H3x+ 5y+ 7z= 0 R3
R3→RH
R2→R3H
H f :R3→R
f(x, y, z)=3x+ 5y+ 7z
(x, y, z)H z =−(3/7)x−(5/7)y H
g:R2→R3g(x, y) = (x, y, −(3/7)x−(5/7)y)
0 = x0< x1<· · · < xn−1< xn= 1 [0; 1] F
f: [0; 1] →R[xi;xi+1]
F
FR
[0; 1] RF f, g
F λ, µ λf +µg
λf µg [xi;xi+1]f
g λf +µg λf +µg ∈F
FRn+ 1
u:F→Rn+1 f(f(x0), f(x1), . . . , f(xn))
[xi;xi+1]f∈F
xixi+1 u
FRn+1
{e0, . . . , en}eii
1F eiu
u i=u−1(ei) 0 xjj6=i
1xii∈ {1, . . . , n −1}[xi−1;xi+1]
1xii= 0 i=n
[0; 1]
{0, . . . , n}F f ∈F
f=f(x0)0+· · · +f(xn)n
F f(xi)xi