TD 1: Bases de l`algèbre linéaire Exercice 8. Soit n ∈ N

C
R+
x0(t) + √t2+e−t5x(t)=0
x0(t) + √t2+e−t5x(t)=1
E F G E
F∩G E
F∪G E F ⊂G G ⊂F
R2R2
{((0, y)|y∈R}
{((0, y)|y>0}
{((0, y)|y∈R}∪{((x, 0)|x∈R}
{((x, y)||x|=|y|}
{((x, y)|x=y}
R3
F=V ect((0,1,0)) G=V ect((1,1,1))
F=V ect((0,1,0),(1,1,0)) G=V ect(1,0,0)
ER R F f ∈E
f(0) = f0(0) = 0 F E F
{(1,0,0),(1,0,1),(0,0,5)} ⊂ R3
{(1,0,0),(1,0,1),(0,2,5)} ⊂ R3
{(1,0),(1,1),(2,5)} ⊂ R2
{x7→ ex}⊂F(R,R)
u=
0
3
1
v=
−1
h
1
3
w=
k
1
h
3
(h, k)∈R2{u, v, w}
n∈N∗n V ect({t7→ cos(kt)|k6n}) = V ect({t7→ cos(t)k|k6n})

C
x0+ 2tx = 0
{(x, y, z)∈R3|z= 3x+ 5y}
n
R R
Rn0 1
2n−1
a0,...ann+ 1 b0, . . . , bnn+ 1
n
n+ 1
(L0,...Ln)i∈[|0, n|]
Li(ai)=1 Li(aj)=0 j6=i
Rn[X]
P∈Rn[X]i P (ai) = bi
p
v
n0
p
4
n1, n2, n3p
P=
3
X
i=0
niXi